Special functions of mathematical (Geo-)physics [without chapter 1] 9783034805636, 3034805632


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Applied and Numerical Harmonic Analysis Series Editor John J. Benedetto University of Maryland College Park, MD, USA

Editorial Advisory Board Akram Aldroubi Vanderbilt University Nashville, TN, USA

Jelena Kovaˇcevi´c Carnegie Mellon University Pittsburgh, PA, USA

Andrea Bertozzi University of California Los Angeles, CA, USA

Gitta Kutyniok Technische Universit¨at Berlin Berlin, Germany

Douglas Cochran Arizona State University Phoenix, AZ, USA

Mauro Maggioni Duke University Durham, NC, USA

Hans G. Feichtinger University of Vienna Vienna, Austria

Zuowei Shen National University of Singapore Singapore, Singapore

Christopher Heil Georgia Institute of Technology Atlanta, GA, USA

Thomas Strohmer University of California Davis, CA, USA

St´ephane Jaffard University of Paris XII Paris, France

Yang Wang Michigan State University East Lansing, MI, USA

For further volumes: www.birkhauser-science.com/series/4968

Willi Freeden



Martin Gutting

Special Functions of Mathematical (Geo-)Physics

Willi Freeden Martin Gutting Geomathematics Group University of Kaiserslautern Germany

ISBN 978-3-0348-0562-9 ISBN 978-3-0348-0563-6 (eBook) DOI 10.1007/978-3-0348-0563-6 Springer Basel Heidelberg New York Dordrecht London Library of Congress Control Number: 2013931107 Mathematical Subject Classification (2010): 33-01, 42C10, 35Q86, 42B37, 42C05, 43A90, 33C10, 33C45, 33C55, 11P21, 42A16, 42B05, 42B05, 65B15, 86A10, 86A25, 86A30 c Springer Basel 2013  This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer Basel is part of Springer Science+Business Media (www.birkhauser-science.com)

Preface

An essential aim of geomathematics is the investigation of qualitative and quantitative structures of the Earth’s system to deepen our understanding of its complexity. In this respect, special functions comprise the essential instruments for mathematical interaction of abstraction and concretization. Special functions enable the formulation of a geoscientific problem by reduction such that a new, more concrete problem can be attacked within a well-structured framework, usually in the context of differential equations. A good understanding of special functions provides the capacity to recognize causality between the abstractness of the geomathematical concept and the impact on, as well as cross-sectional importance to, the geoscientific reality. Our purpose in this work is to present a textbook that allows the reader to concentrate on special fields such as the geosphere, hydrosphere, or atmosphere. In other words, the special functions to be discussed vary widely, depending on the chosen measurement parameters (gravitation, electric and magnetic fields, deformation, climate observables, fluid flow, etc.) and on the field characteristic (potential field, diffusion field, wave field). The differential equation under consideration determines the type of special functions that are needed in the desired reduction process. The diversity of geomathematical problems involves such a large number of scientific manifestations that our approach to any of them has to be selective. In consequence, since greater weight has to be given to some topics than to others, we have chosen to restrict ourselves to gravitation, geomagnetism, elasticity, and fluid flow theory. Gravitational field theory defines a canonical need to generate special function systems for the Laplace equation. Geomagnetism and electric current systems are closely related to the (pre-)Maxwell equation; the deformation of the solid Earth leads to function systems solving the Cauchy–Navier equation (at least when linear material behavior is assumed). Oceanic circulation and wind motion have to be handled in terms of vectorial function systems involving the Navier–Stokes equation or modifications of it. Unfortunately, we are confronted with the difficult challenge to characterize special function systems under adequate consistency in terms of less mathematically structured geometric features of a reference model (such as the geoid or the real Earth’s surface) as well as the intrinsic structure of v

vi

Preface

underlying differential equations involving the laws of physics. Thus, at the present stage of geoscience, no compendium can be expected that is both geometrically consistent with modern navigation results and geophysically reflected by advanced mathematical settings. The complexity of a real “potato-like” Earth model is a striking obstacle that can only be overcome to some extent in today’s mathematics. Accordingly, the principles lie in the suitable transition to a regularly structured geometry for the Earth, namely, the ball in first approximation. This leads us to a prestructured framework, namely, spherically oriented special function systems. Looking at the special functions available in the geophysical literature today, we find that a spherical shape of the Earth is used in almost all publications. Indeed, by modern satellite positioning methods, the maximum deviation of the actual Earth’s surface from the average Earth’s radius (6,371 km) can be determined to be less than 0.4 %. Although a spheriodization, i.e., a mathematical formulation simply in spherical reference geometry, amounts to a strong restriction, it is at least acceptable for a large number of problems. Standard special functions since the time of C.F. Gauß are polynomial trial functions, conventionally called spherical harmonics. Spherical harmonics represent the analogs of trigonometric functions for orthogonal (Fourier) expansions on the sphere. In consequence, the use of spherical harmonics in diverse areas of geosciences is a well-established method, particularly for the purpose of decomposing scalar potentials. Nowadays, reference models for the Earth’s gravitational and magnetic potential, e.g., are widely known by tables of expansion coefficients of the frequency constituents of their potentials. However, it should be mentioned that vectorial potentials—even in a spherical Earth’s reference model— have their own nature. Concerning the mathematical modeling of vector fields, one is usually not interested in their separation into scalar Cartesian component functions. Instead, inherent physical properties should be observed. For example, the external gravitational field is curl-free, the magnetic field is divergence-free, the equations for incompressible flow, i.e., the Navier–Stokes equations, imply divergence-free vector solutions. In a spherical nomenclature as intended in our approach, all these physical constraints result in a formulation by certain operators, such as the surface gradient, surface curl gradient, surface divergence, surface curl. Our types of vector spherical harmonics satisfy these requirements by splitting the tangential part into a curl-free and a divergence-free field, thereby avoiding artificial singularities arising from the use of local coordinates. Basically, two transitions are undertaken in our approach to harmonics: first, the extension from the scalar to the vectorial case is strictly realized under physical constraints and, second, the definition of Legendre functions is canonically described under the phenomenon of rotational invariance on the sphere. The Legendre functions act as constituting elements for zonal functions, i.e., one-dimensional functions only depending on the polar distance of their two arguments. Altogether, the concept of spherical harmonics plays the central role in a geomathematical presentation of special functions, reflecting the significance of a polynomial nature in a spherically shaped Earth. In addition, spherical harmonics comprise the canonical candidates to represent the angular part in a radial/angular decomposition of solution systems for Laplace, Helmholtz, Cauchy–Navier, (pre-)Maxwell, and Navier–Stokes equations.

Preface

vii

It is surprising that, besides the geometrically implied spheriodization, the methodologically oriented periodization should take some space in a modern collection of special function systems of geomathematical importance. The reasons are twofold. First, the periodization leads back to the Fourier transform in Euclidean spaces that has been well understood for a long time and is extremely efficient in numerical computation. Second, the procedure of periodization leads to the Euler summation formula and the Poisson summation formula which show a close relationship to each other. The Green (lattice) functions forming the essential basis of these summation formulas indeed enable us to express key volume integrals in geophysics, such as the Newton integral, Mie potentials, elastic potentials, by mass lattice point conglomerates that discretely fill out the integration domain under consideration in an equidistributed way. A variety of examples for combined periodization and spheriodization occur in the theory of Earth-satellite relations (cf., e.g., Kaula 1966), mixing time-wise obligations on periodic orbits with space-wise approaches on torus and/or sphere. Satellite gravimetry (see, e.g., Pail and Plank (2002), Sneeuw (2000), Xu et al. (2008), and the references therein) is a particularly interesting area of spaceborne technology, where one-dimensional periodization in time is adequately involved in three-dimensional periodization and/or spheriodization in space. This textbook presents material used by the Geomathematics Group, University of Kaiserslautern, during the last several years to set up a contemporary theory of special functions of mathematical (geo-)physics. Our work canonically shows a threefold subdivision. Part I provides preparatory material concerning auxiliary functions such as the Gamma function and important classes of orthogonal polynomials. The general concept of orthogonal polynomials is introduced before we start to consider the classical polynomials, in particular the Jacobi polynomials and— as a special and very important case of them—the ultraspherical or Gegenbauer polynomials. Several basic mathematical and physical applications are included, such as quadrature rules, modeling of the electrostatic potential, and the quantummechanical description of oscillations. Part II deals with spherically structured function systems. It starts with the scalar theory of spherical harmonics in the Euclidean space R3 including the addition theorem, the Funk–Hecke formula, as well as the closure and completeness of spherical harmonics in the space of squareintegrable functions, i.e., the space of functions with finite signal energy. It follows the physically based theory of vector spherical harmonics. The basic tool to establish divergence-free and curl-free tangential fields is the Helmholtz decomposition theorem. An alternative system of vector spherical harmonics is also constructed in such a way that they can be identified as eigenfunctions of the Beltrami operator. This eigenfunction system plays a particular role in geomagnetism to separate, e.g., the crustal field from other magnetic sources. Both vector spherical harmonic systems are shown to be closed and complete in the space of square-integrable vector fields on the sphere. All properties characterize vector spherical harmonics as suitable trial functions to constitute the angular ingredients in a radial/angular decomposition of solutions of the Cauchy–Navier as well as the Navier–Stokes equation. Part III is devoted to the lattice function as the multi-dimensional,

viii

Preface

2.1

3.0

3.1

3.4

4

5

Fig. 1 Selected sections and chapters for a basic one-term course (note that some parts of Sect. 3.3 have to be included to complete Sect. 3.4. Sections 5.5, 5.6, 5.9, and 5.10 can be skipped in the oneterm course)

periodic analog to the well-known Bernoulli function. From a physical point of view, the lattice function is interpreted to be the Green function for the Laplace operator corresponding to the boundary conditions of periodicity. It turns out to be the most essential tool for the process of periodization in the context of Euler and Poisson summation formulas. Lattice point sums such as the Zeta and Theta functions, generated by the interaction of point potentials to each other, conclude our multi-periodic theory. It should be remarked that the whole palette of multi-periodic functions is provided in relation to the Laplace operator and arbitrary lattices so that this approach serves as a prototype for further formulations of more general (elliptic) partial differential equations. Essential ingredients of the textbook are the work of M¨uller (1952, 1969, 1998), Freeden et al. (1998), Freeden and Schreiner (2009), and Freeden (2011). Each chapter of the book is followed by exercises related to the presented material. The exercises reflect significant topics, mostly in computational geoapplications. In doing so, they not only confront the reader directly with the contents of the chapter, but also with additional knowledge in geomathematical fields of research, where special functions play a decisive role in applications. Students who wish to continue further studies should consult the literature given as supplements for each topic worked out by exercises. All in all, the content of the book is equally suitable for an education in geomathematics and a study in applied and harmonic analysis. The book is primarily meant to be a self-consistent introductory text for an advanced undergraduate or graduate course in special functions. The schedule of topics allows a selected subdivision into a one-term course (see Fig. 1) as well as a two-term course. In addition to the proposed sections and chapters in Fig. 1, further contents can be selected from Chap. 3, such as Sect. 3.2 and all details of Sects. 3.3 or 3.5–3.7, if the schedule allows it. The examples of Chap. 1 can be presented at any appropriate time. A two-term course with special emphasis on particular research fields should include additional material from Chaps. 5 and 7 documenting the special interest of a graduate student in gravitation, geomagnetism, deformation, atmospheric/oceanic flow, respectively. Chapters 6 and 8 give multi-dimensional radial/angular decompositions of harmonic and metaharmonic functions as a reference tool, thereby assuming as preparatory material the whole theory of the Gamma function as presented in Chap. 2. Another separate route going exclusively into the field of lattice functions includes Chaps. 9 and 10 while also requiring all the material of Chap. 2.

Preface

ix

In a book of this type, special precautions have been taken to ensure the accuracy of formulas and examples. It is a pleasure to acknowledge with thanks the valuable reading of the manuscript by Dipl.-Math. C. Blick, Dr. C. Gerhards, and Dr. I. Ostermann. We thank our student cand.-phys. Hanna Haug for pointing out some errors and slips in an early version of the manuscript. Finally, we want to thank Prof. Dr. J.J. Benedetto, Series Editor of Applied and Numerical Harmonic Analysis, and Dr. T. Hempfling, Managing Director of Springer Basel and Executive Editor of Birkh¨auser Basel, for valuable remarks and hints. Additionally, it is a pleasure to acknowledge the courtesy and ready cooperation of Dr. B. Hellriegel and all staff members of Birkh¨auser/Springer, Basel. Kaiserslautern, Germany

Willi Freeden, Martin Gutting

About the Authors

Willi Freeden: Studies in mathematics, geography, and philosophy at the RWTH Aachen; 1971, diploma in mathematics; 1972, Staatsexamen in mathematics and geography; 1975, Ph.D. in mathematics; 1979, habilitation in mathematics; 1981/1982, visiting research professor at The Ohio State University, Columbus (Department of Geodetic Science and Surveying); 1984, professor of mathematics at the RWTH Aachen (Institute of Pure and Applied Mathematics); 1989, professor of technomathematics (industrial mathematics); 1994, head of the Geomathematics Group; 2002–2006, vice-president for research and technology at the University of Kaiserslautern; 2009, editor in chief of the International Journal on Geomathematics (GEM); 2010, editor of the Handbook of Geomathematics, member of the editorial board of eight international journals. Martin Gutting: Studies in mathematics at the University of Kaiserslautern; 2003, diploma in mathematics, focus on geomathematics; 2007, Ph.D. in mathematics, postdoc researcher at the University of Kaiserslautern, lecturer in the course of geomathematics (in particular for constructive approximation, special functions, and inverse problems); 2011, lecturer for engineering mathematics at the University of Kaiserslautern and DHBW Mannheim.

xi

Contents

1

Introduction: Geomathematical Motivation .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Example: Gravitation (Laplace and Poisson Equation) . . . . . . . . . . . . 1.2 Example: Geomagnetism (Maxwell’s Equations) . . . . . . . . . . . . . . . . . . 1.3 Example: Fluid Flow (Navier–Stokes Equation) . . . . . . . . . . . . . . . . . . . 1.4 Example: Elastic Field (Cauchy–Navier Equation).. . . . . . . . . . . . . . . .

Part I 2

3

Auxiliary Functions

The Gamma Function .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Definition and Functional Equation . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Euler’s Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Stirling’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Pochhammer’s Factorial . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Exercises (Incomplete Gamma and Beta Function, Applications in Statistics) . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

25 25 29 33 38 43

Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 47 3.1 Properties of Orthogonal Polynomials . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 56 3.2 Quadrature Rules and Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . 66 3.3 The Jacobi Polynomials .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 70 3.4 Ultraspherical Polynomials . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 80 3.5 Application of the Legendre Polynomials in Electrostatics . . . . . . . . 90 3.6 Hermite Polynomials and Applications . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 95 3.7 Laguerre Polynomials and Applications . . . . . . . .. . . . . . . . . . . . . . . . . . . . 98 3.8 Exercises (Gaussian Integration, Legendre Series, Kernel Expansions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 101

Part II 4

1 1 9 13 16

Spherically Oriented Functions

Scalar Spherical Harmonics in R3 . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 113 4.1 Basic Notation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 114 4.2 Orthogonal Invariance.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 119 xiii

xiv

Contents

Homogeneous Polynomials on the Unit Sphere in R3 .. . . . . . . . . . . . . Closure and Completeness of Spherical Harmonics .. . . . . . . . . . . . . . . The Funk–Hecke Formula and the Irreducibility of Scalar Harmonics.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Green’s Function with Respect to the Beltrami Operator . . . . . . . . . . The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Exercises (Low Discrepancy Method, Locally Supported Wavelets, Up Function, Anharmonic Functions for the Ball, Fast Multipole Method, Wigner Matrices, Quaternionic Generation of Spherical Harmonics) .. . . . .

125 144

Vectorial Spherical Harmonics in R3 . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Basic Notation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Definition of Vector Spherical Harmonics . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 The Helmholtz Decomposition Theorem . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 Closure and Completeness of Vector Spherical Harmonics .. . . . . . . 5.5 Homogeneous Harmonic Vector Polynomials . .. . . . . . . . . . . . . . . . . . . . 5.6 Vectorial Beltrami Operator . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.7 Vectorial Addition Theorem . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.8 Vectorial Funk–Hecke Formulas . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.9 Vectorial Counterparts of the Legendre Polynomial .. . . . . . . . . . . . . . . 5.10 Application to Elastic Fields . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.11 Exercises (Uncertainty Principle, Classification of Zonal Functions, Coupling Integrals and Navier–Stokes Equation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

211 212 218 220 223 231 236 239 246 251 254

4.3 4.4 4.5 4.6 4.7 4.8

5

6

7

8

q

154 159 165

169

264

Spherical Harmonics in R . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Nomenclature and Basics . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Integral Theorems for the Laplace Operator . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Integral Theorems for the Laplace–Beltrami Operator .. . . . . . . . . . . . 6.4 Homogeneous Harmonic Polynomials . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5 Spherical Harmonics of Dimension q . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6 Integral Theorems for the Helmholtz–Beltrami Operator . . . . . . . . . . 6.7 Exercises (Cartesian Generation of Spherical Harmonics, Best Approximations) . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

285 286 290 297 302 312 330

Classical Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Derivation and Definition of Bessel Functions .. . . . . . . . . . . . . . . . . . . . 7.2 Orthogonality Relations .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3 Bessel Functions with Integer Index . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4 Exercises (Bessel Function Expansions, Hankel Transform and Discontinuous Integrals) . . . . . . . .. . . . . . . . . . . . . . . . . . . .

347 347 354 355

q

Bessel Functions in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Regular Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Modified Bessel Functions .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3 Hankel Functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

337

359 363 363 370 372

Contents

xv

8.4 8.5 8.6

Kelvin Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 378 Expansion Theorems .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 385 Exercises (Helmholtz Equation, Entire Solutions, Bessel Function Like Asymptotics) . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 390

Part III 9

Periodically Oriented Functions

Lattice Functions in R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Bernoulli Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Periodic Polynomials.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 Lattice Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4 Euler Summation Formula .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.5 Riemann Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.6 Poisson Summation Formula for the Laplace Operator . . . . . . . . . . . . 9.7 Theta Function.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.8 Exercises (Trapezoidal Rule, Periodic Sobolev Spaces, Projection Method) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . q

10 Lattice Functions in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1 Lattices in Euclidean Spaces. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2 Periodic Polynomials.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3 Lattice Function for the Laplace Operator . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4 Euler Summation Formula for the Laplace Operator .. . . . . . . . . . . . . . 10.5 Zeta Functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.6 Integral Asymptotics for Lattice Functions . . . . .. . . . . . . . . . . . . . . . . . . . 10.7 Poisson Summation Formula . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.8 Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.9 Exercises (Algebraic, Periodic, and Spherical Splines, Lattice Point Sums, Lattice Point Distributions) .. . . . . . . . . . . . . . . . . . .

395 395 398 400 404 411 415 419 420 427 427 430 432 436 439 443 448 455 460

11 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 483 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 485 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 495

Chapter 2

The Gamma Function

In what follows, we introduce the classical Gamma function in Sect. 2.1. It is essentially understood to be a generalized factorial. However, there are many further applications, e.g., as part of probability distributions (see, e.g., Evans et al. 2000). The main properties of the Gamma function are explained in this chapter (for a more detailed discussion the reader is referred to, e.g., Artin (1964), Lebedev (1973), M¨uller (1998), Nielsen (1906), and Whittaker and Watson (1948) and the references therein). We briefly consider Euler’s Beta function in Sect. 2.2 and use it to recursively compute the volume of the .q  1/-dimensional unit sphere Sq1  Rq . As outstanding property of the Gamma function the Stirling formula is verified in Sect. 2.3. It leads us to the so-called duplication formula (Lemma 2.3.3) which will simplify a lot of calculations in later chapters. The extension of the Gamma function to complex values is studied in Sect. 2.4. In doing so, we introduce Pochhammer’s factorial and Euler’s constant  . Moreover, we establish product representations for the Gamma function as well as for trigonometric functions. In Sect. 2.5 the incomplete Gamma and Beta functions are briefly presented in form of some exercises and their relation to probability distributions is indicated.

2.1 Definition and Functional Equation For real values x > 0, we consider the integrals Z

1 0

and

Z

et t x1 dt;

1 1

et t x1 dt:

W. Freeden and M. Gutting, Special Functions of Mathematical (Geo-)Physics, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-0348-0563-6 2, © Springer Basel 2013

(2.1.1)

(2.1.2)

25

26

2 The Gamma Function

In order to show the convergence of (2.1.1) we observe that 0 < et t x1  t x1 holds true for all t 2 .0; 1. Therefore, for " > 0 sufficiently small, we have Z

1

t x1

e t

"

Z

1

dt 

t

"

x1

ˇ t x ˇˇ1 1 "x dt D D  : x ˇ" x x

(2.1.3)

Consequently, for all x > 0, the integral (2.1.1) is convergent. To assure the convergence of (2.1.2) we observe that 1 et t x1 D P1

tk kD0 kŠ

t x1 

1

tn nŠ

t x1 D

nŠ t nxC1

(2.1.4)

for all n 2 N and t  1. This shows us that Z

A 1

Z

et t x1 dt  nŠ

A 1

1 t nxC1

dt D nŠ

ˇA   1 t nCx ˇˇ nŠ D  1 x  n ˇ1 x  n Anx (2.1.5)

provided that A is sufficiently large and n is chosen such that n  x C 1. Thus, the integral (2.1.2) is convergent which we summarize in the following lemma. Lemma 2.1.1. For all x > 0, the integral Z

1 0

et t x1 dt

(2.1.6)

is convergent. Definition 2.1.2. The function x 7!  .x/; x > 0, defined by Z  .x/ D

1 0

et t x1 dt

(2.1.7)

is called the Gamma function (see Fig. 2.3 (right) for an illustration). Obviously, we have the following properties: (i)  is positive R 1 for all x > 0, (ii)  .1/ D 0 et dt D 1. We can use integration by parts to obtain for x > 0: Z  .x C 1/ D

1 0

Z

Dx

ˇ1 et t x dt D et t x ˇ0 

1 0

et t x1 dt D x  .x/:

Z 0

1

.et /xt x1 dt

(2.1.8)

2.1 Definition and Functional Equation

27

Lemma 2.1.3. The Gamma function  satisfies the functional equation  .x C 1/ D x  .x/; x > 0:

(2.1.9)

Moreover, by iteration for x > 0 and n 2 N;  .x C n/ D .x C n  1/  : : :  .x C 1/x  .x/ D  .n C 1/ D

.x C i  1/ .x/; (2.1.10)

i D1

Y n  n Y i  .1/ D i D nŠ: i D1

n Y

(2.1.11)

i D1

In other words, the Gamma function can be interpreted as an extension of factorials. Lemma 2.1.4. The Gamma function  is differentiable for all x > 0 and we have Z 1  0 .x/ D et ln.t/t x1 dt: (2.1.12) 0

Proof. For x > jhj > 0, we use the formula t y D eln.t /y , y > 0, t > 0: Z 1 Z 1 t xCh1 e t dt D et Cln.t /.xCh1/ dt:  .x C h/ D

(2.1.13)

By Taylor’s formula we find 0 < # < 1 such that Z 1    .x C h/   .x/ D et e ln.t / eln.t /.xCh/  eln.t /x dt

(2.1.14)

0

Z

0

0

1

D 0

Z

Dh

  et e ln.t / h ln.t/t x C 12 h2 .ln.t//2 t xC#h dt

1

e

t

ln.t/t

x1

dt C

0

1 2 2h

Z

1 0

et .ln.t//2 t xC#h1 dt:

This gives us the differentiability of  if the second integral is bounded. Consider the following estimate (we employ that .ln.t//2  t 2 for t  1 and that et  1 for t 2 Œ0; 1/ Z 0

1

et .ln.t//2 t xC#h1 dt D

Z Z



1

0 1 0

et .ln.t//2 t xC#h1 dt C

.ln.t//2 t xC#h1 dt C

  .2 C x C #h/ C

This provides us with the desired result.

Z 1

Z

1

1

1

et .ln.t//2 t xC#h1 dt

et t 2 t xC#h1 dt

2 < 1: .x C #h/3

(2.1.15) t u

28

2 The Gamma Function

An analogous proof can be given to show that  is infinitely often differentiable for all x > 0 and Z 1  .k/ .x/ D

et .ln.t//k t x1 dt; k 2 N:

0

(2.1.16)

Lemma 2.1.5 (Gauß’ Expression of the Second Logarithmic Derivative). For x > 0,  0 2 (2.1.17)  .x/ <  .x/ 00 .x/: Equivalently, we have 

d dx

2

 00 .x/  ln. .x// D  .x/



 0 .x/  .x/

2

> 0;

(2.1.18)

i.e., x 7! ln. .x//, x > 0, is a convex function or  is logarithmic convex. Proof. We start with 

 0 .x/

2

Z

1

D Z

et ln.t/t x1 dt

2

0 1

D

e

 2t

0

t

x1 2

ln.t/e

 2t

t

x1 2

2 dt

:

(2.1.19)

The Cauchy–Schwarz inequality yields (note that equality cannot occur since the two functions are linearly independent): 

 0 .x/

2

Z
0; dx 2 dx  .x/ . .x//2

(2.1.21) t u

which yields (2.1.18). Note that ln. .// is convex, i.e., for t 2 Œ0; 1 ln . .tx C .1  t/y//  t ln. .x// C .1  t/ ln. .y//     D ln  t .x/ C ln  1t .y/   D ln  t .x/   1t .y/

(2.1.22)

which is equivalent to  .tx C .1  t/y/   t .x/   1t .y/ with x; y > 0.

2.2 Euler’s Beta Function

29

Fig. 2.1 The illustration of the coordinate transformation relating the Beta and the Gamma functions

2.2 Euler’s Beta Function Next, we notice that for x; y > 0; the integral Z

1 0

t x1 .1  t/y1 dt

(2.2.1)

is convergent. Definition 2.2.1. The function .x; y/ 7! B.x; y/, x; y > 0, defined by Z B.x; y/ D

1 0

t x1 .1  t/y1 dt

(2.2.2)

is called the Euler Beta function. For x; y > 0, we see that Z  .x/ .y/ D

1 0

et t x1 dt

Z

1

0

es s y1 ds D

Z

1

Z

0

0

1

e.t Cs/ t x1 s y1 dt ds: (2.2.3)

Note that the transition from one-dimensional to two-dimensional integrals is permitted by Fubini’s theorem. We make a coordinate transformation (see Fig. 2.1) as follows: t D u.1  v/;

0  u < 1;

(2.2.4)

s D uv;

0  v  1:

(2.2.5)

It is not difficult to verify that the functional determinant of the coordinate transformation is given by ˇ ˇ ˇ1  v uˇ @.t; s/ ˇ ˇ D u.1  v/ C uv D u  0: Dˇ (2.2.6) v uˇ @.u; v/

30

2 The Gamma Function

Thus, we find Z

1

Z

0

1 0

e.t Cs/ t x1 s y1 dt ds D

Z Z

Z

1

Z

0

D Z

1

0

0 1 0

1

D

1

eu .u.1  v//x1 .uv/y1 u du dv eu uxCy2 .1  v/x1 vy1 u du dv

eu uxCy1 du

Z

0

0

1

vy1 .1  v/x1 dv: (2.2.7)

This leads us to the following theorem: Theorem 2.2.2. For x; y > 0, B.x; y/ D

 .x/ .y/ :  .x C y/

(2.2.8)

In particular,  B

1 1 ; 2 2



Z

1

D 0

1

1

t  2 .1  t/ 2 dt D 2

D 2 arcsin.1/ D 2 Therefore, we have

This shows that

Z 0

1

1

.1  u2 / 2 du

(2.2.9)

 D : 2

   2 12 D :  .1/

(2.2.10)

  Z 1 p 1 1 D D  et t  2 dt: 2 0

(2.2.11)

Other types of integrals can be derived from Z

1

e 0

t ˛

1 dt D ˛ uDt ˛

Z

1

u

e u 0

1 ˛ 1

1 du D  ˛

  1 ; ˛

˛ > 0:

(2.2.12)

Lemma 2.2.3. For ˛ > 0, Z

1

e 0

t ˛

 dt D 

 ˛C1 : ˛

(2.2.13)

2.2 Euler’s Beta Function

31

In particular, Lemma 2.2.3 yields for ˛ D 2 Z

1 0

2

et dt D 

p     1 1  3 D  D : 2 2 2 2

(2.2.14)

Moreover, we have Z

1 0

and

Z

1 0

˛

t x1 et dt D

2

t x1 e˛t dt D

1 x   ; ˛ ˛

x; ˛ > 0;

1 x x  ˛ 2 ; 2 2

x; ˛ > 0:

(2.2.15)

(2.2.16)

Within the notational framework of polar coordinates (see (6.1.17) and (6.1.18) for details) we are now prepared to give the well-known calculation of the area kSq1 k of the unit sphere Sq1 in Rq : By definition, we set kS0 k D 2. Clearly, S1 is the unit circle in R2 , i.e., S1 D fx 2 R2 W jxj D 1g. Hence, its area is equal to kS1 k D 2:

(2.2.17)

Furthermore, S2 D fx 2 R3 W jxj D 1g is the unit sphere in R3 . Thus, its area is known to be equal to kS2 k D 4:

(2.2.18)

We are interested in deriving the area of the sphere Sq1 in Rq .q > 3/: q1

kS

Z kD

Sq1

dS.q1/ ..q/ /:

(2.2.19)

In terms of spherical coordinates (6.1.17) and (6.1.18) in Rq the surface element dS.q1/ ./ of the sphere Sq1 in Rq admits the representation p    dS.q1/ .q/ D dS.q2/ 1  t 2 .q1/ dt p  C .1/q1 t dV.q1/ 1  t 2 .q1/ :

(2.2.20)

Now, we notice that dV.q1/

p    q3 1  t 2 .q1/ D t.1  t 2 / 2 dt dS.q2/ .q1/ D .1/q1 t.1  t 2 /

q3 2

(2.2.21)

  dS.q2/ .q1/ dt:

32

2 The Gamma Function

In addition, it is not difficult to see that p    q1 1  t 2 .q1/ D .1  t 2 / 2 dS.q2/ .q1/ : dS.q2/

(2.2.22)

Combining our results we are led to the identity   p dS.q1/ t "q C 1  t 2 .q1/ D .1  t 2 /

q3 2



(2.2.23)

   1  t 2 C t 2 dS.q2/ .q1/ dt;

p where we have used the decomposition .q/ D t "q 1  t 2 .q1/ . Note that "1 ; : : : ; "q is the canonical orthonormal system in Rq . In brief, we obtain     q3 dS.q1/ .q/ D .1  t 2 / 2 dS.q2/ .q1/ dt;

(2.2.24)

such that Z

kSq1 k D

1 1

Z Sq2

Z

D kSq2 k

.1  t 2 / 1

1

q3 2

.1  t 2 /

dS.q2/ ..q1/ / dt

q3 2

(2.2.25)

dt:

For the computation of the remaining integral it is helpful to use some facts known from the Gamma function as well as Euler’s Beta function. More explicitly, Z

1 1

2

.1  t /

q3 2

Z dt D 2

0

 DB

1

2

.1  t /

1 q1 ; 2 2

q3 2

t 2 Dv

Z

1

dt D

 D



0

1

v 2 .1  v/

q3 2

dv

(2.2.26)

    p q1 q1   2 2 2     D :  q2  q2

1

By recursion we get the following lemma from (2.2.26): Lemma 2.2.4. For q  2, q

q1

kS

2 k D 2 q :  2

(2.2.27)

q1

The area of the sphere SR .y/ with center y 2 Rq and radius R > 0 is given by q

q1

kSR .y/k D kSq1 k Rq1 D 2

2   Rq1 :  q2

(2.2.28)

2.3 Stirling’s Formula

33

Furthermore, using .q/ D radius R > 0 is given by q kBR .y/k

x , jxj

q

the volume of the ball BR .y/ with center y 2 Rq and

Z

Z

D

q

BR .y/

dV.q/ .x/ D

Z

R

 q1

Sr

rD0 q

2 D 2 q   2

Z 0

.y/

R

dS.q1/ ..q/ /

dr

(2.2.29)

q

r

q1

2  Rq : dr D  q  2 C1

2.3 Stirling’s Formula Next, we are interested in the behavior of the Gamma function  for large positive values x. This provides us with the so-called Stirling’s formula, a result which we apply to verify the helpful duplication formula and to extend the Gamma function in Sect. 2.4. Theorem 2.3.1 (Stirling’s Formula). For x > 0, ˇ ˇ r ˇ ˇ 2 ˇ p  .x/ ˇ ˇ 2x x1=2 ex  1ˇ  x :

(2.3.1)

Proof. Regard x as fixed and substitute t D x.1 C s/;

1  s < 1;

dt Dx ds

(2.3.2)

in the defining integral of the Gamma function. We obtain Z  .x/ D

1 0

et t x1 dt D

x x

Dx e

Z

1 1

Z

1 1

exxs x x1 .1 C s/x1 x ds

(2.3.3)

.1 C s/x1 exs ds D x x ex I.x/:

Our aim is to verify that I.x/ satisfies ˇ r ˇˇ ˇ 2 ˇ 2 ˇ ˇI.x/  ˇ : ˇ x ˇ x such that ˇ ˇ ˇ  .x/ r 2 ˇ 2 ˇ ˇ ˇ x x  ˇ ; ˇx e x ˇ x

ˇ ˇ r ˇ ˇ 2  .x/ ˇ i:e:; ˇ : p  1ˇˇ  x1=2 x x x e 2

(2.3.4)

(2.3.5)

34

2 The Gamma Function 5 4 3 2 1 0 −1 −2 −3 −1

0

1

2

3

4

5

6

Fig. 2.2 The functions s 7! u2 .s/ D s  ln.1 C s/ (blue) and u.s/ defined by (2.3.7) (red)

For that purpose we write 2 .s/

.1 C s/x exs D exp .x.s  ln.1 C s/// D exu

;

(2.3.6)

where (cf. Fig. 2.2) ( u.s/ D

1

js  ln.1 C s/j 2 js  ln.1 C s/j

1 2

;

s 2 Œ0; 1/;

;

s 2 .1; 0/:

(2.3.7)

We set up the Taylor expansion of u2 for s 2 .1; 1/ at 0: d2 u2 du2 s2 .0/s C .#s/ ds ds 2 2   1 1 s2 sC ; D0C 1 1C0 .1 C s#/2 2

u2 .s/ D u2 .0/ C

(2.3.8)

where # 2 .0; 1/. Therefore, u2 .s/ D

1 s2 2 .1 C s#/2

(2.3.9)

with 0 < # < 1. We interpret # as a uniquely defined function of s, i.e., # W s 7! #.s/; such that

2.3 Stirling’s Formula

35

1 u.s/ 1 D p s .1 C s#.s// 2

(2.3.10)

is a positive continuous function for s 2 .1; 1/ with the property ˇ ˇ ˇ ˇ ˇ  ˇ ˇ u.s/ ˇ 1 ˇˇ s#.s/ ˇˇ 1 1 ˇˇ ˇˇ 1 ˇ ˇ ˇ s  p2 ˇ D ˇ p2 1 C s#.s/  1 ˇ D p2 ˇ 1 C s#.s/ ˇ ˇ ˇ ˇ s#.s/u.s/ ˇ ˇ ˇ D j#.s/j  ju.s/j  ju.s/j: Dˇ ˇ s

(2.3.11)

From u2 .s/ D s  ln.1 C s/ follows that 2u du D

s ds: 1Cs

(2.3.12)

Obviously, s W u 7! s.u/; u 2 R, is of class C.1/ .R/ and thus, Z I.x/ D

1 1

.1 C s/x1 exs ds D 2

Z

C1

2

exu

1

u du: s.u/

(2.3.13)

We are able to deduce that ˇ ˇ ˇ ˇ p Z 1 xu2 ˇ ˇ Z 1 xu2 u p Z C1 xu2 ˇ ˇ ˇI.x/  2 2 ˇ D ˇ2 ˇ du  e du e 2 e du ˇ ˇ ˇ ˇ s.u/ 0 1 1 ˇ Z 1   ˇ ˇ ˇ 1 u 2 D ˇˇ2 p duˇˇ exu s.u/ 2 1 ˇ ˇ Z 1 1 ˇ 2 ˇ u  p ˇˇ du exu ˇˇ 2 s.u/ 2 1 Z 1 Z 1 2 xu2 e exu u du: (2.3.14) 2 juj du D 4 1

0

Note that we can use the integral (2.2.16) with ˛ D x and x D 1; Z

1

2

u11 exu du D

Z

0

1

2

exu du D

0

1 2

x 1=2 

1 2

p  D p ; 2 x

(2.3.15)

as well as with ˛ D x and x D 2 in (2.2.16) Z

1

u 0

21 xu2

e

Z

1

du D 0

2

exu u du D

1 2

x 1  .1/ D

1 : 2x

(2.3.16)

36

2 The Gamma Function

This yields: r ˇˇ r ˇ ˇ ˇ p 1  ˇ ˇˇ ˇ ˇ D ˇI.x/  2 ˇˇ  4 1 D 2 : ˇI.x/  2 2 ˇ 2 xˇ ˇ x ˇ 2x x

(2.3.17) t u

This leads to the desired result. Remark 2.3.2. Stirling’s formula can be rewritten in the form lim p

x!1

 .x/ 2x x1=2 ex

D 1:

(2.3.18)

a > 0:

(2.3.19)

An immediate application is the limit relation lim

x!1

 .x C a/ D 1; x a  .x/

This can be seen from Stirling’s formula by lim p

x!1

 .x C a/ 1

2.x C a/xCa 2 exa

D1

(2.3.20)

due to the relation 1

1

1

.x C a/xCa 2 D x xCa 2 .1 C xa /xCa 2

(2.3.21)

and the limits .1 C xa /x D 1; x!1 ea lim

1

lim .1 C xa /a 2 D 1:

x!1

(2.3.22)

Next, we prove the so-called Legendre relation or duplication formula. Lemma 2.3.3 (Duplication Formula). For x > 0 we have 2

x1



x  2

 

xC1 2

 D

p   .x/:

(2.3.23)

Proof. We consider the function x 7! ˚.x/; x > 0; defined by ˚.x/ D

2x1  . x2 / . xC1 2 /  .x/

(2.3.24)

for x > 0. Setting x C 1 instead of x we find the following functional equation for the numerator     x  x C 1 x xC1 2x   C 1 D 2x1 x  ; (2.3.25) 2 2 2 2

2.3 Stirling’s Formula

37

such that the numerator satisfies the same functional equation as the denominator. This means ˚.x C 1/ D ˚.x/; x > 0. By repetition we get for all n 2 N and x fixed ˚.x C n/ D ˚.x/. We let n tend to 1. For the numerator of ˚.x C n/ we then find by use of the result in Remark 2.3.2, i.e., by using (2.3.19) twice, that xCnC1 2xCn1  . xCn / 2 / . 2 x    n 2 D 1: n!1 xCn1 n 2 n xC1 2 .2/ 2  .2/ 2

lim

(2.3.26)

For the denominator we consider lim

n!1 xCn1 2

 .x C n/  n  x2 n xC1  n 2 .2/ 2  .2/ 2

D lim

n!1

D lim

n!1

2xCn1

n!1

 .x C n/

2

2

. . n / / . n2 /n1 en 2 . n /n12en 2 2

en 

2 !1  . n2 / . n2 /n1 en 2

p 1 2nnCx 2 en 

 n 2 !1  .2/ n n1 n . 2 / e 2

2xCn

D lim p

 n xC 12

 .x C n/  n nCx 12 2



 .x C n/

1 Dp ; 

(2.3.27)

since Stirling’s formula yields that lim n!1 p i.e.,

 . n2 / n

1

n

2. n2 / 2  2 e 2

D 1;

(2.3.28)

 n 2 .2/ D 1; lim n!1 2. n /n1 en 2

(2.3.29)

and by the same arguments as in Remark 2.3.2 (set a in (2.3.20) to x and x in (2.3.20) to n) we find that  .x C n/ lim p D 1: 1 2nnCx 2 en

(2.3.30)

n!1

Therefore, we get for every x > 0 and all n 2 N; ˚.x/ D ˚.x C n/ D lim ˚.x C n/ D n!1

p :

(2.3.31)

38

2 The Gamma Function

A periodic function with this property must be constant. This proves the lemma. u t A generalization of the Legendre relation (“duplication formula”) is the Gauß multiplication formula that can be verified by analogous arguments. Lemma 2.3.4. For x > 0 and n  2,



x n

 

xC1 n



  ::: 

 xCn1 n1 p nx D .2/ 2 n  .x/ : (2.3.32) n

2.4 Pochhammer’s Factorial Thus far, the Gamma function  is defined for positive values, i.e., x 2 R>0 . We are interested in an extension of  to the real line R (or even to the complex plane C) if possible. Definition 2.4.1. The so-called Pochhammer factorial .x/n with x 2 R and n 2 N is defined by .x/n D x.x C 1/ : : : .x C n  1/ D

n Y

.x C i  1/:

(2.4.1)

i D1

For x > 0, it is clear that .x/n D or

 .x C n/  .x/

.x/n 1 D :  .x C n/  .x/

(2.4.2)

(2.4.3)

The left-hand side is defined for x > n and gives the same value for all n 2 N 1 with n > x. We may use this relation to define  .x/ for all x 2 R, and we see that this function vanishes for x D 0; 1; 2; : : : (see Fig. 2.3 (left)). We know that the Gamma integral is absolutely convergent for x 2 C with Re.x/ > 0 and represents a holomorphic function for all x 2 C with Re.x/ > 0. Moreover, the Pochhammer factorial .x/n can be defined for all complex x. Because of (2.4.3) with n chosen 1 sufficiently large, we have a definition of  .x/ for all x 2 C. This is summarized in the following lemma. Lemma 2.4.2. The  -function is a meromorphic function that has simple poles in 1 is an entire analytic 0; 1; 2; : : : (see Fig. 2.3 (right)). The reciprocal x 7!  .x/ function (see Fig. 2.3 (left) for an illustration).

2.4 Pochhammer’s Factorial

39

5

10

4

8 6

3

4

2

2

1

0

0

−2 −4

−1

−6

−2

−8

−3 −4

−3

−2

−1

0

1

2

3

4

−10 5 −5

−4

−3

−2

−1

0

1

2

3

4

5

Fig. 2.3 The reciprocal of the Gamma function (left) and the Gamma function (right) on the real line R

Lemma 2.4.3. For x 2 C, n1 Y  1 D lim nx x 1 C xk : n!1  .x/

(2.4.4)

kD1

Proof. Because of (2.4.3) the identity .x/n  .n/ 1 D  .n/  .x C n/  .x/

(2.4.5)

is valid for all x 2 C with Re.x/ > n. Furthermore it is easy to see that n1 Y .x/n x .x C 1/.x C 2/ : : : .x C n  1/ 1C Dx Dx :  .n/ 1  2 : : : .n  1/ k

(2.4.6)

kD1

Combining the two and multiplying with nx we find that n1 Y  .x C n/ x x 1C D n : x x n  .n/ .x/ k

(2.4.7)

kD1

Now, we use (2.3.19) with x D n and a D x, i.e., lim

n!1

 .x C n/ D 1;  .n/nx

x > 0;

(2.4.8)

on the left-hand side and obtain for x > 0; n1 Y x 1  .x C n/ 1 x 1 C D lim D lim ; n x n!1 nx  .n/  .x/ n!1  .x/ k kD1

(2.4.9)

40

2 The Gamma Function

which proves Lemma 2.4.3 for x > 0. To determine if this limit is also defined for x  0 we consider once again s  ln.1 C s/ (cf. Fig. 2.2) with 1 < s < 1 from (2.3.9) in the proof of Theorem 2.3.1: 0  s  ln.1 C s/ D Therefore, we can put s D i.e., with # D 0:

1 s2 2 .1 C #s/2 1 k

   ln 1 C k1 

1 k

This immediately proves that lim

n  P 1

n!1 kD1

lim

n!1

n X  kD1

1 k

# D #.s/ 2 .0; 1/:

(2.4.10)

and estimate the right-hand side with its maximum,

0

Moreover,

;

k

1 : 2k 2

(2.4.11)

   ln 1 C k1 exists and is positive.

n X  1   1  ln 1 C k D lim k  ln.k C 1/ C ln.k/ n!1

D lim

n!1

kD1

X n kD1

1 k

  ln.n C 1/ D ;

(2.4.12)

where  denotes Euler’s constant  D lim

m!1

 m1 X kD1

 1  ln m  0:577215665 : : : : k

(2.4.13)

Assume now that x 2 R. If k  2jxj, then 0

x k

   ln 1 C xk
e k 2 ;

(2.4.16)

2.4 Pochhammer’s Factorial

41

which shows that n1 Y

lim

n!1

kD1

1 Y   x   x x k 1C k e 1 C xk e k D

(2.4.17)

kD1

exists for all x. Furthermore, n1 Y

x ej

j D1

 X  X  n1  n1 1 1 D exp x D exp x j j  x ln.n/ C x ln.n/ j D1

j D1

 X  n1 x 1  x ln.n/ ; D n exp x j

(2.4.18)

j D1

where

 X  n1 1 lim exp x  x ln.n/ D ex : j

n!1

Therefore, lim nx x n!1

n1 Q kD1

1C

lim nx x

n!1

(2.4.19)

j D1

x k



exists for all x 2 R and it holds that

n 1 Y Y     x 1 C xk D xex 1 C xk e k ;

kD1

(2.4.20)

kD1

where the infinite product is also convergent for all x 2 R. By similar arguments these results can be extended for all x 2 C. t u The proof of Lemma 2.4.3 also shows us the following lemma. Lemma 2.4.4. For x 2 C, 1  Y 1 x  x x 1C D xe e k:  .x/ k

(2.4.21)

kD1

Let us consider the expression Q.x/ D

1  .x/ .1  x/ sin.x/; 

(2.4.22)

which has no singularities and is holomorphic for all x 2 C. It is not difficult to show that

42

2 The Gamma Function

1  .1 C x/ .1  x/ sin.x/ x 1 D  .x/ .2  x/ sin.x/: .1  x/

Q.x/ D

(2.4.23) (2.4.24)

Obviously, by using (2.4.23) for x D 0 and (2.4.24) for x D 1, we obtain sin.x/ D 1; x!0 x sin.x/ D 1: Q.1/ D  .1/ .1/ lim x!1 .1  x/

Q.0/ D  .1/ .1/ lim

(2.4.25) (2.4.26)

In the interval Œ0; 1 the function Q is positive and twice continuously differentiable. With the duplication formula (Lemma 2.3.3) we get x 



xC1 Q Q 2 2

 D Q.x/;

(2.4.27)

which is easily verified. Setting R.x/ D ln.Q.x//, we see that x 



xC1 R CR 2 2

 D R.x/:

(2.4.28)

By differentiation we obtain 1 00  x  1 00 R C R 4 2 4



xC1 2



D R00 .x/:

(2.4.29)

As the second order derivative R00 is continuous on the compact interval Œ0; 1, there is a value  2 Œ0; 1 such that jR00 ./j  jR00 .x/j; x 2 Œ0; 1: Therefore, we obtain from (2.4.29) ˇ  ˇ ˇ  ˇ 1 ˇˇ 00  ˇˇ 1 ˇˇ 00  C 1 ˇˇ 1 00 00 jR ./j  ˇR C R ˇ  2 jR ./j; 4 2 ˇ 4ˇ 2

(2.4.30)

(2.4.31)

which implies jR00 ./j D 0, i.e., R00 .x/ D 0. From R.1/ D R.0/ D 0 we then deduce R.x/ D 0. Therefore, Q.x/ D 1. This result can be written in the form  .x/ .1  x/ D

 : sin.x/

(2.4.32)

2.5 Exercises

43

It establishes an identity between the meromorphic functions  ./,  .1  /, and .sin /1 . Altogether, we have  1  Y 1 x2 1 1 2 : Dx  .x/  .1  x/ k

(2.4.33)

kD1

In connection with (2.4.32) we obtain Lemma 2.4.5. For x 2 C, sin.x/ D x

 1  Y x2 1 2 : k

(2.4.34)

kD1

2.5 Exercises (Incomplete Gamma and Beta Function, Applications in Statistics) In this section the discussion of the so-called incomplete Gamma functions and their relation to the error functions erf and erfc as well as the incomplete Beta function is left to the reader in the form of some exercises. The results demonstrate an immediate transition to probability distributions in statistics.

Incomplete Gamma Function Definition 2.5.1. By definition, we let for x; a > 0; Z  .a; x/ D

1 x

et t a1 dt Z

and .a; x/ D  .a/   .a; x/ D

x 0

et t a1 dt:

(2.5.1)

(2.5.2)

The functions  .; x/, .; x/ are called the incomplete Gamma functions related to x. Exercise 2.5.2. Prove that .a; x/ D x a

Z

1 0

a x

 .a; x/ D x e

ext t a1 dt; Z

1 0

ext dt:

(2.5.3) (2.5.4)

44

2 The Gamma Function

Exercise 2.5.3. Show that the so-called error functions 2 erf.x/ D p 

Z

x

2

et dt;

0

2 erfc.x/ D 1  erf.x/ D p 

(2.5.5) Z

1

2

et dt

(2.5.6)

x

admit the representations 1 erf.x/ D p . 21 ; x 2 /; 

(2.5.7)

1 erfc.x/ D p  . 12 ; x 2 /: 

(2.5.8)

Exercise 2.5.4. Verify that  .n C 1; x/ D nŠex

n X xm ; mŠ mD0

.n C 1; x/ D nŠ 1  e

x

(2.5.9)

n X xm mŠ mD0

! (2.5.10)

hold true for all n 2 N0 . Exercise 2.5.5. Prove the following recurrence relations .a C 1; x/ D a.a; x/  x a ex ; a x

 .a C 1; x/ D a .a; x/ C x e

(2.5.11) :

(2.5.12)

Incomplete Beta Function Definition 2.5.6. The function .x; y/ 7! B.x; y; ˛/ defined by Z ˛ B.x; y; ˛/ D t x1 .1  t/y1 dt

(2.5.13)

0

is called incomplete Beta function relative to ˛ 2 .0; 1. Exercise 2.5.7. Show that I.x; y; ˛/ D satisfies

B.x; y; ˛/ B.x; y/

(2.5.14)

2.5 Exercises

45

I.x; y; 1/ D 1;

(2.5.15)

I.x; y; ˛/ D 1  I.y; x; 1  ˛/:

(2.5.16)

Exercise 2.5.8. Prove the recurrence relation .x C y/I.x; y; ˛/ D xI.x C 1; y; ˛/ C yI.x; y C 1; ˛/:

(2.5.17)

Exercise 2.5.9. Verify the binomial expansion ! n X n l ˛ .1  ˛/nl ; I.m; n C 1  m; ˛/ D l

n; m 2 N:

(2.5.18)

lDm

As already announced, the incomplete Gamma and Beta functions possess a variety of applications in probability theory and statistics, from which we mention only two examples. We restrict ourselves to continuous random variables with nonnegative realizations. Definition 2.5.10 (Gamma Distribution). A random variable X with density distribution ( 0 ; if x  0; F .x/ D ˛p p1 ˛x (2.5.19) e ; if x > 0;  .p/ x is called a Gamma distribution with p > 0 and ˛ > 0. Clearly, F .x/  0 for all x 2 R. An easy calculation gives Z R

F .x/ dx D 1:

(2.5.20)

The probability density function of the Gamma distribution reads as follows: Z x 7!

x 0

F .t/ dt D

˛p  .p/

Z 0

x

t p1 e˛t dt D

1  .p/

Z

˛x 0

t p1 et dt D .p; ˛x/: (2.5.21)

The Gamma distribution is widely used as a conjugate prior in Bayesian statistics (for more details see, e.g., Papoulis and Pillai 2002).

Beta Distribution The Beta distribution is a family of continuous probability distributions defined on the interval .0; 1/ and parameterized by two positive values p and q.

46

2 The Gamma Function

Definition 2.5.11. A random variable X with density distribution ( F .x/ D

0

; if x … .0; 1/;

1 p1 .1 B.p;q/ x

 x/

q1

; if x 2 .0; 1/;

(2.5.22)

is called a Beta distribution with p; q > 0. The probability density function of the Beta distribution reads as follows: Z x 7!

x 0

1 F .t/ dt D B.p; q/

Z

x 0

t p1 .1  t/q1 dt D

B.p; q; x/ B.p; q/

(2.5.23)

for x 2 .0; 1/. The expectation value and the variance of a Beta distributed random variable X corresponding to the parameters p and q are given by  D E.X / D

p ; pCq

Var.X / D E.X  2 / D

pq : .p C q/2 .p C q C 1/

(2.5.24) (2.5.25)

Beta distributions are often used in Bayesian inference, since Beta distributions provide a family of conjugate prior distributions for binomial distributions (more details can be found, e.g., in van der Waerden 1969).

Chapter 3

Orthogonal Polynomials

In this chapter, we introduce polynomial function systems that are orthogonal with respect to a scalar product characterized by a measure d. We start with some very general results from Fourier analysis (see, e.g., Davis 1963; Reed and Simon 1972; Rudin 1991; Yoshida 1980), before we begin to specifically consider polynomials. As means of normalization we deal with monic polynomials, i.e., polynomials whose leading coefficient is equal to 1, and consider their properties (symmetry, zeros, best approximation) in Sect. 3.1. This enables us to find a three-term recurrence whose coefficients yield matrices that possess the zeros of the monic orthogonal polynomials as eigenvalues. Finally, the Christoffel–Darboux formula is included. In Sect. 3.2 we deal with one highly important application of orthogonal polynomials, namely quadrature rules, in particular Gauß quadratures. For more on the general orthogonal polynomials, in particular on computational techniques, we refer to Gautschi (2004) and the references therein. Starting with the Jacobi polynomials in Sect. 3.3, we concentrate on classical orthogonal polynomials which possess additional properties such as a defining differential equation and Rodrigues’ representation. We also investigate their relation to the hypergeometric function. In Sect. 3.4 we restrict ourselves to the ultraspherical (or Gegenbauer) polynomials which are a particular type of Jacobi polynomials. Later on, they are of special interest in geomathematics. They form the coefficients of the power series of their generating function and we establish a set of recurrence relations for the ultraspherical polynomials and their derivatives. Moreover, we provide their connection with the Legendre polynomials of dimension q (see, e.g., Freeden 2011; M¨uller 1998) which will be of great importance in Chap. 6. In Sect. 3.5, we develop the application of Legendre polynomials in electrostatics, briefly introduce Hermite polynomials in Sect. 3.6 as well as Laguerre polynomials in Sect. 3.7. We also give examples for their application and significance in physics. Finally, some exercises deal with error estimates for Gaussian integration, stable evaluation algorithms for Legendre series, and error estimates for kernel expansions in Sect. 3.8. For further details on classical orthogonal polynomials the reader is

W. Freeden and M. Gutting, Special Functions of Mathematical (Geo-)Physics, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-0348-0563-6 3, © Springer Basel 2013

47

48

3 Orthogonal Polynomials

referred to, e.g., Abramowitz and Stegun (1972), Lebedev (1973), Magnus et al. (1966), Sneddon (1980), and Szeg¨o (1967). We consider weighted Hilbert spaces of square-integrable functions on intervals in R. In general, these spaces are denoted by L2 .Œa; b/, where a; b can also be 1; 1, respectively. They possess the scalar product, for F; G 2 L2 .Œa; b/, Z

b

hF; Gid D

F .x/G.x/ d.x/:

(3.0.1)

a

 is a nondecreasing function on R which has finite limits as x ! ˙1 and whose induced positive measure d has finite moments of all orders, i.e., Z r D r .d/ D

x r d.x/ < 1

(3.0.2)

R

for r 2 N0 with 0 > 0. If  is absolutely continuous, the scalar product becomes Z

b

hF; Gid D

F .x/G.x/w.x/ dx:

(3.0.3)

a

Here w is a non-negative function which is Lebesgue-measurable and for which Rb a w.x/ dx > 0. It is called a weight function. Definition 3.0.1. Two functions F; G 2 L2 .Œa; b/ are orthogonal if hF; Gid D 0. If additionally kF kd D kGkd D 1, they are called orthonormal. Example 3.0.2. The trigonometric functions Fn .x/ D cos.nx/, n 2 N0 , form a system of orthogonal functions on the interval .0; / with the weight function w.x/ D 1 since Z 

cos.mx/ cos.nx/ dx D 0

(3.0.4)

0

for n ¤ m. Definition 3.0.3. Let fxk gk2N0 be a given orthonormal set (either finite or infinite). To an arbitrary real-valued function F let there correspond the formal Fourier expansion 1 X Fk xk .x/ (3.0.5) F .x/  kD0

with the coefficients Fk defined by Z

b

Fk D hF; xk id D

F .x/xk .x/ d.x/;

k 2 N0 :

(3.0.6)

a

These coefficients are called Fourier coefficients of F with respect to the given orthonormal system.

3 Orthogonal Polynomials

49

Theorem 3.0.4. Let xk , Fk be as before with F 2 L2 .Œa; b/. Let l  0 be a fixed integer and ak 2 R, k D 0; : : : ; l. The integral Z

b

 2 l X F .x/  ak xk d.x/

a

(3.0.7)

kD0

becomes minimal if and only if ak D Fk for k D 0; : : : ; l. P Proof. We consider the integral in question with G D lkD0 ak xk : kF  Gk2d D hF; F id  2Re.hF; Gid / C hG; Gid D kF k2d  2Re

X l kD0

D kF k2d C

l X

 l X ak hF; xk id C ak aj hxk ; xj id „ ƒ‚ … k;j D0

 X l jak j2  2Re ak Fk

kD0

D kF k2d 

l X

(3.0.8)

ık;j

kD0

jFk j2 C

kD0

l X

jak  Fk j2

kD0

which is minimal if and only if ak D Fk for all k D 0; : : : ; l. Therefore, the truncated Fourier series is the best approximating element. t u Remark 3.0.5. We also find Bessel’s equality for F , Fk , and xk as in Theorem 3.0.4: l l  2 X X   Fk xk  D kF k2d  jFk j2 F  d

kD0

(3.0.9)

kD0

and Bessel’s inequality (the left-hand side in (3.0.9) is non-negative): l X

jFk j2  kF k2d :

(3.0.10)

kD0

Lemma 3.0.6. For F , Fk , and xk as in Theorem 3.0.4 the Fourier series X Fk xk (3.0.11) k2N0

converges in the L2 -norm (in the L2 -sense) to an L2 -function S and .F  S /?xk for all k 2 N0 .

50

3 Orthogonal Polynomials

Proof. For n  m we have n n n 2 X X X   Fk xk  D jFk j2 kFk xk k2d D  kDm

d

kDm

(3.0.12)

kDm

due to the Theorem of Pythagoras and the orthonormality of the functions P xk . Bessel’s inequality (3.0.10) gives us the convergence of the series k2N0 jFk j2 . Thus, for all " > 0 there exists an m 2 N, such that for n  m; n X

jFk j2 < ":

(3.0.13)

kDm

Therefore, the sequence fSn gn with Sn D

n X

Fk xk

(3.0.14)

kD0

is a Cauchy sequence with respect to the L2 -norm. Since L2 .Œa; b/ is complete, there exists S 2 L2 .Œa; b/ which is the limit of this sequence, i.e., S D lim Sn D n!1

X

Fk xk

(3.0.15)

k2N0

in the sense of L2 .Œa; b/. It remains to show the orthogonality of S  F : If j 2 f0; : : : ; lg, then hS  F; xj id D

DX k2N0

E Fk xk ; xj

d

 hF; xj id D

X k2N0

Fk hxk ; xj id Fj D 0; „ ƒ‚ … ık;j

(3.0.16) since for Fn ! F , Gn ! G we have hFn ; Gn id ! hF; Gid .

t u

Now, we only have to show completeness of the system fxk gk . Then, F  S D 0 (in the L2 -sense), i.e., the limit of the Fourier series is the function F . Therefore, we consider properties of Fourier series in a general inner product space. Theorem 3.0.7. Let fxk gk2N  X be a sequence of orthonormal elements of the inner product space X . Consider the following statements: (A) The elements xk are closed in X , i.e., for any element x 2 X and for all " > 0 there exist an n 2 N and coefficients a1 ; : : : ; an 2 K such that n   X   ak xk   ": x  kD1

X

(3.0.17)

3 Orthogonal Polynomials

51

(B) The Fourier series of any y 2 X converges to y (in the norm of X ), i.e., n   X   hy; xk iX xk  D 0: lim y 

n!1

X

kD1

(3.0.18)

(C) Parseval’s identity holds, i.e., for all y 2 X , kyk2X D hy; yiX D

1 X

jhy; xk iX j2 :

(3.0.19)

kD1

(D) The extended Parseval identity holds, i.e., for all x; y 2 X , hx; yiX D

1 X hx; xk iX hxk ; yiX :

(3.0.20)

kD1

(E) There is no strictly larger orthonormal system containing the set fxk gk2N . (F) fxk gk2N is complete, i.e., if y 2 X and hy; xk iX D 0 for all k 2 N, then y D 0. (G) An element of X is determined uniquely by its Fourier coefficients, i.e., if hx; xk iX D hy; xk iX for all k 2 N, then x D y. Then, it holds that .A/ ” .B/ ” .C / ” .D/ H) .E/ ” .F / ” .G/:

(3.0.21)

If X is a complete inner product space, then also .E/ H) .D/ and all statements are equivalent. Proof. Assume (A). Due to the minimizing property of the truncated Fourier series (see Theorem 3.0.4 which holds for any orthonormal system fxk gk ) n n     X X     hy; xk iX xk   y  ak xk   "; y  X

kD1

kD1

X

(3.0.22)

where the last estimate is provided by .A/. If on the other hand .B/ holds, we can approximate any y by its truncated Fourier series which shows closure. Thus, .A/ ” .B/. By orthogonality, D

x

n X kD1

hx; xk iX xk ; y 

n X kD1

hy; xk iX xk

E X

D hx; yiX 

n X hx; xk iX hxk ; yiX : kD1

(3.0.23)

52

3 Orthogonal Polynomials

Using the Schwarz inequality, n n n ˇ ˇ     X X X ˇ ˇ     hx; xk iX hxk ; yiX ˇ  x  hx; xk iX xk   y  hy; xk iX xk  : ˇhx; yiX  kD1

X

kD1

kD1

X

(3.0.24) If .B/ holds, the right-hand members both tend to 0, hence .B/ H) .D/. Selecting x D y in .D/ shows .C /, i.e., .D/ H) .C /. Since n n  2 X X   2 0  y  hy; xk iX xk  D kykX  jhy; xk iX j2 ; kD1

X

(3.0.25)

kD1

we see .C / H) .B/ and thus, .A/ ” .B/ ” .C / ” .D/. Now, assume .A/ and suppose that fxk gk2N [fwg, w ¤ xk , is also an orthonormal system. This system is also closed in X . Since .A/ H) .C / kwk2X D

1 X

jhw; xk iX j2 C jhw; wiX j2 ; and kwk2X D

kD1

1 X

jhw; xk iX j2 :

kD1

(3.0.26) Thus, hw; wiX D 0 which contradicts kwkX D 1. This means that .A/ H) .E/. Suppose there is a 0 ¤ y 2 X , such that hy; xk iX D 0 for all k. Then, the set fxk gk2N [ fy= kykX g would be a strictly larger orthonormal system thanfxk gk2N . Therefore, .E/ ” .F /. Suppose that hw; xk iX D hy; xk iX , k 2 N. Then, hw  y; xk iX D 0, k 2 N. Assuming .F /, w  y D 0 and .F / H) .G/. If .F / were false, we could find z ¤ 0 with hz; xk iX D 0, k 2 N. For any y, hy; xk iX D hy Cz; xk iX , k 2 N. So y and y Cz would be two distinct elements with the same Fourier coefficients. Thus, .G/ would be false and we obtain .G/ H) .F /. This completes the chain of implications (3.0.21). Assume now that additionally X is complete. We want to show .G/ H) .B/ which is the missing implication. Let w 2 X and consider Sn D

n X hw; xk iX xk :

(3.0.27)

kD1

For n > m, we find that kSn  Sm k2X D

n X kDmC1

jhw; xk iX j2 :

(3.0.28)

3 Orthogonal Polynomials

53

Bessel’s inequality gives us 1 X

jhw; xk iX j2 < 1:

(3.0.29)

kD1

Thus, for a given " > 0, we can find an N."/ such that the right-hand side of (3.0.28) is less than " for all m; n  N."/. Thus, fSn gn is a Cauchy sequence and since X is complete, it converges to an element S 2 X . If l be fixed and n  l, then hS  Sn ; xl iX D hS; xl iX  hSn; xl iX D hS; xl iX  hw; xl iX

(3.0.30)

and by the Schwarz inequality jhS; xl iX  hw; xl iX j D jhS  Sn ; xl iX j  kS  Sn kX kxl kX D kS  Sn kX : (3.0.31) Together with the convergence of Sn to S this gives us hS; xl iX D hw; xl iX

for l 2 N:

(3.0.32)

By .G/, this implies that S D w, such that the convergence reads n   X   hw; xk iX xk  D 0: lim w 

n!1

kD1

X

(3.0.33) t u

This is precisely .B/.

From here on, we consider polynomials. At first, we have to investigate the positive definiteness of the inner product on the space of polynomials. Definition 3.0.8. The space of real polynomials up to degree n is denoted by ˘n . The space of real polynomials of all degrees is ˘ , ˘n  ˘ for all n 2 N0 . For P; Q 2 ˘ we use the scalar product Z hP; Qid D

P .x/Q.x/ d.x/

(3.0.34)

R

and the induced norm. Note that these integrals exist by definition of  (finite moments). Definition 3.0.9. The scalar product (3.0.34) is called positive definite on the space of all polynomials ˘ if kP kd > 0 for all P 2 ˘ , P 6 0. It is called positive definite on the space ˘n (polynomials of degree less than or equal to n) if kP kd > 0 for all P 2 ˘n , P 6 0.

54

3 Orthogonal Polynomials

Theorem 3.0.10. The scalar product (3.0.34) is positive definite on the space ˘ if and only if for the corresponding moments r (see (3.0.2)), ˇ ˇ ˇ 0 1    k1 ˇ ˇ ˇ ˇ 1 2    k ˇ ˇ ˇ det Mk D ˇ : :: :: ˇˇ > 0; ˇ :: : : ˇ ˇ ˇ ˇ k1 k    2k2

k 2 N:

(3.0.35)

It is positive definite on the space ˘n if and only if det Mk > 0 for k D 1; 2; : : : ; n C 1. n P ck x k . Proof. At first, we treat the case of ˘n . Let P 2 ˘n , i.e., P .x/ D kD0

kP k2d

D

n X k;lD0

Z ck cl

n X

x kCl d.x/ D R

ck cl kCl D c T MnC1 c;

(3.0.36)

k;lD0

where c D Œc0 ; c1 ; : : : ; cn T 2 RnC1 . Therefore, positive definiteness on ˘n is equivalent to positive definiteness of the matrix MnC1 , i.e., det Mk > 0 for k D 1; 2; : : : ; n C 1:

(3.0.37) t u

Taking n to infinity yields the result for ˘ .

Definition 3.0.11. Polynomials whose leading coefficient is 1, i.e., Pk .x/ D x k C : : : with k 2 N0 , are called monic polynomials. The set of monic polynomials of degree n is denoted by ˘nı . Monic real polynomials Pk , k 2 N0 , are called monic orthogonal polynomials with respect to the measure d if hPk ; Pl id D 0 for k ¤ l; k; l 2 N0 and kPk kd > 0 for k 2 N0 :

(3.0.38)

Normalization yields the orthonormal polynomials PQk .x/ D Pk .x/= kPk kd . Lemma 3.0.12. Let fPk gkD0;1;:::;n be monic orthogonal polynomials. If Q 2 ˘n satisfies hQ; Pk id D 0 for k D 0; 1; : : : ; n, then Q  0. Proof. We write Q as Q.x/ D

n P i D0

ai x i . Then,

0 D hQ; Pn id D

n X i D0

ai hx i ; Pn id

(3.0.39)

3 Orthogonal Polynomials

55

and x i can be written as Pi .x/ C Ri 1 .x/ with Ri 1 2 ˘i 1 : Thus hx i ; Pn id D hPi ; Pn id C hRi 1 ; Pn id D ıi;n hPn ; Pn id ;

(3.0.40)

since Ri 1 can be written as a linear combination of monic orthogonal polynomials whose degree is i 1. Thus, they are orthogonal to Pn and so is Ri 1 . This gives us 0 D hQ; Pn id D an hPn ; Pn id . Since hPn ; Pn id > 0, we obtain an D 0. Similarly, we can show that an1 D 0, an2 D 0; : : : ; a0 D 0. t u Lemma 3.0.13. A set P0 ; : : : ; Pn of monic orthogonal polynomials is linearly independent. Moreover, any polynomial P 2 ˘n can be uniquely represented in P the form P D nkD0 ck Pk for some real coefficients ck , i.e., P0 ; : : : ; Pn is a basis of ˘n . P Proof. If nkD0 k Pk  0, taking the scalar product on both sides of the equation with Pj , j D 0; : : : ; n, yields that j D 0. This gives us linear P independence. If we take the scalar product with Pj on both sides of P D nkD0 ck Pk , we find that hP; Pj id D

n X

ck hPk ; Pj id D cj hPj ; Pj id ;

j D 0; : : : ; n:

(3.0.41)

kD0

P The difference P  nkD0 ck Pk is orthogonal to P0 ; : : : ; Pn and by Lemma 3.0.12 has to be the zero polynomial. t u Theorem 3.0.14. If the scalar product h; id is positive definite on ˘ , there exists a unique infinite sequence fPk gk2N0 of monic orthogonal polynomials. Proof. The polynomials Pk can be constructed by applying Gram-Schmidt orthogonalization to the sequence of powers fEk gk2N0 , where Ek .x/ D x k . Therefore, we choose P0 D 1 and for k 2 N we use the recursion Pk D Ek 

k1 X lD0

cl Pl ;

cl D

hEk ; Pl id : hPl ; Pl id

(3.0.42)

Since the scalar product is positive definite, hPl ; Pl id > 0. Thus, the monic polynomial Pk is uniquely defined and by construction orthogonal to all Pj , j < k. t u The prerequisites of Theorem 3.0.14 are fulfilled if  has many points of increase, i.e., points t0 such that .t0 C "/ > .t0  "/ for all " > 0. The set of all points of increase of  is called support of the measure d, its convex hull is the support interval of d. Theorem 3.0.15. If the scalar product h; id is positive definite on ˘n , but not on ˘m for all m > n, there exist only n C 1 orthogonal polynomials P0 ; : : : ; Pn .

56

3 Orthogonal Polynomials

Proof. We apply Gram–Schmidt orthogonalization, i.e., (with Ek .x/ D x k ) Pk D Ek 

k1 X hEk ; Pl id lD0

hPl ; Pl id

k 2 N0 ;

Pl ;

(3.0.43)

as long as hPl ; Pl id > 0, i.e., for k  n C 1. In the end we obtain PnC1 which is orthogonal to Pj for all j  n and all Pj , j D 0; : : : ; n, are orthogonal polynomials with positive norm. We now show that kPnC1 kd D 0. Since by assumption h; id is not positive definite on ˘nC1 , there exists a monic polynomial w 2 ˘nC1 with kwkd D 0. Moreover, deg.w  PnC1 /  n and thus, w D PnC1 C

n X

aj Pj

(3.0.44)

j D0

with coefficients aj 2 R. Since 0 D kwk2d D kPnC1 k2d C

n X j D0

 2 aj2 Pj d

(3.0.45)

  and Pj d > 0, we find that aj D 0, j D 0; : : : ; n, and kPnC1 kd D 0. Hence, PnC1 is not an orthogonal polynomial. t u Theorem 3.0.16. If the moments of d exist only for r D 0; 1; : : : ; r0 , there exist only n C 1 orthogonal polynomials P0 ; : : : ; Pn , where n D br0 =2c. Proof. The Gram–Schmidt procedure can be performed as long as the scalar products in (3.0.42), in particular hPk ; Pk id , exist. This is the case as long as 2k  r0 , i.e., k  n D br0 =2c. t u

3.1 Properties of Orthogonal Polynomials For this section we assume that the measure d is a positive measure on R which possesses an infinite number of points of increase and finite moments of all orders. Definition 3.1.1. An absolutely continuous measure d.x/ D w.x/dx is symmetric (with respect to the origin) if its support interval is Œa; a with 0 < a  1 and w.x/ D w.x/ for all x 2 R. Remark 3.1.2. Note that if no confusion is likely to arise, we skip the subscript d for norms and scalar products and we just use the notation L2 .Œa; b/ for the spaces. Theorem 3.1.3. If d is symmetric, then Pk .x/ D .1/k Pk .x/;

k 2 N0 :

(3.1.1)

3.1 Properties of Orthogonal Polynomials

57

Thus, Pk is either an even or an odd polynomial depending on the degree k. Proof. Set POk .x/ D .1/k Pk .x/. We compute for k ¤ l; hPOk ; POl id D .1/kCl

Z R

Pk .x/Pl .x/ d.x/ D .1/kCl hPk ; Pl id D 0:

(3.1.2) Since all POk are monic, POk .x/ D Pk .x/ by the uniqueness of monic orthogonal polynomials. t u Theorem 3.1.4. If d is symmetric on Œa; a, 0 < a  1, and the polynomials PkC and Pk are related to the monic orthogonal polynomials with respect to d such that P2k .x/ D PkC .x 2 /; P2kC1 .x/ D xPk .x 2 /; (3.1.3) then fPk˙ g are the monic orthogonal polynomials with respect to the measure 1

1

1

dC .x/ D x  2 w.x 2 / dx

1

or d .x/ D x C 2 w.x 2 / dx;

(3.1.4)

respectively, on Œ0; a2 . Proof. We restrict ourselves to the proof for PkC , the other case follows analogously. Obviously, the polynomials PkC are monic. By symmetry it holds that Z

a

0 D hP2k ; P2l id D 2

P2k .x/P2l .x/w.x/ dx

(3.1.5)

0

for k ¤ l. Therefore, we obtain for k ¤ l Z 0D2 0

a

PkC .x 2 /PlC .x 2 /w.x/

Z

a2

dx D 0

1

1

PkC .t/PlC .t/t  2 w.t 2 / dt;

where the substitution t D x 2 has been used.

(3.1.6) t u

Theorem 3.1.5. All zeros of Pk , k 2 N, are real, simple, and located in the interior of the support interval Œa; b of d. R Proof. Since R Pk .x/ d.x/ D 0 for k  1, there must exist at least one point in the interior of Œa; b at which Pk changes sign. Let x1 ; x2 ; : : : ; xn , n  k, be all these points. If we had n < k, then due to orthogonality Z R

Pk .x/

n Y

.x  xj / d.x/ D 0;

(3.1.7)

j D1

which is impossible since the integrand does no longer change sign. Therefore, we obtain that n D k. t u

58

3 Orthogonal Polynomials

Theorem 3.1.6. The zeros of PkC1 alternate with those of Pk , i.e., .kC1/

.k/

.kC1/

xkC1 < xk < xk .kC1/

where xj

.k/

, xi

.k/

.k/

.kC1/

< xk1 <    < x1 < x1

;

(3.1.8)

are the zeros of PkC1 and Pk , respectively, in descending order.

Proof. See Remark 3.1.21. The proof uses the Christoffel–Darboux formula, i.e., Theorem 3.1.19. t u Theorem 3.1.7. In any open interval .c; d / in which d  0 there can be at most one zero of Pk . .k/

Proof. We perform a proof by contradiction. Suppose there are two zeros xi ¤ .k/ .k/ xj in .c; d /. Let all the other zeros (within .c; d / or not) be denoted by xn . Then, Z R

Y

Pk .x/ Z

D R

.x  xn.k/ / d.x/

n¤i;j

Y

.k/

.k/

.x  xn.k/ /2 .x  xj /.x  xi / d.x/ > 0;

(3.1.9)

n¤i;j

since the integrand is non-negative outside of .c; d /. This is a contradiction to the Q .k/ orthogonality of Pk to polynomials of lower degree such as .x  xn /. t u n¤i;j

Theorem 3.1.8. For any monic polynomial P 2 ˘nı we have Z

Z P .x/ d.x/  2

R

R

Pn2 .x/ d.x/;

(3.1.10)

where equality is achieved only for P D Pn . In other words, Pn minimizes the integral on the left-hand side of (3.1.10) over all P 2 ˘nı , i.e., Z

Z P 2 .x/ d.x/ D

minı

P 2˘n

R

R

Pn2 .x/ d.x/:

(3.1.11)

Proof. Due to Lemma 3.0.13 the polynomial P can be represented as P .x/ D Pn .x/ C

n1 X

ck Pk .x/:

(3.1.12)

kD0

Therefore, Z

Z P 2 .x/ d.x/ D

R

R

Pn2 .x/ d.x/ C

n1 X kD0

Z ck2

R

Pk2 .x/ d.x/:

(3.1.13)

3.1 Properties of Orthogonal Polynomials

59

This shows the desired inequality. Equality in (3.1.10) holds if and only if the coefficients c0 D c1 D : : : D cn1 D 0, i.e., for P D Pn . u t Remark 3.1.9. If we consider the function Z  n1 2 X ˚.a0 ; a1 ; : : : ; an1 / D xn C ak x k d.x/; R

(3.1.14)

kD0

we can compute the partial derivatives and set them equal to zero which yields Z P .x/x k d.x/ D 0; k D 0; 1; : : : ; n  1: (3.1.15) R

These are exactly the conditions of orthogonality that Pn has to satisfy. Furthermore, the Hessian of ˚ is 2Mn , where Mn is the matrix of Theorem 3.0.10 which is positive definite. This confirms that Pn gives us a minimum. Theorem 3.1.10. Let 1 < p < 1. Then, the extremal problem of determining Z minı

P 2˘n

jP .x/jp d.x/

(3.1.16)

R

possesses the unique solution Pn . Proof. The search for the desired minimum is equivalent to the problem of best approximation of x n by polynomials of degree n  1 in the Lp -norm. The problem of best approximation in normed spaces is uniquely solvable if the space is strictly convex, i.e., if from kxk D kyk D 1 and x ¤ y we can conclude that kx C yk < 2. A normed space X is strictly convex if and only if kx C yk D kxk C kyk yields x D ˛y or y D ˛x with an ˛  0. The Minkowski inequality guarantees this for the Lp -norm. For further details see, e.g., Davis (1963), Reed and Simon (1972), Rudin (1991), and Yoshida (1980) or other books on functional analysis. t u Orthogonal polynomials fulfill a three-term recurrence relation which can be used for: • Generating values of the polynomials and their derivatives, • Computation of the zeros as eigenvalues of a symmetric tridiagonal matrix via the recursion coefficients, • Normalization of the orthogonal polynomials, • Efficient evaluation of expansions in orthogonal polynomials. The reason for the existence of these three-term recurrences is the shift property of the scalar product, i.e., Z hxU; V i D

xU.x/V .x/ d.x/ D hU; xV i R

for all U; V 2 ˘:

(3.1.17)

60

3 Orthogonal Polynomials

Note that there are other scalar products that do not possess this property (even though they are positive definite). Theorem 3.1.11. Let Pk , k 2 N0 , be the monic orthogonal polynomials with respect to the measure d (see Definition 3.0.11). Then, P1 .x/ D 0; P0 .x/ D 1; PkC1 .x/ D .x  ˛k /Pk .x/  ˇk Pk1 .x/; k 2 N0 ; (3.1.18) where ˛k D

hxPk ; Pk i ; hPk ; Pk i

k 2 N0 ;

ˇk D

hPk ; Pk i ; hPk1 ; Pk1 i

k 2 N:

(3.1.19)

The index range for k is infinite if the scalar product is positive definite on ˘ . The range is finite (k  d  1) if the scalar product is positive definite on ˘d , but not on ˘n with n > d . Remark 3.1.12. Although ˇ0 in (3.1.18) can be arbitrary since it is multiplied with P1  0, we define it for later purposes as Z ˇ0 D hP0 ; P0 i D

d.x/:

(3.1.20)

R

Proof (of Theorem 3.1.11). Since PkC1  xPk is a polynomial of degree k, we can write PkC1 .x/  xPk .x/ D ˛k Pk .x/  ˇk Pk1 .x/ C

k2 X

k;j Pj .x/

(3.1.21)

j D0

for certain constants ˛k , ˇk , k;j , where P1 .x/ D 0 and empty sums are also zero. We take the scalar product with Pk on both sides which gives us (using orthogonality): hPkC1 ; Pk i  hxPk ; Pk i D ˛k hPk ; Pk i  ˇk hPk1 ; Pk i C

k2 X

k;j hPj ; Pk i:

j D0

(3.1.22) Therefore, we find that  hxPk ; Pk i D ˛k hPk ; Pk i; or ˛k D

hxPk ; Pk i : hPk ; Pk i

(3.1.23)

(3.1.24)

3.1 Properties of Orthogonal Polynomials

61

This proves the relation for ˛k . For ˇk we need to take the scalar product with Pk1 , where k  1: hPkC1 ; Pk1 i  hxPk ; Pk1 i D ˛k hPk ; Pk1 i  ˇk hPk1 ; Pk1 i C

k2 X

k;j hPj ; Pk1 i:

j D0

(3.1.25) Thus,  hxPk ; Pk1 i D ˇk hPk1 ; Pk1 i:

(3.1.26)

Since hxPk ; Pk1 i D hPk ; xPk1 i D hPk ; Pk C Rk1 i with Rk1 2 ˘k1 , we find that hxPk ; Pk1 i D hPk ; Pk i which provides us with ˇk D

hPk ; Pk i : hPk1 ; Pk1 i

(3.1.27)

As a last step we take the scalar product on both sides with Pi , i < k  1, and obtain hPkC1 ; Pi i  hxPk ; Pi i D ˛k hPk ; Pi i  ˇk hPk1 ; Pi i C

k2 X

k;j hPj ; Pi i:

j D0

(3.1.28) This immediately leads to  hxPk ; Pi i D k;i hPi ; Pi i:

(3.1.29)

Here we make use of the shift property of the scalar product, i.e., hxPk ; Pi i D hPk ; xPi i D 0 since xPi 2 ˘k1 . Therefore, k;i D 0 for i < k  1 which finally proves (3.1.18). t u Remark 3.1.13. If the index range in Theorem 3.1.11 is finite (k  d  1), the relations for ˛d and ˇd still make sense, ˇd > 0, but the polynomial Pd C1 that results from (3.1.18) has norm kPd C1 k D 0 (see also Theorem 3.0.15). Remark 3.1.14. Note that ˇk > 0 for all k 2 N0 and for n 2 N0 ; kPn k2 D ˇn  ˇn1  : : :  ˇ1  ˇ0 :

(3.1.30)

There is a converse result to Theorem 3.1.11 saying that any sequence of polynomials is orthogonal with respect to a positive measure with infinite support if the polynomials satisfy a three-term recurrence relation of the form (3.1.18) and all ˇk are positive. Theorem 3.1.15. Let the support interval Œa; b of d be finite. Then, a  ˛k  b;

k 2 N0 ;

0 < ˇk  maxfa2 ; b 2 g;

k 2 N;

(3.1.31)

where the index range is k  1 or k  d (d as in Theorem 3.1.11), respectively.

62

3 Orthogonal Polynomials

Proof. Since for x in the support of d we know that a  x  b, the definition of ˛k in Theorem 3.1.11 immediately yields the desired estimates. By definition 0 < ˇk and it remains to show the upper bound. We notice that kPk k2 D hPk ; Pk i D jhxPk1 ; Pk ij

(3.1.32)

and apply the Cauchy–Schwarz inequality to obtain kPk k2  maxfjaj; jbjg kPk1 k kPk k :

(3.1.33)

Therefore, kPk k2 kPk k  maxfa2 ; b 2 g:  maxfjaj; jbjg and ˇk D 2 kPk1 k kPk1 k

(3.1.34) t u

Definition 3.1.16. If the index range in Theorem 3.1.11 is infinite, the Jacobi matrix associated with the measure d is the infinite, symmetric, tridiagonal matrix

J1

p 3 2 ˛ ˇ1 p 0 0 p 7 6 ˇ1 ˛1 ˇ2 p 7 6 p 7 6 ˇ2 ˛2 ˇ3 D6 7: 6 :: :: :: 7 4 : : :5 0

(3.1.35)

Its leading principal minor matrix of size n n is denoted by 2

Jn D ŒJ1 Œ1Wn;1Wn

3 p ˇ1 p 0 ˛0 p 6 ˇ ˛ 7 ˇ2 p 1 p1 6 7 6 7 ˇ2 ˛2 ˇ3 6 7 6 7: D6 :: :: :: 7 : : : 6 7 p p 6 7 4 ˇn2 p˛n2 ˇn1 5 0 ˇn1 ˛n1

(3.1.36)

If the index range in Theorem 3.1.11 is finite (k  d  1), then Jn is well-defined for 0  n  d . Remark 3.1.17. Based on the monic orthogonal polynomials Pk , k 2 N0 , we can define orthonormal polynomials by PQk .x/ D Pk .x/= kPk k ;

such that hPQk ; PQj i D ık;j :

(3.1.37)

3.1 Properties of Orthogonal Polynomials

63

.n/ Theorem 3.1.18. The zeros xi of Pn (or the orthonormal version PQn ) are the eigenvalues of the Jacobi matrix Jn of order n. The corresponding eigenvectors are .n/ given by p.x Q i /, where

 T p.x/ Q D PQ0 .x/; PQ1 .x/; : : : ; PQn1 .x/ :

(3.1.38)

Proof. A simple calculation transforms the three-term recurrence relation (3.1.18) for the Pk into a relation for the PQk , i.e., p p ˇkC1 PQkC1 .x/ D .x  ˛k /PQk .x/  ˇk PQk1 .x/

(3.1.39)

for k 2 N0 , where we use the ˛k , ˇk of Theorem 3.1.11. Together with the usual convention PQ1 D 0 we obtain the following system of equations x p.x/ Q D Jn p.x/ Q C

p ˇn PQn .x/"n ;

(3.1.40)

where "n D Œ0; 0; : : : ; 1T is the n-th unit vector in Rn . .n/ If we put x D xi in (3.1.40), the second summand on the right-hand side drops p .n/ out. Since the first component of the vector p.x Q i / is 1= ˇ0 , the vector cannot be .n/ 0 and is indeed the eigenvector to the eigenvalue xi . t u Theorem 3.1.19 (Christoffel–Darboux Formula). Let PQk denote the orthonormal polynomials with respect to the measure d. Then, n X

PQk .x/PQk .t/ D

kD0

p PQnC1 .x/PQn .t/  PQn .x/PQnC1 .t/ ˇnC1 xt

(3.1.41)

and n X

PQk .x/

2

D

p 0

ˇnC1 PQnC1 .x/PQn .x/  PQn0 .x/PQnC1 .x/ :

(3.1.42)

kD0

Proof. We use the recurrence relation for the orthonormal polynomials (3.1.39), where PQ1 .x/ D 0 and PQ0 .x/ D p1ˇ . It is multiplied by PQk .t/ to obtain: 0

p p ˇkC1 PQkC1 .x/PQk .t/ D .x  ˛k /PQk .x/PQk .t/  ˇk PQk1 .x/PQk .t/:

(3.1.43)

Now, we interchange the roles of t and x and subtract that from the first relation, i.e.,

64

3 Orthogonal Polynomials

p

ˇkC1 PQkC1 .x/PQk .t/ 

p

ˇkC1 PQkC1 .t/PQk .x/

(3.1.44) p

D .x  ˛k /PQk .x/PQk .t/  ˇk PQk1 .x/PQk .t/   p  .t  ˛k /PQk .t/PQk .x/  ˇk PQk1 .t/PQk .x/ D .x  t/PQk .x/PQk .t/ 

p

ˇk PQk1 .x/PQk .t/  PQk1 .t/PQk .x/ :

Therefore, .x  t/PQk .x/PQk .t/ D

p

ˇkC1 PQkC1 .x/PQk .t/  PQkC1 .t/PQk .x/ p

 ˇk PQk1 .t/PQk .x/  PQk1 .x/PQk .t/ :

(3.1.45)

We sum up both sides from k D 0 to k D n and make use of the telescoping sum on the right-hand side (note that PQ1 D 0): n X

PQk .x/PQk .t/ D

kD0

p

ˇnC1

PQnC1 .x/PQn .t/  PQn .x/PQnC1 .t/ : xt

(3.1.46)

For the second part we take the limit t ! x on both sides. Thus, on the right-hand side we find PQnC1 .x/PQn .t /  PQn .x/PQnC1 .t / PQn .t /  PQn .x/ PQn .x/ D PQnC1 .x/ C PQnC1 .x/ xt xt xt  PQn .x/ D PQnC1 .x/

(3.1.47)

PQnC1 .t /  PQnC1 .x/ PQnC1 .x/  PQn .x/ xt xt PQn .t /  PQn .x/ PQnC1 .t /  PQnC1 .x/  PQn .x/ xt xt

Letting t tend to x we obtain the limit



0 0 PQnC1 .x/ PQn0 .x/  PQn .x/ PQnC1 .x/ D PQnC1 .x/PQn .x/  PQn0 .x/PQnC1 .x/; (3.1.48) t u

which is the desired result.

Corollary 3.1.20. Let Pk denote the monic orthogonal polynomials with respect to the measure d. Then, for x ¤ t, n  Y n X kD0

 n X ˇi Pk .x/Pk .t/ D ˇn  ˇn1  : : :  ˇkC1 Pk .x/Pk .t/

i DkC1

kD0

D

PnC1 .x/Pn .t/  Pn .x/PnC1 .t/ : xt

(3.1.49)

3.1 Properties of Orthogonal Polynomials

65

Proof. Put PQk D Pkp = kPk k in the first formula of Theorem 3.1.19 and use from Theorem 3.1.11 that ˇnC1 D kPnC1 k = kPn k: n X

PQk .x/PQk .t/ D

kD0

D

n X

1

kD0

kPk k2

Pk .x/Pk .t/

(3.1.50)

kPnC1 k PnC1 .x/Pn .t/  Pn .x/PnC1 .t/ : kPn k kPnC1 k kPn k .x  t/

This yields n X kPn k2 2

kD0

kPk k

Pk .x/Pk .t/ D

PnC1 .x/Pn .t/  Pn .x/PnC1 .t/ : xt

Together with kPn k2 D ˇn  ˇn1      ˇ1  ˇ0 D

n Y

ˇi

(3.1.51)

(3.1.52)

i D0

this provides us with n  Y n X kD0

i DkC1

 PnC1 .x/Pn .t/  Pn .x/PnC1 .t/ : ˇi Pk .x/Pk .t/ D xt

(3.1.53) t u

Remark 3.1.21. From the second part of Theorem 3.1.19 we obtain the inequality 0 PQnC1 .x/PQn .x/  PQn0 .x/PQnC1 .x/ > 0:

(3.1.54)

This can be used to prove Theorem 3.1.6 in the following way: Let  and  be consecutive zeros of PQn , such that PQn0 ./PQn0 ./ < 0 (which holds since all zeros are simple). Then, (3.1.54) tells us that  PQn0 ./PQnC1 ./ > 0

and

 PQn0 ./PQnC1 ./ > 0:

(3.1.55)

This implies that PQnC1 has opposite signs at  and . Therefore, there is at least one zero of PQnC1 between  and . In this way we find at least n  1 zeros of PQnC1 . .n/ .n/ For the largest zero of PQn , i.e., for 1 , it holds that PQn0 .1 / > 0 and by (3.1.55) .n/ .n/ we have PQnC1 .1 / < 0. The polynomial PQnC1 has another zero at the right of 1 Q since PnC1 .x/ > 0 for x sufficiently large. .n/ A similar argument holds for the smallest zero of PQn , i.e., for n . This proves Theorem 3.1.6.

66

3 Orthogonal Polynomials

3.2 Quadrature Rules and Orthogonal Polynomials In this section, let d be a measure with bounded or unbounded support, positive definite or not. An n-point quadrature rule for d is a formula of the type Z F .x/ d.x/ D R

n X

wj F .xj / C Rn .F /:

(3.2.1)

j D1

The mutually distinct points xj are called nodes, the coefficients wj are the weights of the quadrature rule. Rn .F / is the remainder or error term. Definition 3.2.1. The quadrature rule (3.2.1) possesses degree of exactness d if Rn .P / D 0 for all P 2 ˘d . It has precise degree of exactness d if it has degree of exactness d , but not d C 1. A quadrature rule (3.2.1) with degree of exactness d D n  1 is called interpolatory.

Interpolatory Quadrature Rules A quadrature rule is interpolatory if and only if it is obtained by integration of the Lagrange interpolation, i.e., by integrating F .x/ D

n X

F .xj /Lj .x/ C rn1 .F I x/;

(3.2.2)

j D1

where Lj .x/ D

n Y x  xi : x  xi i D1 j

(3.2.3)

i ¤j

Thus, we obtain Z wj D

Z R

Lj .x/ d.x/; j D 1; 2; : : : ; n; and Rn .F / D

R

rn1 .F I x/ d.x/: (3.2.4)

It is well-known that for P 2 ˘n1 the interpolation error rn1 .P I /  0, i.e., the remainder term Rn .P / D 0 and d D n  1. Given any n distinct nodes an interpolatory quadrature can always be constructed.

3.2 Quadrature Rules and Orthogonal Polynomials

67

Theorem 3.2.2. Let 0  k  n be an integer. The quadrature rule (3.2.1) has degree of exactness d D n  1 C k if and only if both of the following two conditions are fulfilled: (i) (3.2.1) is interpolatory, (ii) The node polynomial !n .x/ D

n Q

.x  xj / satisfies

j D1

Z R

!n .x/P .x/ d.x/ D 0 for all P 2 ˘k1 :

(3.2.5)

Proof. First, let the degree of exactness be d D n1Ck. Obviously, the quadrature rule is interpolatory. For P 2 ˘k1 we know that !n P 2 ˘nCk1 . Therefore, Z R

!n .x/P .x/ d.x/ D

n X

wj !n .xj /P .xj / D 0

(3.2.6)

j D1

by definition of !n . Let now the conditions .i / and .i i / be fulfilled. Let P 2 ˘nCk1 . We have to prove that the remainder term Rn .P / D 0. Polynomial long division by the node polynomial !n yields that P D Q!n C R, where Q 2 ˘k1 and R 2 ˘n1 . Then, Z

Z

Z

P .x/ d.x/ D R

R

Q.x/!n .x/ d.x/ C

R.x/ d.x/ D R

n X

wj R.xj /;

j D1

(3.2.7) since the first integral vanishes due to .i i / and the second integral is evaluated exactly by the quadrature rule due to .i /. However, R.xj / D P .xj /  Q.xj /!n .xj / D P .xj /: Therefore,

Z P .x/ d.x/ D R

i.e., Rn .P / D 0.

n X

wj P .xj /;

(3.2.8)

(3.2.9)

j D1

t u

Remark 3.2.3. If d is positive definite on ˘n , then k D n is optimal. k D n C 1 requires that !n is orthogonal to all elements of ˘n , i.e., also to itself, which is not possible.

Gauß Quadratures Definition 3.2.4. The quadrature rule (3.2.1) with k D n is called the Gauß quadrature rule with respect to the measure d. Its degree of exactness is d D 2n  1.

68

3 Orthogonal Polynomials

Remark 3.2.5. The second condition in Theorem 3.2.2 shows that for a Gauß quadrature (i.e., k D n) we have !n D Pn . Therefore, the nodes xjG are the zeros of the n-th orthogonal polynomial with respect to d. The weights wG j can be found by interpolation. Note that for the Gauß quadratures we assume that d is positive definite on ˘n . Theorem 3.2.6. All nodes xjG of the Gauß quadrature rule are mutually distinct and contained in the interior of the support interval Œa; b of d. All weights wG j are positive. Proof. Since the nodes xjG are the zeros of Pn , the first part follows directly from Theorem 3.1.5. Now, we consider the weights: Z 0< R

L2i .x/

d.x/ D

n X

2 G G wG j Li .xj / D wi

for i D 1; 2; : : : ; n, since L2i ./ 2 ˘2n2  ˘2n1 . Theorem 3.2.7. Let rule. Then,

wG j n X

(3.2.10)

j D1

be the weights and

xjG

t u

be the nodes of a Gauß quadrature

Z G wG j F .xj /

j D1

D R

P2n1 .F I x/ d.x/;

(3.2.11)

where P2n1 .F I / is the Hermite interpolation polynomial of degree 2n  1 which satisfies the equations 0 P2n1 .F I xjG / D F 0 .xjG /

P2n1 .F I xjG / D F .xjG /;

for j D 1; 2; : : : ; n: (3.2.12)

Proof. The Hermite interpolation polynomial is given by P2n1 .F I x/ D

n  X

 Hj .x/F .xjG / C Kj .x/F 0 .xjG / ;

(3.2.13)

j D1

where   Hj .x/ D 1  2.x  xjG /L0j .xjG / L2j .x/;

(3.2.14)

Kj .x/ D.x  xjG /L2j .x/;

(3.2.15)

Lj .x/ D

n Y i D1 i ¤j

x  xiG : xjG  xiG

(3.2.16)

3.2 Quadrature Rules and Orthogonal Polynomials

69

Note that Hj .xiG / D ıj;i , Hj0 .xiG / D 0, as well as Kj .xiG / D 0, Kj0 .xiG / D ıj;i . Obviously, Hj ; Kj 2 ˘2n1 . Hence, Z R

P2n1 .F I x/ d.x/ D

n Z X j D1

R

Z Hj .x/ d.x/F .xjG / C

R

 Kj .x/ d.x/F 0 .xjG / :

(3.2.17) Moreover, Z n   X Hj .x/ d.x/ D wi 1  2.xiG  xjG /L0j .xjG / L2j .xiG / R

i D1

D

n X

  wi 1  2.xiG  xjG /L0j .xjG / ıj;i D wj ;

(3.2.18)

i D1

and Z R

Kj .x/ d.x/ D

n X

wi .xiG xjG /L2j .xiG / D

i D1

n X

wi .xiG xjG /ıj;i D 0: (3.2.19)

i D1

t u

This finally gives us the desired result, i.e., (3.2.11).

Corollary 3.2.8. If F 2 C .Œa; b/ and Œa; b is the support interval of d, then the remainder term of the Gauß quadrature rule can be expressed as Z F .2n/ ./ Rn .F / D .Pn .x//2 d.x/ (3.2.20) .2n/Š R .2n/

with  2 .a; b/. Proof. We know from the theory of interpolation (see, e.g., Davis 1963) that F .x/ D P2n1 .F I x/ C r2n1 .F I x/ D P2n1 .F I x/ C

n F .2n/ ..x// Y .x  xj /2 .2n/Š j D1

(3.2.21) with .x/ 2 .a; b/. This yields for the remainder of the numerical integration that Z Rn .F / D

Z R

r2n1 .F I x/ d.x/ D

R

n F .2n/ ..x// Y .x  xj /2 d.x/: .2n/Š j D1

The mean value theorem of integration completes the proof.

(3.2.22) t u

Theorem 3.2.9. The first n orthogonal polynomials Pk , k D 0; : : : ; n  1, are discretely orthogonal in the sense that n X j D1

2 G G wG j Pk .xj /Pl .xj / D ık;l kPk k ;

k; l D 0; 1; : : : ; n  1;

(3.2.23)

70

3 Orthogonal Polynomials

where xjG , wG j are the nodes and weights of the n-point Gauß quadrature rule. Proof. This follows directly since the degree of exactness of the n-point Gauß quadrature is 2n  1 and the product of Pk and Pl with k; l 2 f0; 1; : : : ; n  1g is a polynomial of degree 2n  2. t u Remark 3.2.10. If a > 1, b  1, it can be desirable to have x0 D a. To achieve this, we replace n by n C 1 in (3.2.1) and write !nC1 .x/ D .x  a/!n .x/. The optimal formula (see Theorem 3.2.2) requires !n to satisfy Z R

!n .x/P .x/.x  a/ d.x/ D 0 for all P 2 ˘n1 :

(3.2.24)

Therefore, !n .x/ D Pn .xI da / with da .x/ D .x  a/ d.x/. The remaining n nodes have to be zeros of Pn .I da /. The resulting rule is called the Gauß–Radau rule. If also b < 1 and a as well as b are both desired as nodes, we find the .n C 2/-point Gauß–Lobatto rule similarly with the measure da;b .x/ D .x  a/.b  x/ d.x/:

(3.2.25)

These rules have degrees of exactness equal to 2n and 2n C 1, respectively. For further details, the reader is referred to, e.g., Davis and Rabinowitz (1967), Gautschi (2004), H¨ammerlin and Hoffmann (1992).

3.3 The Jacobi Polynomials In this section we consider the Jacobi polynomials which are classical orthogonal polynomials depending on two parameters ˛; ˇ. For special choices of these parameters we obtain other important classes such as the ultraspherical polynomials which are treated in more detail in Sect. 3.4. The exercises in Sect. 4.8 come back to the general Jacobi polynomials and show their importance for geomathematical applications. For further details on Jacobi polynomials we refer to e.g., Abramowitz and Stegun (1972), Lebedev (1973), Magnus et al. (1966), Sneddon (1980), and Szeg¨o (1967). Definition 3.3.1. Let d.x/ D w.x/ dx with w.x/ D .1  x/˛ .1 C x/ˇ , ˛; ˇ > 1, and the support interval Œ1; 1. The corresponding orthogonal polynomials are the .˛;ˇ/ .˛;ˇ/ .˛;ˇ/ Jacobi polynomials Pn 2 ˘n which fulfill hPn ; Pm id D 0 if n ¤ m and the normalization condition ! nC˛ .n C ˛ C 1/ .˛;ˇ/ D Pn .1/ D : (3.3.1) n .n C 1/ .˛ C 1/ Note that we generalize the notation of the binomial coefficients naturally via the Gamma function.

3.3 The Jacobi Polynomials

71

Remark 3.3.2. Similar to Theorem 3.1.3 we find the identity Pn.˛;ˇ/ .x/ D .1/n Pn.ˇ;˛/ .x/ and so we obtain Pn.˛;ˇ/ .1/

D .1/

n

! nCˇ : n

(3.3.2)

(3.3.3)

Theorem 3.3.3. For ˛ > 1, we have .˛;˛/

P2n

.x/ D

.2n C ˛ C 1/ .n C 1/ .˛;1=2/ P .2x 2  1/ .n C ˛ C 1/ .2n C 1/ n

D .1/n .˛;˛/

P2nC1 .x/ D

.2n C ˛ C 1/ .n C 1/ .1=2;˛/ P .1  2x 2 /; .n C ˛ C 1/ .2n C 1/ n

.2n C ˛ C 2/ .n C 1/ x Pn.˛;1=2/ .2x 2  1/ .n C ˛ C 1/ .2n C 2/

D .1/n

(3.3.4)

(3.3.5)

.2n C ˛ C 2/ .n C 1/ x Pn.1=2;˛/ .1  2x 2 /: .n C ˛ C 1/ .2n C 2/

Proof. At first, we show (3.3.4). For that we use the notation d1 .x/ D .1  x/˛ .1 C x/˛ dx D .1  x 2 /˛ dx;

(3.3.6)

d2 .x/ D .1  x/˛ .1 C x/1=2 dx:

(3.3.7)

It suffices to prove that Z I D Z

R

Pn.˛;1=2/ .2x 2  1/P .x/ d1 .x/

1

D 1

Pn.˛;1=2/ .2x 2  1/P .x/.1  x 2 /˛ dx D 0;

(3.3.8)

where P 2 ˘2n1 . If P is an odd polynomial, this is fulfilled. Therefore, let P be even, i.e., P .x/ D R.x 2 / with R 2 ˘n1 . Then, Z

1

I D 1

Z

D2 0

Z

1

D 0

Pn.˛;1=2/ .2x 2  1/R.x 2 /.1  x 2 /˛ dx 1

Pn.˛;1=2/ .2x 2  1/R.x 2 /.1  x 2 /˛ dx Pn.˛;1=2/ .2u  1/R.u/.1  u/˛ u1=2 du;

(3.3.9)

72

3 Orthogonal Polynomials

where we have substituted x 2 by u. Applying the substitution rule another time with u D 1Cx gives us: 2 I D 2˛1=2 D2

˛1=2

Z Z

1 1

R

˛ 1=2 Pn.˛;1=2/ .x/R. xC1 dx 2 /.1  x/ .1 C x/

Pn.˛;1=2/ .x/R. xC1 / d2 .x/ D 0: 2

(3.3.10) t u

A similar argument can be used to prove (3.3.5). Remark 3.3.4 (Special Cases).

(i) For ˛ D ˇ, we are dealing with the special case of ultraspherical polynomials (or Gegenbauer polynomials) which due to Theorem 3.3.3 are even or odd polynomials (depending on n being even or odd). Cn./ .x/ D D

.˛ C 1/ .n C 2˛ C 1/ .˛;˛/ P .x/ .2˛ C 1/ .n C ˛ C 1/ n . C 12 / .n C 2/ .2/ .n C  C 12 /

Pn.1=2;1=2/ .x/;

(3.3.11)

where ˛ D ˇ D   12 ,  >  12 since ˛ > 1. .0/ If ˛ D  12 (or  D 0), the Gegenbauer polynomial Cn vanishes identically for n  1 (see Lemma 3.4.1). Another consequence of Theorem 3.3.3 is that Jacobi polynomials with ˛ D ˙ 12 or ˇ D ˙ 12 can be expressed by ultraspherical polynomials. We will treat these classical orthogonal polynomials in detail in Sect. 3.4. (ii) For ˛ D ˇ D  12 , we obtain the Chebyshev polynomials of first kind Tn , i.e., Qn

i D1 .2i  2n nŠ

Pn.1=2;1=2/ .x/ D

1/

Qn

i D1 .2i  2n nŠ

Tn .x/ D

1/

cos.n#/; (3.3.12)

where x D cos.#/. (iii) For ˛ D ˇ D 12 , we obtain the Chebyshev polynomials of second kind Un , i.e., Qn Pn.1=2;1=2/ .x/ D

i D0 .2i C 1/ Un .x/ 2n .n C 1/Š

Qn D

i D0 .2i C 1/ sin..n C 1/#/ ; 2n .n C 1/Š sin.#/ (3.3.13)

where x D cos.#/. (iv) The mixed variants ˛ D  12 , ˇ D 12 and ˛ D 12 , ˇ D  12 yield the Chebyshev polynomials of third kind Vn and of fourth kind Wn , respectively, i.e.,

3.3 The Jacobi Polynomials

73

Qn Pn.1=2;1=2/ .x/

i D1 .2i  2n nŠ

D

1/

Qn Vn .x/ D

Qn Pn.1=2;1=2/ .x/ D

i D1 .2i  2n nŠ

i D1 .2i  1/ Wn .x/ D 2n .n C 1/Š

1/ cos

Qn

.2nC1/#

cos

#2

;

2

(3.3.14) .2nC1/#

i D1 .2i  1/ sin 2 ; 2n .n C 1/Š sin #2 (3.3.15)

where x D cos.#/. For details on Chebyshev polynomials see, e.g., Rivlin (1990). (v) For ˛ D ˇ D 0, we find the Legendre polynomials Pn , i.e., Pn.0;0/ .x/ D Cn.1=2/ .x/ D Pn .x/:

(3.3.16)

Note that in this case the weight function is w.x/ D 1, x 2 Œ1; 1. .˛;ˇ/

Theorem 3.3.5. The Jacobi polynomials y D Pn .x/ satisfy the following linear, homogeneous differential equation of the second order: .1  x 2 /y 00 C .ˇ  ˛  .˛ C ˇ C 2/x/y 0 C n.n C ˛ C ˇ C 1/y D 0; (3.3.17) or equivalently

d .1  x/˛C1 .1 C x/ˇC1 y 0 C n.n C ˛ C ˇ C 1/.1  x/˛ .1 C x/ˇ y D 0: dx (3.3.18) Proof. We note that since y 2 ˘n we have that

d .1  x/˛C1 .1 C x/ˇC1 y 0 D .1  x/˛ .1 C x/ˇ z; dx

(3.3.19)

where z 2 ˘n . To show that z is a constant multiple of y, i.e., z D Cy, we have to prove the orthogonality to any P 2 ˘n1 , i.e., Z

1 1

d .1  x/˛C1 .1 C x/ˇC1 y 0 P .x/ dx D 0: dx

(3.3.20)

We use integration by parts on the left-hand side which yields (since ˛ C 1 > 0 and ˇ C 1 > 0): Z

d .1  x/˛C1 .1 C x/ˇC1 y 0 P .x/ dx dx 1 Z 1 D .1  x/˛C1 .1 C x/ˇC1 y 0 P 0 .x/ dx 1

1

74

3 Orthogonal Polynomials

Z

1

D

y 1

d .1  x/˛C1 .1 C x/ˇC1 P 0 .x/ dx; dx „ ƒ‚ …

(3.3.21)

CQ .1x/˛ .1Cx/ˇ R.x/

where we performed another integration by parts and R 2 ˘n1 . Therefore, the integral vanishes and we find that z D Cy. The constant factor C can be calculated by comparing the highest terms, i.e., y D kn x n C : : : ;

y 0 D nkn x n1 C : : : ;

y 00 D n.n  1/kn x n2 C : : : ; (3.3.22)

and 0D

d .1  x/˛C1 .1 C x/ˇC1 y 0 C C.1  x/˛ .1 C x/ˇ y dx

(3.3.23)

D .˛ C 1/.1  x/˛ .1 C x/ˇC1 y 0 C .ˇ C 1/.1  x/˛C1 .1 C x/ˇ y 0 C.1  x/˛C1 .1 C x/ˇC1 y 00 C C.1  x/˛ .1 C x/ˇ y D .˛ C 1/.1  x/˛ .1 C x/ˇ .1 C x/y 0  .ˇ C 1/.1  x/˛ .1 C x/ˇ .x  1/y 0 .1  x/˛ .1 C x/ˇ .x 2  1/y 00 C C.1  x/˛ .1 C x/ˇ y: Thus, C D .n.˛ C 1/  n.ˇ C 1/  n.n  1// D n.n C ˛ C ˇ C 1/;

(3.3.24) t u

which proves the differential equation (3.3.18). Theorem 3.3.6. Let ˛; ˇ > 1. The differential equation .1  x 2 /y 00 C .ˇ  ˛  .˛ C ˇ C 2/x/y 0 C y D 0;

(3.3.25)

where  2 R is a parameter, has a polynomial solution not identically zero if and .˛;ˇ/ only if  D n.n C ˛ C ˇ C 1/, n 2 N0 . This solution is A  Pn .x/, A ¤ 0, and .˛;ˇ/ no solution which is linearly independent of Pn can be a polynomial. P1 k Proof. Substitute y D kD0 ak .x  1/ in the differential equation. This gives us 0 D  .x C 1/

1 X

k.k  1/ak .x  1/k1  .2.˛ C 1/ C .˛ C ˇ C 2/.x  1//

kD2



1 X kD1

kak .x  1/k1 C 

1 X kD0

ak .x  1/k

(3.3.26)

3.3 The Jacobi Polynomials

D  .x  1 C 2/

1 X

75

k.k  1/ak .x  1/k1  2.˛ C 1/

kD2

 .˛ C ˇ C 2/

1 X

D

kak .x  1/k C 



1 X

ak .x  1/k

kD0

k.k  1/ak .x  1/k  2

kD2 1 X

kak .x  1/k1

kD1

kD1 1 X

1 X

1 X

.k C 1/kakC1 .x  1/k  2.˛ C 1/

kD1

.k C 1/akC1 .x  1/k  .˛ C ˇ C 2/

kD0

1 X

kak .x  1/k C 

kD1

1 X

ak .x  1/k :

kD0

Thus, the coefficients have to fulfill the relation .  k.k C ˛ C ˇ C 1// ak  2.k C 1/.k C ˛ C 1/akC1 D 0

(3.3.27)

for k 2 N0 . If we assume that y is a polynomial, we can suppose that an denotes the last nonzero coefficient, i.e., anC1 D 0. Therefore, the factor in front of an in the recurrence relation above has to vanish, i.e.,  D n.n C ˛ C ˇ C 1/:

(3.3.28)

On the other hand, if (3.3.28) holds for  , we find that ai D 0 for i  n C 1 since the factor of akC1 in (3.3.28) differs from zero. Let  D n.n C ˛ C ˇ C 1/ and let z be a second solution of the differential equation (3.3.18), i.e.,

d .1  x/˛C1 .1 C x/ˇC1 y 0 C n.n C ˛ C ˇ C 1/.1  x/˛ .1 C x/ˇ y D 0; dx (3.3.29)

d .1  x/˛C1 .1 C x/ˇC1 z0 C n.n C ˛ C ˇ C 1/.1  x/˛ .1 C x/ˇ z D 0: dx (3.3.30) Multiply the first equation by z and the second by y and subtract them: d dx d D dx

0D





d .1  x/˛C1 .1 C x/ˇC1 y 0 z  .1  x/˛C1 .1 C x/ˇC1 z0 y dx

(3.3.31) .1  x/˛C1 .1 C x/ˇC1 .y 0 z  z0 y/ :

Thus, for all x 2 Œ1; 1, .1  x/˛C1 .1 C x/ˇC1 .y 0 z  yz0 / D c D const:

(3.3.32)

76

3 Orthogonal Polynomials

If we let x ! ˙1, we see that y and z cannot both be polynomials unless c D 0. Therefore, y 0 z D z0 y for all x 2 .1; 1/, i.e., the two solutions y and z are linearly .˛;ˇ/ dependent. Therefore, z.x/ D cP Q n .x/. t u Remark 3.3.7. The Jacobi polynomials can also be defined as the polynomial solutions of the corresponding differential equation that additionally take the value ! nC˛ .˛;ˇ/ Pn .1/ D : (3.3.33) n Definition 3.3.8. The hypergeometric function F (sometimes denoted by 2 F1 ) is defined by 1 1 X .c/ X .a C k/ .b C k/ x k .a/k .b/k x k D F .a; bI cI x/ D .c/k kŠ .a/ .b/ .c C k/ kŠ kD0 kD0 (3.3.34)

or its analytic continuation with a; b 2 R, c 2 R n fN0 g, x 2 .1; 1/. If a D n or b D n, n 2 N0 , the hypergeometric function reduces to a polynomial in x whose degree is n. Theorem 3.3.9. For the Jacobi polynomials the following representations hold: !

nC˛ .˛;ˇ/ Pn .x/ D (3.3.35) F n; n C ˛ C ˇ C 1I ˛ C 1I 1x 2 n !  n X .n C ˛ C 1/ x1 k .n C ˛ C ˇ C 1 C k/ n D : nŠ .n C ˛ C ˇ C 1/ .˛ C 1 C k/ k 2 kD0

(3.3.36) This can be reformulated as ! n 1 X n .n C ˛ C 1/ .n C ˇ C 1/ .˛;ˇ/ .x  1/nk .x C 1/k Pn .x/ D n k .n C ˛ C 1  k/ .ˇ C 1 C k/ 2 nŠ kD0 ! ! n 1 X nC˛ nCˇ .x  1/nk .x C 1/k : D n (3.3.37) k nk 2 kD0

Proof. This can be shown via the properties of the hypergeometric function, in particular its differential equation for which we refer to Abramowitz and Stegun (1972), Magnus et al. (1966), and Szeg¨o (1967). Another way uses Rodrigues’ representation of the Jacobi polynomials which can be found in Theorem 3.3.12. t u .˛;ˇ/

Corollary 3.3.10. The leading coefficient of the Jacobi polynomial Pn n is given by

of degree

3.3 The Jacobi Polynomials

77

kn.˛;ˇ/

D2

n

! 2n C ˛ C ˇ : n

(3.3.38)

Proof. Consider the summand for k D n in representation (3.3.36) of Theorem 3.3.9: kn.˛;ˇ/

! .n C ˛ C ˇ C 1 C n/ n 1 .n C ˛ C 1/ D nŠ .n C ˛ C ˇ C 1/ .˛ C 1 C n/ n 2n ! n 2n C ˛ C ˇ D2 : (3.3.39) n t u

Corollary 3.3.11. For the derivative of the Jacobi polynomials we have n C ˛ C ˇ C 1 .˛C1;ˇC1/ d .˛;ˇ/ Pn .x/ D Pn1 .x/: dx 2

(3.3.40)

Proof. This follows immediately if both sides are expanded according to Theorem 3.3.9. u t Theorem 3.3.12 (Rodrigues’ Formula). Let ˛; ˇ > 1. Then, .1  x/ .1 C x/ ˛

ˇ

Pn.˛;ˇ/ .x/

.1/n D n 2 nŠ



d dx

n



.1  x/nC˛ .1 C x/nCˇ : (3.3.41)

Proof. We use Leibniz’ rule for the differentiation of products to find 

d dx

n 

!   nk n  X n d k d .1  x/nC˛ .1 C x/nCˇ D .1  x/nC˛ .1 C x/nCˇ dx dx k kD0

! n X n D .1  x/˛ Rnk .x/.1 C x/ˇ Sk .x/ k kD0

D .1  x/˛ .1 C x/ˇ %.x/;

(3.3.42)

where Rj ; Sj 2 ˘j , deg.Rj / D deg.Sj / D j , j D 0; : : : ; n, and % 2 ˘n , deg.%/ D n. In detail (we will need that later): (3.3.43) .1  x/˛ .1 C x/ˇ %.x/ ! n X n .n C ˛ C 1/ .n C ˇ C 1/ .1/k .1  x/nC˛k .1 C x/ˇCk : D k .n C ˛  k C 1/ .ˇ C k C 1/ kD0

78

3 Orthogonal Polynomials

Therefore, ! n X n .n C ˛ C 1/ .n C ˇ C 1/ .1/k .1  x/nk .1 C x/k : %.x/ D k .n C ˛  k C 1/ .ˇ C k C 1/ kD0 (3.3.44) .˛;ˇ/

We now have to show that % D C  Pn Z

1

with a constant C . It suffices to show that

.1  x/˛ .1  x/ˇ %.x/R.x/ dx D 0

(3.3.45)

1

for all R 2 ˘n1 . We use (3.3.42) and apply integration by parts n times: Z

Z

 d n   .1  x/nC˛ .1 C x/nCˇ R.x/ dx 1 1 dx   ˇ1  ˇ d n1  D .1  x/nC˛ .1 C x/nCˇ R.x/ ˇˇ dx 1 „ ƒ‚ … 1

.1  x/˛ .1  x/ˇ %.x/R.x/ dx D

Z

1

D0

 d n1   .1  x/nC˛ .1 C x/nCˇ R0 .x/ dx  1 dx Z 1  D : : : D .1/n .1  x/nC˛ .1 C x/nCˇ R.n/ .x/ dx D 0; 1

1

(3.3.46)

since deg.R/  n1, i.e., R.n/  0. Now, we just have to determine the constant C . Consider x D 1 in %, i.e., only the summand k D n remains, i.e., ! ! n .n C ˛ C 1/ .n C ˇ C 1/ nC˛ n n n n .1/ 2 D .1/ 2 nŠ %.1/ D : .˛ C 1/ .n C ˇ C 1/ n n (3.3.47) .˛;ˇ/

Thus, %.1/ D .1/n 2n nŠ Pn

.1/, i.e., C D .1/n 2n nŠ.

t u

Remark 3.3.13. The Theorem 3.3.12 also gives us the explicit representation (3.3.37) of the second part of Theorem 3.3.9. Theorem 3.3.14. Let ˛; ˇ > 1. Then, for n 2 N, 2  h.˛;ˇ/ D Pn.˛;ˇ/  D n D and

Z

1 1

.˛;ˇ/ 2 Pn .x/ .1  x/˛ .1 C x/ˇ dx

2 .n C ˛ C 1/ .n C ˇ C 1/ 2n C ˛ C ˇ C 1 .n C 1/ .n C ˛ C ˇ C 1/ ˛CˇC1

(3.3.48)

3.3 The Jacobi Polynomials

.˛;ˇ/

h0

79

Z    .˛;ˇ/ 2 D P0  D

1

1

D 2˛CˇC1

 2 .˛;ˇ/ P0 .x/ .1  x/˛ .1 C x/ˇ dx

.˛ C 1/ .ˇ C 1/ : .˛ C ˇ C 2/

(3.3.49)

Proof. Let n 2 N. We have, from the proof of Theorem 3.3.12, Z

1

1

D



.˛;ˇ/

Pn

.x/

2

Z

1

1

.˛;ˇ/

Pn

.x/x n .1  x/˛ .1 C x/ˇ dx

 d n   .1  x/nC˛ .1 C x/nCˇ x n dx 1 dx !Z 1 2n C ˛ C ˇ .1  x/nC˛ .1 C x/nCˇ dx; n 1

.1/n .˛;ˇ/ kn 2n nŠ

D 22n

Z

.˛;ˇ/

.1  x/˛ .1 C x/ˇ dx D kn 1

(3.3.50)

where we used integration by parts n times and Corollary 3.3.10. Z

1

1

.˛;ˇ/ 2 Pn .x/ .1  x/˛ .1 C x/ˇ dx

D2

D2 D

˛Cˇ

2n C ˛ C ˇ n

˛CˇC1

!Z

2n C ˛ C ˇ n



1 1

!Z

1x 2

nC˛ 

(3.3.51) 1Cx 2

nCˇ dx

1

y nCˇ .1  y/nC˛ dy 0

2˛CˇC1 .n C ˛ C 1/ .n C ˇ C 1/ : 2n C ˛ C ˇ C 1 .n C 1/ .n C ˛ C ˇ C 1/

The case n D 0 follows similarly.

t u

Theorem 3.3.15. The Jacobi polynomials fulfill the following three-term recurrence relation: .˛;ˇ/

(3.3.52) 2.n C 1/.n C ˛ C ˇ C 1/.2n C ˛ C ˇ/PnC1 .x/

D .2n C ˛ C ˇ C 1/ .2n C ˛ C ˇ C 2/.2n C ˛ C ˇ/x C ˛ 2  ˇ 2 Pn.˛;ˇ/ .x/ .˛;ˇ/

 2.n C ˛/.n C ˇ/.2n C ˛ C ˇ C 2/Pn1 .x/ .˛;ˇ/

for n 2 N with P0

.˛;ˇ/

.x/ D 1 and P1

.x/ D 12 .˛ C ˇ C 2/x C 12 .˛  ˇ/.

80

3 Orthogonal Polynomials

Proof. Division by the leading coefficient (see Corollary 3.3.10) gives us POn.˛;ˇ/ D

1 .˛;ˇ/ kn

Pn.˛;ˇ/ ;

n 2 N0 ;

(3.3.53)

which are monic orthogonal polynomials. Therefore, they fulfill (3.1.18) of Theorem 3.1.11 with

ˇn.˛;ˇ/

  2   O .˛;ˇ/ 2 .˛;ˇ/ .˛;ˇ/ kn1 hn Pn  D   D 2  O .˛;ˇ/ 2 .˛;ˇ/ .˛;ˇ/ kn hn1 Pn1  4n.n C ˛/.n C ˇ/.n C ˛ C ˇ/ ; n 2 N; (3.3.54) .2n C ˛ C ˇ/2 .2n C ˛ C ˇ C 1/.2n C ˛ C ˇ  1/ Z 1 2˛CˇC1 .˛ C 1/ .ˇ C 1/ ; (3.3.55) D .1  x/˛ .1 C x/ˇ dx D .˛ C ˇ C 2/ 1 D

.˛;ˇ/

ˇ0

.˛;ˇ/

where we use the values of hn from Theorem 3.3.14. A lengthy, but uneventful calculation yields the missing values of the recurrence relation, i.e., for n 2 N0 , .˛;ˇ/

˛n.˛;ˇ/ D

hxPn

.˛;ˇ/

; Pn

.˛;ˇ/

hn

i

D

ˇ2  ˛2 : .2n C ˛ C ˇ/.2n C ˛ C ˇ C 2/

(3.3.56)

.˛;ˇ/

Note that in the case of ˇ1 , i.e., n D 1, the last factors in the numerator and in the denominator cancel each other out. The same happens with the factor ˛ C ˇ in .˛;ˇ/ .˛;ˇ/ ˛0 . By multiplying (3.1.18) with knC1 we obtain .˛;ˇ/ PnC1 .x/



D x

˛n.˛;ˇ/

.˛;ˇ/

knC1

P .˛;ˇ/ .x/ .˛;ˇ/ n kn



.˛;ˇ/ .˛;ˇ/ .˛;ˇ/ knC1 ˇn P .x/; .˛;ˇ/ n1 kn1

which can easily be transformed into the desired expression.

(3.3.57) t u

3.4 Ultraspherical Polynomials In Remark 3.3.4 we have already presented the ultraspherical (or Gegenbauer) ./ polynomials Cn and have established their connection to the general Jacobi polynomials, i.e.,

3.4 Ultraspherical Polynomials

Cn./ .x/ D D

81

.˛ C 1/ .n C 2˛ C 1/ .˛;˛/ P .x/ .2˛ C 1/ .n C ˛ C 1/ n . C 12 / .n C 2/ .2/ .n C  C 12 /

1 1 . ; / 2 2

Pn

.x/;

(3.4.1)

where ˛ D ˇ D   12 ,  >  12 since ˛ > 1. This relation gives us the following properties for  ¤ 0:

./ (i) The value at 1 is Cn .1/ D nC21 , the symmetry relation is given by n Cn./ .x/ D .1/n Cn./ .x/;

(3.4.2)

the polynomials are either even or odd (depending on n) due to Theorem 3.3.3. ./ (ii) The differential equation (3.3.17) whose solution is y D Cn becomes .1  x 2 /y 00  .2 C 1/xy 0 C n.n C 2/y D 0:

(3.4.3)

(iii) We know the explicit representation from Theorem 3.3.9, namely !

n C 2  1 Cn./ .x/ D F n; n C 2I  C 12 I 1x 2 n !   n . C 12 / X n .n C 2 C k/ x  1 k D : k . C 12 C k/ .2/nŠ 2

(3.4.4)

(3.4.5)

kD0

(iv) Due to Corollary 3.3.10 the leading coefficient is kn./ D lim x n Cn./ .x/ D 2n x!1

! nC1 : n

(3.4.6)

(v) The derivative is again an ultraspherical polynomial, i.e., by Corollary 3.3.11: d ./ .C1/ C .x/ D 2Cn1 .x/: dx n

(3.4.7)

(vi) Rodrigues’ representation (see Theorem 3.3.12) for the Gegenbauer polynomials reads as follows: ./

Cn .x/ D D

.1/n . C 12 / .n C 2/ 2n nŠ .2/ .n C  C 12 /

1

.1  x 2 / 2 

1 .2/n .n C / .n C 2/ .1  x 2 / 2  nŠ ./ .2n C 2/

 

d dx d dx

n  n 

1



1



.1  x 2 /nC 2

.1  x 2 /nC 2

:

(3.4.8)

82

3 Orthogonal Polynomials

(vii) Their L2 -norm is given by Theorem 3.3.14, i.e., 12  ./ 2 .n C 2/ C  D 2 h./ D : n n nŠ.n C /. .//2

(3.4.9)

(viii) The corresponding three-term recurrence (see Theorem 3.3.15) is given by ./

./

.n C 1/CnC1.x/ D 2.n C /xCn./ .x/  .n C 2  1/Cn1 .x/; ./

(3.4.10)

./

for n 2 N, with C0 .x/ D 1 and C1 .x/ D 2x. .0/

Lemma 3.4.1. If n  1, Cn  0, but ./

2 Cn .x/ D Tn .x/ !0  n lim

(3.4.11)

with the Chebyshev polynomial of first kind Tn . Proof. We consider ./ Cn .x/ . C 12 / .n C 2/ D P .1=2;1=2/ .x/   .2/ .n C  C 12 / n

D

(3.4.12)

2 . C 12 / .n C 2/ P .1=2;1=2/ .x/: .2 C 1/ .n C  C 12 / n

Now, we take the limit  ! 0 and obtain: ./ Cn .x/ 2 . 12 / .n/ D P .1=2;1=2/ .x/ !0  .1/ .n C 12 / n

lim

D

2 nŠ . 12 / P .1=2;1=2/ .x/ n .n C 12 / n

nŠ 2n 2 2 P .1=2;1=2/ .x/ D Tn .x/; D Qn n i D1 .2i  1/ n n where we have used (3.3.12) in the final step.

(3.4.13) t u

./

Note that C0 D 1, even as  ! 0. Combining Theorems 3.3.3 and 3.3.9 for the ultraspherical polynomials we obtain the following representations. Lemma 3.4.2. For n 2 N0 , we have the following representations in terms of the hypergeometric function: ./ C2n .x/

!

2n C 2  1 D F n; n C I  C 12 I 1  x 2 ; 2n

(3.4.14)

3.4 Ultraspherical Polynomials

./ C2nC1 .x/

83

!

2n C 2 D xF n; n C  C 1I  C 12 I 1  x 2 : 2n C 1

(3.4.15)

Proof. Consider the even case (3.4.14) first. We start with the relation between Gegenbauer and Jacobi polynomials and apply Theorem 3.3.3. ./

C2n .x/ D D

. C 12 / .2n C 2/ .2/ .2n C  C 12 /

.1=2;1=2/

P2n

(3.4.16)

.x/

. C 12 / .2n C 2/ .2n C  C 12 / .n C 1/ .2/ .2n C  C 12 / .n C  C 12 / .2n C 1/

Pn.1=2;1=2/.2x 2  1/

Now, we use the representation (3.3.35) of Theorem 3.3.9: ./ C2n .x/

!   . C 12 / .2n C 2/ .n C 1/ n C   12 2 1 D I 1  x F n; n C I  C 2 n .2/ .n C  C 12 / .2n C 1/ !   2n C 2  1 F n; n C I  C 12 I 1  x 2 : D (3.4.17) 2n

t u

The odd case (3.4.15) can be proved analogously.

Lemma 3.4.3. For n 2 N0 , the ultraspherical polynomials possess the following representations in terms of the hypergeometric function F : ! nC1 D .1/ F .n; n C I 12 I x 2 /; n ! n C  ./ C2nC1 .x/ D .1/n 2 x F .n; n C  C 1I 32 I x 2 /: n ./ C2n .x/

n

(3.4.18)

(3.4.19)

Proof. Consider the even case. This time we apply the second variant of Theorem 3.3.3. ./

C2n .x/ D

. C 12 / .2n C 2/ .2/ .2n C  C 12 /

.1=2;1=2/

P2n

D

. C 12 / .2n C 2/

.2n C  C 12 / .n C 1/

D

. C 12 / .2n C 2/ .n C 1/

.1=2;1=2/

Pn .1  2x 2 / .n C  C 12 / .2n C 1/ ! 1   2/ . C 12 / .2n C 2/ .n C 1/ n n 2 .1/ F n; n C I 12 I 1.12x D 2 1 n .2/ .n C  C 2 / .2n C 1/ .2/ .2n C  C 12 /

.1/n

(3.4.20)

.x/

.2/ .n C  C 12 / .2n C 1/

.1/n

.n C 12 / .n C 1/ . 21 /

  F n; n C I 12 I x 2 :

84

3 Orthogonal Polynomials

Due to the duplication formula of Lemma 2.3.3, we find that .1/ . C 12 / D 21 2 ; .2/ 2 ./

(3.4.21)

.1/ .n C  C 12 / D 2nC21 2 ; .2n C 2/ 2 .n C / .n C 12 / . 12 /

D

.2n/ 22n1 .n/

(3.4.22)

:

(3.4.23)

Therefore, ./

C2n .x/ D D

. 12 / 22nC21 .n C / .2n/ 21 22n1 .n/ 2 ./ . 12 /  2 .n C /.1/n  F n; n C I 12 I x 2 2 n ./ .n/

D .1/n

  .1/n F n; n C I 12 I x 2 .2n C 1/

!   nC1 F n; n C I 12 I x 2 : n

(3.4.24) t u

The odd case can be verified analogously.

Corollary 3.4.4. For N 2 N0 , the ultraspherical polynomials possess the following explicit representation: bN=2c ./

CN .x/ D

X

.1/m

mD0

.N  m C / .2x/N 2m ; ./ .m C 1/ .N  2m C 1/

where b N2 c is the largest integer that is less than or equal to

N 2

(3.4.25)

.

Proof. Let N D 2n, i.e., bN=2c D n. We use the definition of the hypergeometric function F (Definition 3.3.8) in Lemma 3.4.3. ./

CN .x/ D .1/n



.n C / F n; n C I 12 I x 2 ./ .n C 1/ .n C / X .n/k .n C /k x 2k : ./ .n C 1/ kŠ . 12 /k n

D .1/n

(3.4.26)

kD0

We resolve the Pochhammer symbols as follows (note that the duplication formula of Lemma 2.3.3 is used in (3.4.29)): .n/k D .1/k

.n C 1/ ; .n C 1  k/

(3.4.27)

3.4 Ultraspherical Polynomials

.n C /k D . 12 /k D

85

.n C  C k/ ; .n C / .k C 12 / . 12 /

D

(3.4.28) .2k/

22k1 .k/

:

(3.4.29)

We obtain: ./

CN .x/ D

n X .1/nCk kD0

D

n X .1/nk kD0

n X D .1/nk kD0

.n C 1/ .n C  C k/ . 12 / x 2k .n C / ./ .n C 1/ .n C 1  k/ .n C / .k C 12 / kŠ 22k1 .k/ 2k .n C  C k/ x ./ .n  k C 1/ .k C 1/ .2k/ .2x/2k .n C  C k/ : ./ .n  k C 1/ .2k C 1/

(3.4.30)

Now, we shift the index using k D n  m or m D n  k: ./

CN .x/ D

n X

.1/m

mD0

.2x/2n2m .n C n  m C / ./ .m C 1/ .2n  2m C 1/

bN=2c

D

X

mD0

.1/m

.N  m C / .2x/N 2m : ./ .m C 1/ .N  2m C 1/

(3.4.31)

For N D 2n C 1, the proof can be performed analogously.

t u

Theorem 3.4.5. For  > 0 and n 2 N0 , we have that ! ˇ ./ ˇ n C 2  1 : max ˇCn .x/ˇ D Cn./ .1/ D 1x1 n

(3.4.32)

For  < 0 and n 2 N0 , it holds that ˇ ˇ ˇ ˇ max ˇCn./ .x/ˇ D ˇCn./ .x 0 /ˇ ;

(3.4.33)

1x1

where x 0 is one of the two maximum points nearest to 0 if n is odd, x 0 D 0 if n is even. Proof. Let n  1. We consider

2 n.n C 2/F .x/ D n.n C 2/ Cn./ .x/ C .1  x 2 /



d ./ C .x/ dx n

2 :

(3.4.34)

86

3 Orthogonal Polynomials

 2 ./ Then, F .x/ D Cn .x/ if

./ d C .x/ dx n

max

1x1

D 0 or x D ˙1, i.e.,

./ 2 Cn .x/  max F .x/: 1x1

(3.4.35)

Now, we differentiate equation (3.4.34) and use the differential equation (3.4.3):   2 d ./ d ./ Cn .x/  2x Cn .x/ dx dx !   d ./ d2 ./ C .1  x 2 /2 Cn .x/ Cn .x/ dx dx 2       d ./ d ./ d ./ D2 .2 C 1/x Cn .x/ Cn .x/  x Cn .x/ dx dx dx 2  d ./ (3.4.36) Cn .x/ : D 4x dx

n.n C 2/F 0 .x/ D n.n C 2/2Cn .x/ ./



Therefore, we find that, if  > 0, F is increasing in Œ0; 1. If  < 0, F is decreasing in Œ0; 1. This (together with the symmetry relation of Theorem 3.1.3) gives us the desired result. t u Remark 3.4.6. For the Legendre polynomials, i.e.,  D 12 , Theorem 3.4.5 gives us jPn .x/j  Pn .1/ D 1

(3.4.37)

and ! ! ˇ 0 ˇ ˇˇ .3=2/ ˇˇ n1C31 nC1 n.n C 1/ ˇP .x/ˇ D ˇC D D n n1 .x/ˇ  2 n1 n1

(3.4.38)

for x 2 Œ1; 1. Theorem 3.4.7 (Generating Function). For  > 0 and h 2 .1; 1/, the ultraspherical polynomials form the coefficients of the following power series: 1 X

hn Cn./ .x/ D .1  2hx C h2 / ;

(3.4.39)

nD0

where x 2 Œ1; 1. Proof. First, we check the convergence of the series. ! ˇX ˇ 1 1 X ˇ 1 n ./ ˇ X ˇ ˇ n C 2  1 ˇ : h Cn .x/ˇˇ  jhjn ˇCn./ .x/ˇ  jhjn ˇ n nD0

nD0

nD0

(3.4.40)

3.4 Ultraspherical Polynomials

87

! n C 2  1 .n C 2/  C n2 D .n C 1/ .2/ n

Since

with a constant C > 0 and

1 X

n2 jhjn < 1;

(3.4.41)

(3.4.42)

nD0

we have absolute and uniform convergence of the series. Now, we consider the recurrence relation (3.4.10): 1 X

nCn./ .x/hn1 D 2x

nD1

1 1 X X ./ ./ .nC1/Cn1 .x/hn1  .nC22/Cn2 .x/hn1 ; nD1

nD1

(3.4.43) where

./ C1

 0. For a fixed x 2 Œ1; 1, we introduce the notation ˚.h/ D

1 X

Cn./ .x/hn :

(3.4.44)

nD0

Then, we obtain by differentiating with respect to h: 0

˚ .h/ D

1 X

nCn./ .x/hn1 ;

(3.4.45)

nD1

and for the first term on the right-hand side of (3.4.43)

0 h1 h ˚.h/ D ˚.h/ C h˚ 0 .h/ D

1 X

Cn./ .x/hn C

nD0

D

1 X

1 X

nCn./ .x/hn

nD1 ./

.n C   1/Cn1 .x/hn1 :

(3.4.46)

nD1

The second term on the right-hand side can be computed as

0 h22 h2 ˚.h/ D 2h˚.h/ C h2 ˚ 0 .h/ D

1 X

2Cn./ .x/hnC1 C

nD0

D

1 X

nCn./ .x/hnC1

nD1

1 X

./

.n C 2  2/Cn2 .x/hn1 ;

nD1

(3.4.47)

88

3 Orthogonal Polynomials ./

where in each case C1  0. Therefore,

0

0 ˚ 0 .h/ D 2xh1 h ˚.h/  h22 h2 ˚.h/

D 2x h1 h1 ˚.h/ C h1 h ˚ 0 .h/

 2h21 h22 ˚.h/ C h22 h2 ˚ 0 .h/



D 2x ˚.h/ C h˚ 0 .h/  2h˚.h/ C h2 ˚ 0 .h/ :

(3.4.48)

.1  2hx C h2 /˚ 0 .h/ D .2x  2h/˚.h/

(3.4.49)

2h  2x ˚ 0 .h/ D  : ˚.h/ 1  2hx C h2

(3.4.50)

This yields or equivalently

We perform integration with respect to the variable h:

ln j˚.h/j D  ln.1  2hx C h2 / C C D ln .1  2hx C h2 / C C: (3.4.51) Finally,

˚.h/ D CQ .1  2hx C h2 / ;

where CQ D 1, since ˚.0/ D 1.

(3.4.52) t u

Remark 3.4.8. For  D 0, we can find the generating function of the Chebyshev polynomials of first kind, namely  ln.1  2hx C h2 / D

1 1 n X X 2 2 Y 2k Tn .x/hn D Pn.1=2;1=2/ .x/hn : n n 2k  1 nD1 nD1 kD1 (3.4.53)

Remark 3.4.9. The generating function can also be used to define the corresponding orthogonal polynomials as coefficients of the series expansion. Corollary 3.4.10. For  > 0, x 2 Œ1; 1 and jhj < 1, 1 X nC nD0



hn Cn./ .x/ D

1  h2 : .1 C h2  2hx/C1

(3.4.54)

Proof. Use again (3.4.44) and (3.4.45) of the proof of Theorem 3.4.7 and find that 1 X nC nD0



hn Cn./ .x/ D

1 0 1  h2 h˚ .h/ C ˚.h/ D : (3.4.55)  .1 C h2  2hx/C1 t u

3.4 Ultraspherical Polynomials

89

Theorem 3.4.11. We can derive the following relations for n 2 N0 : .1  x 2 /

 d ./ 1 ./ Cn .x/ D .n C 2  1/.n C 2/Cn1 .x/ dx 2.n C /  ./ n.n C 1/CnC1 .x/ (3.4.56) ./

(3.4.57)

./

(3.4.58)

D nxCn./ .x/ C .n C 2  1/Cn1 .x/ D .n C 2/xCn./ .x/  .n C 1/CnC1 .x/ .C1/

D 2.1  x 2 /Cn1 .x/;

(3.4.59)

d ./ d ./ Cn .x/  C .x/; dx dx n1 d ./ d CnC1 .x/  x Cn./ .x/; .n C 2/Cn./ .x/ D dx dx nCn./ .x/ D x

(3.4.60) (3.4.61)

as well as (for n 2 N)  d  ./ ./ CnC1 .x/  Cn1 .x/ D2.n C /Cn./ .x/ dx   .C1/ D2 Cn.C1/ .x/  Cn2 .x/ :

(3.4.62) (3.4.63)

Proof. Relation (3.4.56) is proved by induction. The case n D 0 is obvious. Taking ./ n to n C 1 we then use the three-term recurrence (3.4.10) for CnC1 and obtain .1  x 2 /

d ./ 2.n C / ./ 2.n C / 2 ./ CnC1 .x/ D Cn .x/  x Cn .x/ (3.4.64) dx nC1 nC1 2.n C / d n C 2  1 d ./ C x.1  x 2 / Cn./ .x/  .1  x 2 / Cn1 .x/: nC1 dx nC1 dx ./

d Now, the induction assumption is applied twice, for .1  x 2 / dx Cn .x/ as well as ./ d for .1  x 2 / dx Cn1 .x/. Next, the three-term recurrence (3.4.10) helps to substitute ./ ./ ./ xCnC1 , xCn1 , and x 2 Cn (here we use it twice). This leads to a formula that only ./ ./ ./ contains the ultraspherical polynomials CnC2 , Cn , and Cn2 . Simplifying their ./ coefficients concludes the induction and yields (3.4.56) (the coefficient of Cn2 turns ./ out to be 0). The three-term recurrence (3.4.10) for .n C 1/CnC1.x/ in (3.4.56) gives us (3.4.58) and the step between (3.4.58) and (3.4.57) is once again only (3.4.56). Equation (3.4.59) is a direct consequence of (3.4.7). To prove (3.4.60), apply the derivative to (3.4.57) and rearrange the terms such that

90

3 Orthogonal Polynomials

nCn./ .x/  x

d ./ d ./ C .x/ C C .x/ D dx n dx n1

.n C 2/

(3.4.65)

d d2 d ./ Cn1 .x/  .n  1/x Cn./ .x/  .1  x 2 / 2 Cn./ .x/: dx dx dx

Using (3.4.7) on each term of the right-hand side as well as (3.4.57), the righthand side becomes 0 as desired. Equation (3.4.61) is shown analogously to (3.4.60), only now (3.4.58) takes the role of (3.4.57) before. Finally, (3.4.62) is obtained by adding (3.4.60), (3.4.61) and (3.4.63) follows directly from (3.4.59) or (3.4.7). u t Remark 3.4.12. For q 2 N, q  3, and t 2 Œ1; 1, 1

q2

Pn .qI t/ D nCq3 Cn

2

n



.n C 1/ .q  2/ q2 2 Cn .t/ D .t/ .n C q  2/

(3.4.66)

denotes the Legendre polynomials of dimension q and degree n (see, e.g., M¨uller 1998 and Sect. 6.4, in particular Remark 6.4.8).

3.5 Application of the Legendre Polynomials in Electrostatics R Let x 2 R3 and let %.x/ be a charge distribution with total charge R3 %.x/ dx. Then, the fundamental equations of electrostatics are the electrostatic pre-Maxwell equations. Let E denote the electric field, i.e., for x 2 R3 , rx  E.x/ D 4%.x/; rx ^ E.x/ D 0:

(3.5.1) (3.5.2)

We introduce the electric potential ˚ and solve the equations by E.x/ D r˚.x/ which gives us the following Poisson equation

˚.x/ D 4%.x/:

(3.5.3)

Example 3.5.1. A point charge q at x0 2 R3 yields %.x/ D qı.x  x0 /

(3.5.4)

with the delta-distribution ı in R3 (equality holds in the weak sense). We obtain

˚.x/ D 0 The solution is given by ˚.x/ D field can be calculated to be

q jxx0 j

for x 2 R3 n fx0 g:

(3.5.5)

for x 2 R3 nfx0 g. The corresponding electric

3.5 Application of the Legendre Polynomials in Electrostatics

E.x/ D q

x  x0 jx  x0 j3

91

for x 2 R3 n fx0 g:

(3.5.6)

Another approach to (3.5.5) introduces polar coordinates .r; '; t/ for x 2 R3 nf0g by p x1 D r 1  t 2 cos ';

p x2 D r 1  t 2 sin ';

x3 D rt;

(3.5.7)

where r D jxj > 0, ' 2 Œ0; 2/, t 2 Œ1; 1 (see also (4.1.2)). This gives us the representation of the Laplace equation in polar coordinates, namely 

@ @r

2

1 @ 1 1 2 @ @ C 2 .1  t 2 / C 2 C r @r r @t @t r 1  t2



@ @'

2 ! ˚.r; '; t/ D 0 (3.5.8)

for x 2 R3 n f0g. By separation of variables we set ˚.r; '; t/ D U.r/V .'/W .t/

(3.5.9)

and insert this in the equation above, i.e., 

@ V .'/W .t/ @r C

2 U.r/ C V .'/W .t/

2 @ U.r/ r @r

1 1 @ @ 1 U.r/V .'/ .1  t 2 / W .t/ C 2 U.r/W .t/ 2 r @t @t r 1  t2

(3.5.10) 

@ @'

2 V .'/ D 0:

Next, we multiply by r 2 .1  t 2 / and divide by U.r/V .'/W .t/: r 2 .1  t 2 /

U 00 .r/ U 0 .r/ C 2r.1  t 2 / U.r/ U.r/

C .1  t 2 /2

(3.5.11)

V 00 .'/ W 0 .t/ W 00 .t/  2t.1  t 2 / C D 0; W .t/ W .t/ V .'/

which can be rewritten as  r 2 .1  t 2 /  .1  t 2 /2

U 00 .r/ U 0 .r/  2r.1  t 2 / U.r/ U.r/

(3.5.12)

V 00 .'/ W 00 .t/ W 0 .t/ C 2t.1  t 2 / D : W .t/ W .t/ V .'/

The left-hand side only depends on r and t, the right-hand side only on '. Thus, the equation can only be fulfilled if both sides are equal to a constant  2 R. Therefore,

92

3 Orthogonal Polynomials

V 00 .'/ D  , V 00 .'/  V .'/ D 0; V .'/

' 2 Œ0; 2/:

(3.5.13)

The non-trivial solutions p of this differential equation p are linear combinations of the functions V1 D exp.i '/ and V2 D exp.i '/. V1 and V2 have to be 2-periodic. This leads to a discretization of , namely  D m2 , m 2 N0 . The linearly independent solutions are given by Vm .'/ D exp.im'/;

' 2 Œ0; 2/; m 2 Z:

(3.5.14)

Now, we consider the left-hand side of (3.5.12) for these values of : 

m2 W 0 .t/ U 0 .r/ U 00 .r/ W 00 .t/  .1  t 2 / C 2t D r2 C 2r : 2 .1  t / W .t/ W .t/ U.r/ U.r/

(3.5.15)

By the same argument as before we find that this equation can only hold if both Q i.e., sides are equal to a constant , 

m2 W 0 .t/ W 00 .t/  .1  t 2 / C 2t D Q 2 .1  t / W .t/ W .t/

(3.5.16)

which is equivalent to   m2 .1  t 2 /W 00 .t/  2tW 0 .t/ C Q C W .t/ D 0; 1  t2

(3.5.17)

for t 2 Œ1; 1 and U 0 .r/ U 00 .r/ Q C 2r , 0 D r 2 U 00 .r/ C 2rU 0 .r/  U.r/ Q D r 2 U.r/ U.r/

(3.5.18)

for r > 0.

Symmetric Problems For the case of a point charge located at x0 we can assume (after a certain rotation of the coordinate system) that x0 is on the "3 -axis, i.e., x0 D Œ0; 0; r0 T , where r0 D jx0 j > 0. In this case it is clear that the solution of (3.5.5) has a rotational symmetry, i.e., ˚ does not dependent on '. We can use ˚.r; '; t/ D U.r/W .t/

(3.5.19)

3.5 Application of the Legendre Polynomials in Electrostatics

93

with r > 0 and t 2 Œ1; 1 (this corresponds to m D 0 before). Therefore, we find the equation for W : Q .t/ D 0; Lt W .t/ C W

t 2 Œ1; 1;

(3.5.20)

where we have introduced the Legendre operator Lt D .1  t 2 /

d2 d  2t : dt 2 dt

(3.5.21)

Equation (3.5.20) is the Legendre differential equation (or the Gegenbauer differential equation (3.4.3) with  D 1=2) which possesses a polynomial solution if and only if Q D n.n C 1/; n 2 N0 : (3.5.22) The solutions are the Legendre polynomials Wn .t/ D Pn .t/, t 2 Œ1; 1, n 2 N0 . Q Now, we consider the second equation for these values of : r 2 U 00 .r/ C 2rU 0 .r/  n.n C 1/U.r/ D 0;

r > 0:

(3.5.23)

This ordinary differential equation possesses two linearly independent solutions given by U1;n .r/ D r n ;

n 2 N0 ;

U2;n .r/ D

1 r nC1

;

n 2 N0 :

(3.5.24)

Together we find the two linearly independent solutions of the Laplace equation in the rotational symmetric case to be ˚1;n .r/ D r n Pn .t/;

˚2;n .r/ D

1 r nC1

Pn .t/;

(3.5.25)

for r > 0, t 2 Œ1; 1, n 2 N0 . Every solution can be expressed as a linear combination of these two, i.e., there exist coefficients An , Bn such that ˚.r; t/ D

 1  X 1 An r n C Bn nC1 Pn .t/: r nD0

(3.5.26)

For the case of a point charge located in x0 D Œ0; 0; r0 T we know a solution, namely ˚.x/ D

q ; jx  x0 j

x 2 R3 n fx0 g:

(3.5.27)

We use this solution to compute the coefficients An and Bn in the expansion (3.5.26). Restricting the solutions to the "3 -axis, the point x possesses the polar coordinates r D jxj, t D 1, ' D 0, i.e.,

94

3 Orthogonal Polynomials

˚.x/ D

q ; jr  r0 j

r > 0; r ¤ r0 :

(3.5.28)

We use the well-known geometric series and obtain

˚.x/ D

8 ˆ q ˆ < rr0 D ˆ q ˆ : r0 r D

q 1 r 1 r0 r q 1 r0 1 r r

q r

D D

0

1

P r0 n

r nD0  n 1 q P r r0 r0 nD0

for r > r0 ; (3.5.29) for r < r0 :

Comparing this to the expansion for t D 1 (remember that Pn .1/ D 1), i.e., ˚.r; 1/ D

 1  X Bn An r n C nC1 ; r nD0

r > 0;

(3.5.30)

results in the following coefficients: An D

8 r0 ; q

if r < r0 ;

r0nC1

( Bn D

qr0n

if r > r0 ;

0

if r < r0 :

(3.5.31)

Thus, we finally obtain

˚.x/ D ˚.r; t/ D

8 1 P ˆ ˆ r0 D jx0 j; for jxj D r < r0 D jx0 j:

0

(3.5.32) From a mathematical point of view there is another interesting aspect. Canceling the charge q leads to (x ¤ x0 ) 81 P ˆ ˆ
r0 D jx0 j; for jxj D r < r0 D jx0 j:

(3.5.33)

0

The value jx  x0 j can be expressed using polar coordinates (see (4.1.2) in Sect. 4.1 for details) by ˇh iT ˇˇ2 p ˇ p jx  x0 j2 D ˇˇ r 1  t 2 cos '; r 1  t 2 sin '; rt  r0 ˇˇ   r2 r0 D r 2 1 C 02  2 t : r r

(3.5.34)

3.6 Hermite Polynomials and Applications

95

Take r > r0 and we obtain, for r > r0 ,

r

1

1

q 1C

r02 r2

D  2 rr0 t

X rn 1 0 D P .t/; nC1 n jx  x0 j r nD0

(3.5.35)

which is equivalent to 1

q 1C

D

r02 r2

 2 rr0 t

1   X r0 n nD0

r

Pn .t/:

(3.5.36)

Pn .t/:

(3.5.37)

Analogously, for r < r0 , q

1 1C

r2 r02

 2 rr0 t

D

1  n X r nD0

r0

Substituting h D r0 =r, respectively h D r=r0 , we get the result of Theorem 3.4.7 for the Legendre polynomials, i.e.,  D 12 . For a more detailed treatment see, e.g., Jackson (1998).

3.6 Hermite Polynomials and Applications The Hermite polynomials Hn are the unique orthogonal polynomials in L2 .R/ with the measure d.x/ D w.x/dx D exp.x 2 /dx. Definition 3.6.1. The Hermite polynomials Hn are defined as follows: (i) H R n is a polynomial of degree n, (ii) R Hn .x/Hm .x/ exp.x 2 / dx D 0 for m ¤ n, p (iii) kHn k2 D 2n nŠ, n 2 N0 . By this definition we can calculate the first few polynomials H0 .x/ D 1, H1 .x/ D 2x, H2 .x/ D 4x 2  2, H3 .x/ D 8x 3  12x. Rodrigues’ representation and the explicit representation are given by 

d Hn .x/ D .1/ exp.x / dx n

bn=2c

Hn .x/ D

X kD0

n

2



exp.x 2 / ;

.1/k nŠ .2x/n2k : kŠ.n  2k/Š

(3.6.1)

(3.6.2)

The following recurrence relation holds with H0  1, H1  0: Hn .x/ D 2xHn1 .x/  2nHn2 .x/

for n 2 N; x 2 R:

(3.6.3)

96

3 Orthogonal Polynomials

The derivative is given by Hn0 .x/ D 2nHn1 .x/ for n 2 N0 , x 2 R, and the following differential equation is solved by the Hermite polynomials: Hn00 .x/  2xHn0 .x/ C 2nHn .x/ D 0;

(3.6.4)

where n 2 N0 , x 2 R. Furthermore, for un .x/ D exp.x 2 =2/Hn .x/ we have u00n .x/ C .2n C 1  x 2 /un .x/ D 0;

n 2 N0 ;

x 2 R:

(3.6.5)

For further details and properties of Hermite polynomials we refer to, e.g., Lebedev (1973), Sneddon (1980), and Szeg¨o (1967).

Application in Quantum Mechanics The starting point of any system is its Hamilton function E D H.pi ; qj ; t/;

(3.6.6)

where qj are (unified) coordinates, pi are the impulse coordinates and t is the time. H is the total energy of the system. The step from classical mechanics to quantum mechanics is performed by substitution rules: E ! i„

@ ; @t

p ! i„r:

(3.6.7)

h Note that „ D 2

1:05457  1034 Js is Plank’s constant. The coordinates qj are substituted by the wave function, where j j2 is the probability density, i.e., for all t,

Z j .x; t/j2 dx D 1:

(3.6.8)

R3

For example, let be the wave function of a particle in a potential V , the classical Hamilton function is p2 C V .x/: (3.6.9) ED 2m This becomes by our substitutions i„

@ „2 .x; t/ D 

x .x; t/ C V .x/ .x; t/ @t 2m

or equivalently i„ P D H

with H D 

„2

x C V .x/: 2m

(3.6.10)

(3.6.11)

3.6 Hermite Polynomials and Applications

97

If V and, thus, H are time-space separating, we use the approach .x; t/ D '.x/ exp. iEt „ /;

(3.6.12)

such that for all t; i„

@ iEt iEt .x; t/ D i„'.x/ iE „ exp. „ / D H'.x/ exp. „ / @t

(3.6.13)

which is equivalent to H'.x/ D E'.x/. This is the time-independent Schr¨odinger equation. H is the time-independent Hamilton operator and E is the energy of the system, which is an unknown. Furthermore, Z Z Z 2 iEt 2 j .x; t/j dx D j'.x/ exp. „ /j dx D j'.x/j2 dx; (3.6.14) 1D R3

R3

R3

i.e., the probability density is time-independent. Therefore, the eigenvalue problem R H' D E' has to be solved with R3 j'.x/j2 dx D 1.

One-Dimensional Oscillation Let m be a mass connected to a wall by a spring with spring constant f . The classical Hamilton function of this one-dimensional system is H.p; x/ D

m! 2 x 2 p2 C 2m 2

(3.6.15)

p with ! D f =m the eigenfrequency of the system. By our substitution rules we obtain the operator „2 d 2 m! 2 x 2 HD : (3.6.16) C 2 2m dx 2 This operator H describes, e.g., the oscillation of a molecule with two atoms and m is the relative mass of the atoms. Thus, the eigenvalue problem for the oscillation is:   „2 d 2 m! 2 x 2 '.x/ D E'.x/;  C 2m dx 2 2

x 2 R:

(3.6.17)

q „ 2E Defining a new coordinate by y D x=b, where b D m! , and setting " D „! this becomes d2 '.by/ C."  y 2 / '.by/ D 0: (3.6.18) „ƒ‚… dy 2 „ƒ‚… u.y/

u.y/

98

3 Orthogonal Polynomials

This equation possesses a solution if and only if " D 2n C 1, where n 2 N0 . The solution is of the form un .y/ D cn exp 

y2

2 Hn .y/

(3.6.19)

for n 2 N0 using the Hermite polynomial Hn of degree n. The constant cn is determined by the condition Z

Z R

j'n .x/j2 dx D 1 ,

R

jun .y/j2 dy D

1 : b

(3.6.20)

Thus, we can summarize 'n .x/ D cn Hn . xb / exp 

x2 2b 2



(3.6.21)

for n 2 N0 and x 2 R. These functions are the eigenfunctions of the onedimensional oscillation corresponding to the eigenvalues En D .n C 12 /„!;

n 2 N0 :

(3.6.22)

This shows that the energy of a quantum-mechanical oscillator can only take discrete values, i.e., the quantization of energy. For further information the reader is referred to, e.g., Landau and Lifshitz (2004).

3.7 Laguerre Polynomials and Applications In this section we consider orthogonal polynomials with respect to the measure d.x/ D w.x/dx D exp.x/x ˛ dx with a parameter ˛ > 1 and the support interval R0 D Œ0; 1/. .˛/

Definition 3.7.1. For ˛ > 1, the generalized Laguerre polynomials Ln , n 2 N0 , are uniquely defined by .˛/

(i) Ln is a polynomial of degree n defined on R0 , R .˛/ .˛/ (ii) R0 Lm .x/Ln .x/ exp.x/x ˛ dx D 0 for n ¤ m,  .˛/ 2

(iii) Ln  D .˛ C 1/ nC˛ D .nC˛C1/ . n

.nC1/

Note that for ˛ D 0 these are the classical Laguerre polynomials. The Laguerre polynomials admit the following Rodrigues’ representation and explicit representation for n 2 N0 and x 2 R0 : L.˛/ n .x/ D exp.x/

x ˛ nŠ



d dx

n



exp.x/x nC˛ ;

(3.7.1)

3.7 Laguerre Polynomials and Applications

L.˛/ n .x/ D

99

n X .n C ˛ C 1/ kD0

.x/k : .k C ˛ C 1/ kŠ.n  k/Š

(3.7.2) .˛/

By these properties we obtain the first three polynomials, namely L0 .x/ D 1,

.˛/ .˛/ L1 .x/ D 1C˛x and L2 .x/ D 12 .1 C ˛/.2 C ˛/  2.2 C ˛/x C x 2 . We find the recursion formula for n 2 N, n  2: .˛/

.˛/

nL.˛/ n .x/ D .2n C ˛  1  x/ Ln1 .x/  .n C ˛  1/ Ln2 .x/;

x 2 R0 (3.7.3)

and the recursion including the derivative for n 2 N; x

d .˛/ .˛/ L .x/ D nL.˛/ n .x/  .n C ˛/Ln1 .x/; dx n

x 2 R0 :

(3.7.4)

The differential equation with ˛ > 1, x 2 R0 , xy 00 C .1 C ˛  x/y 0 C y D 0

(3.7.5)

possesses a polynomial solution if and only if  D n 2 N0 . This solution is given .˛/ by y.x/ D cn Ln .x/, x 2 R0 , with a constant cn 2 R. For further details and properties of Laguerre polynomials we refer to, e.g., Lebedev (1973), Sneddon (1980), and Szeg¨o (1967).

Eigenoscillations of an n-Fold Pendulum Let l be the total length of the n-fold pendulum, each section has length a D l=n. The angles of each section are collected in the vector ' D Œ'1 ; : : : ; 'n T . The dynamic of the pendulum is described by M'R C C' D 0;

(3.7.6)

where we have linearized near a stable equilibrium, i.e., 'j D 0, j D 1; : : : ; n, and M is the mass matrix with entries Mi;j D a2 minfi; j g

(3.7.7)

and C is the matrix of restitutional forces with Ci;j D igaıi;j , g is the gravitational acceleration. Small vibrations are given by ' D 'O sin.!t/. M'R C C' D 0;

i:e:; .C  ! 2 M/'O D 0:

(3.7.8)

100

3 Orthogonal Polynomials

We decompose M by Cholesky decomposition, i.e. M D UT U. This gives us 2

1  6 :: U D a4 : 0

3 1 :: 7 : :5

(3.7.9)

1

Now, we set xO D U', O where xO D ŒxO 0 ; : : : ; xO n1 T . We find that

H) H) H)



0 D C  ! 2 UT U '; O

1 O 0 D CU  ! 2 UT x; T 1 1

0 D .U / CU  ! 2 I x; O   O 0 D ga .UT /1 CU1  ga ! 2 I xO D .A  I/x:

(3.7.10) (3.7.11) (3.7.12) (3.7.13)

This is an eigenvalue problem for the matrix 2

1 61 6 6 60 6 AD6 6 6 6 4

1 0 3 2 0

0

::

3

7 7 7 7 7 7: 0 7 7 7 n C 15

: 2 5 3 :: :: :: :: : : : : :: :: :: : : : 0 0 n C 1 2n  1

(3.7.14)

Writing out the eigenvalue problem explicitly, we obtain 0xO 1 C 1xO 0  1xO 1 D xO 0 ;

(3.7.15)

k xO k1 C .2k C 1/xO k  .k C 1/xO kC1 D xO k ;

(3.7.16)

.n  1/xO n2 C .2n  1/xO n1 D xO n1 ;

(3.7.17)

where k D 1; : : : ; n  2. We have to append xO 1 and xO n such that (3.7.16) holds for k D 0; : : : ; n  1, i.e., xO 1 can be chosen arbitrarily and xO n D 0 to satisfy (3.7.17). We take a look at the recurrence relation for the classical Laguerre polynomials, i.e., the case ˛ D 0:  kLk1 .x/ C .2k C 1/Lk .x/  .k C 1/LkC1 .x/ D xLk .x/:

(3.7.18)

We can identify x with  and Lk .x/ with xO k . Hence, xO k D Lk ./ for k D 0; : : : ; n satisfies (3.7.15)–(3.7.17) for all . Since we have to fulfill xO n D 0, we find Ln ./ D xO n D 0:

(3.7.19)

3.8 Exercises

101

The eigenfrequencies 1 ; : : : ; n of the system AxO D xO

(3.7.20)

are determined by (3.7.19). Therefore, let n;1 ; : : : ; n;n be the zeros of Ln in R0 . Then, the eigenfrequencies of the system are q (3.7.21) !k D ga n;k ; k D 1; : : : ; n: The corresponding eigenmodes to the eigenvalue !k can be calculated from xO j D Lj .n;k /;

j D 0; : : : ; n  1;

(3.7.22)

and 'O D U 1 x. O All interesting properties of the system, such as energy etc., can be expressed using Laguerre polynomials. For further details the reader is referred to Braun (1997).

3.8 Exercises (Gaussian Integration, Legendre Series, Kernel Expansions) Next, we remember the conditions in Gauß quadrature rules which must be satisfied by the weights and knots in order to get an optimal degree of polynomial precision. Restricting ourselves to the weight function w.x/ D 1 and the interval Œ1; 1 we are canonically led to the theory of Legendre polynomials, i.e., Gauß–Legendre integration. Our exercises do not only provide the classical theory of Gaussian integration, but also new types of error estimates involving the Green function with respect to the Legendre operator as an essential tool. Moreover, the Clenshaw algorithm for the staple evaluation of Legendre series and its modified version are discussed with the help of some exercises. Finally, we develop error estimates for kernel expansions that arise in numerical algorithms involving truncations of expansions in terms of ultraspherical polynomials.

Gaussian Integration Revisited Definition 3.8.1. The Green function G.L C I ; / W Œ1; 1 Œ1; 1 ! R with respect to the Legendre operator L C ,  2 R, (see (3.5.21) for a definition of the operator L) is uniquely defined by the following properties: (i) (Boundedness) For arbitrary, but fixed x 2 Œ1; 1, G.L C I x; / is a continuous function on Œ1; 1 satisfying the conditions jG.L C I x; 1/j < 1;

(3.8.1)

jG.L C I x; 1/j < 1:

(3.8.2)

102

3 Orthogonal Polynomials

(ii) (Differential equation) For each fixed x 2 Œ1; 1, G.L C I x; / is twice continuously differentiable in Œ1; 1 n fxg. For  … SpectL , we have .Lt C /G.L C I x; t/ D 0; t 2 Œ1; 1 n fxg: q For n 2 SpectL , we have with PQn D 2nC1 Pn that 2 .Lt C n /G.L C n I x; t/ D PQn .x/PQn .t/;

t 2 Œ1; 1 n fxg;

(3.8.3)

(3.8.4)

where SpectL D fn D n.n C 1/ W n 2 N0 g. (iii) (Characteristic singularity) .1  x 2 /

ˇt DxC0 d ˇ G.L C I x; t/ˇ D 1: t Dx0 dt

(3.8.5)

(iv) (Normalization) For each x 2 Œ1; 1 and n 2 SpectL , Z

1 1

G.L C n I t/PQn .t/ dt D 0:

(3.8.6)

The Green function is uniquely determined by these four conditions. Definition 3.8.1 enables us to develop the following integral formulas: Exercise 3.8.2. Let x be a point in .1; 1/. Suppose that F is of class C.2/ .Œ1; 1/. (a) Use the Green-Lagrange formula to prove for  … SpectL , Z

1

F .x/ D 1

G.L C I x; t/.Lt C /F .t/ dt:

(3.8.7)

Furthermore, for  2 SpectL , i.e.,  D n D n.n C 1/, F .x/ D PQn .x/

Z

1 1

F .t/PQn .t/ dt C

Z

1 1

G.L C I x; t/.Lt C /F .t/ dt: (3.8.8)

(b) Use (a) to show that for  2 R, t 2 Œ1; 1, Z 1 Q .1  ı;n /Pn .t/ D .  n / G.L C I t; u/PQn .u/ du:

(3.8.9)

1

Exercise 3.8.3. (a) Implement the Gaussian integration formula Z

1 1

F .t/ dt Gn .F / D

n X i D1

wi F .xi /;

(3.8.10)

3.8 Exercises

103

with xi , i D 1; : : : ; n, the zeroes of the Legendre polynomial Pn and wi , i D 1; : : : ; n the weights determined by n X

wi PQj .xi / D ı0;j

p 2;

j D 0; : : : ; n  1:

(3.8.11)

i D1

(b) Show that (3.8.10) is exact for all polynomials of degree 2n  1. (c) Give an example of a polynomial of degree 2n such that (3.8.10) is not exact. (d) Test your routine by approximating the following integrals for n D 2; 4; 8: Z

1

Z

1 dx; 1 C x4

1

1

e

x2

Z

1

dx;

1

1

x dx: ex  1

(3.8.12)

Exercise 3.8.4. Calculate the error terms for the Gaussian integration formula to test the exactness of your results in Exercise 3.8.3. Do this by implementing routines for the following two estimates of the error or remainder term: (a) For n  1 and p  1; 22.nC1/ .2p/ .t/j: 2n 2 sup jF t 2Œ1;1 .2n C 1/Š n

Rn .F / 

(3.8.13)

(b) For n  1, p  1 and  ¤ k.k C 1/, k D 2n; 2n C 1; : : : ; q Rn .F /  

.p/ A .n/

s Z

1 1

..Lt C /p F .t//2 dt

p q .p/ 2 A .n/ sup j.Lt C /p F .t/j

(3.8.14)

t 2Œ1;1

with .p/ A .n/

D

n n X X i D1 j D1

2

1 X kD2n

.p/

wi wj

1 X kD2n

1 PQk .xi /PQk .xj / .  k.k C 1//2p

2k C 1 : .  k.k C 1//2p

(3.8.15)

Note that lim A .n/ D 0. n!1 (c) In case of (3.8.13), apply the estimate for different n; p to at least one of the integrals from Exercise 3.8.3. In case (3.8.14), apply the estimate for fixed p D 1 but different n; . For which  does the error reach its minimum?

104

3 Orthogonal Polynomials

Remark 3.8.5. Relation (3.8.13) is a Peano-type error estimate that can be found in every standard textbook on numerical integration (see, e.g., Deuflhard and Hohmann 1991; Freund and Hoppe 2007; H¨ammerlin and Hoffmann 1992). Estimate (3.8.14) is due to Freeden (1980b).

Staple Evaluation of Legendre Series (Involving Derivatives) Exercise 3.8.6. Let fTk gk2N0 satisfy the linear, second order recurrence relation Tk .x/  ak .x/Tk1 .x/  bk .x/Tk2 .x/ D 0;

k D 2; : : : ; N;

(3.8.16)

where T0 .x/ ¤ 0 and T1 .x/ are given and bk .x/ ¤ 0 for all k. Show that for N 2 N0 the sum N X SN .x/ D Ak Tk .x/; (3.8.17) kD0

where fAk gkD0;:::;N are given, can be calculated by the Clenshaw algorithm, i.e., Algorithm 3.8.7 using ak , bk of (3.8.16). Algorithm 3.8.7 (Clenshaw). Set UN C1 D UN C2 D 0. Do for k D N; : : : ; 1 Uk .x/ D akC1 .x/UkC1 .x/ C bkC2 .x/UkC2 .x/ C Ak enddo Return SN .x/ D .A0 C b2 .x/U2 .x//T0 .x/ C U1 .x/T1 .x/. Exercise 3.8.8. Specify Algorithm 3.8.7 for the special case of the Legendre polynomials Pk , k 2 N0 . Consider the following tests: (a) Implement the resulting algorithm and compute the functions Fh;N .x/ D

N X

hk Pk .x/;

x 2 Œ1; 1;

(3.8.18)

kD0

Gh;N .x/ D

N X .2k C 1/hk Pk .x/;

x 2 Œ1; 1;

(3.8.19)

kD0

on an equidistant grid in Œ1; 1 for N D 100; 200; 400; 800 and for the values h D 0:1; 0:5; 0:9; 0:95; 0:99. (b) Calculate the differences ˇ ˇ ˇ ˇ 1 ˇ ˇ; RF .h; N; t/ D ˇFh;N .x/  p (3.8.20) 2 1 C h  2hx ˇ

3.8 Exercises

105

ˇ ˇ ˇ ˇ 1  h2 ˇ: ˇ RG .h; N; t/ D ˇGh;N .x/  2 3=2 .1 C h  2hx/ ˇ .m/

Exercise 3.8.9. Show that for N 2 N0 the sum of derivatives Tk .m/

SN .x/ D

N X

(3.8.21) of order m 2 N;

.m/

Ak Tk .x/;

(3.8.22)

kD0

where fAk gkD0;:::;N are given, can be calculated by the modified Clenshaw algorithm, i.e., Algorithm 3.8.10 using ak , bk of (3.8.16). Algorithm 3.8.10 (Modified Clenshaw). .0/ .0/ Set stage 0, i.e., UN C1 D UN C2 D 0. Do for k D N; : : : ; 0 .0/ .0/ .0/ Uk .x/ D akC1 .x/UkC1 .x/ C bkC2 .x/UkC2 .x/ C Ak enddo Do for l D 1; : : : ; m .l/ .l/ Set stage l, i.e., UN C1 D UN C2 D 0. Do for k D N; : : : ; 0   l P l  .lj / .l/ .j / .lj / .j / akC1 .x/UkC1 .x/ C bkC2 .x/UkC2 .x/ Uk .x/ D j D0 j enddo enddo .m/ .m/ Return SN .x/ D U0 .x/T0 .x/. .m/

Remark 3.8.11. Note that Algorithm 3.8.10 not just calculates SN .x/, but all .l/ SN .x/ for l D 0; : : : ; m. For an implementation the derivatives of ak , bk of (3.8.16) have to be computed in advance up to order m. If ak , bk are polynomials of a low degree, the sum in Algorithm 3.8.10 simplifies drastically. Exercise 3.8.12. Specify Algorithm 3.8.10 for the special case of the Legendre polynomials Pk , k 2 N0 . Exercise 3.8.13. Specify Algorithm 3.8.7 for the second derivatives of the Legendre polynomial Pk00 , k 2 N, by using (3.4.7) for the derivative of the ultraspherical polynomials and their three-term recurrence relation. Compare the results of the summation with the results of Algorithm 3.8.10 for Pk00 , k 2 N, using Exercise 3.8.12. Remark 3.8.14. See Clenshaw (1955), Deuflhard (1976), and Tscherning (1976) for further details and Fengler (2005) for an application of these algorithms in the computation of vectorial/tensorial scaling functions and wavelets for a non-linear Galerkin scheme to solve the Navier–Stokes equation on the sphere.

106

3 Orthogonal Polynomials

Error Estimates of Kernel Expansions Exercise 3.8.15. Compute the first derivative of the hypergeometric function with respect to its last argument, i.e., for z 2 R: ab d F .a; bI cI z/ D F .a C 1; b C 1I c C 1I z/: dz c

(3.8.23)

Exercise 3.8.16. Let s 2 N, p 2 N0 and z 2 R with jzj < 1. Show that   p1  1  X 1 nCs1 n X nCs1 n z z  D s1 s1 .1  z/s nD0 nDp D

zp .1  z/s

(3.8.24)

  pCs1 F .s C 1; pI p C 1I z/; s1 (3.8.25)

where F denotes the hypergeometric function of Definition 3.3.8. Exercise 3.8.17. Let s 2 N, p 2 N0 . Show that F .s C 1; pI p C 1I / D

s1 X kD0

p pCk

  s1 k  k

(3.8.26)

if p > 0. Show also that F .s C 1; 0I 1I / D 1 holds for p D 0. Remark 3.8.18. For more details the standard literature on the hypergeometric function, e.g., Abramowitz and Stegun (1972) and Magnus et al. (1966). A proof and further applications of Exercise 3.8.16 can be found, e.g., in Cherrie et al. (2002). Now, we investigate the error term s EpC1 .; / D 1 

p p sX n s 1 C  2  2  Cn2 . /

(3.8.27)

nD0

for s 2 N, p 2 N0 ,  2 .0; 1/ and 2 Œ1; 1. Our aim is to prove that ˇ ˇ ˇ ˇ ˇ s ˇ ˇ s ˇ .; 1/ˇ : ˇEpC1 .; /ˇ  ˇEpC1

(3.8.28)

Exercise 3.8.19. Use Exercise 3.8.16 to transform the right-hand side of (3.8.28), i.e., show that   ˇ ˇ pCs ˇ ˇ s F .s C 1; p C 1I p C 2I /: ˇEpC1 .; 1/ˇ D  pC1 s1

(3.8.29)

3.8 Exercises

107

This leads us to a new version of (3.8.28), i.e.,   ˇ ˇ ˇ s ˇ pC1 p C s F .s C 1; p C 1I p C 2I /: ˇEpC1 .; /ˇ   s1

(3.8.30)

Before we can go into the details of this estimate and its proof we are concerned with three helpful tools. Exercise 3.8.20. Let s; p 2 N with s  3 and p  1. Prove that the following hypergeometric function at 1 satisfies the estimate F .s C 1; p C 1I p C 2I 1/ 

p

s

2:

(3.8.31)

Exercise 3.8.21. Let s 2 N, p 2 N0 . Show that the sum of Gegenbauer polynomials at 0 satisfies ( p X s 0 ; for p D 4m C 2 or p D 4m C 3; 2 (3.8.32) Cn .0/  . 2s C2m/ ; for p D 4m or p D 4m C 1; . s / .2mC1/ nD0 2

and moreover,

p X nD0

  pCs Cn .0/ < : s1 s 2

(3.8.33)

Exercise 3.8.22. Let s 2 N, p 2 N0 . Show that the sum of Gegenbauer polynomials at 0 satisfies p X

( s 2

Cn .0/ 

nD0

; for p D 4m or p D 4m C 1;

1 1

. 2s C2mC1/ . 2s / .2mC2/

and moreover,

p X nD0

s

; for p D 4m C 2 or p D 4m C 3;

Cn2 .0/ > 1 

  pCs : s1

(3.8.34)

(3.8.35)

Finally, we are prepared to prove the major theorem necessary for error estimate (3.8.30) in three exercises. Exercise 3.8.23. Let s 2 N, p D 0. Let D D .0; 1/ .1; 1/ and the functions h˙ W D ! R be defined as h˙ .; / D pC1

  pCs F .s C 1; p C 1I p C 2I / s1

p p sX n s 2  Cn2 . / ˙ 1 1 C   2 nD0

DsF .s C 1; 1I 2I / ˙ 1

p s 1 C  2  2 :

(3.8.36)

108

3 Orthogonal Polynomials

Prove that, for all .; / 2 D, it holds that h .; /  0 and hC .; /  0. Exercise 3.8.24. Let s 2 N, s > 1, p 2 N. Let D D .0; 1/ .1; 1/ and the functions h˙ W D ! R be defined as h˙ .; / D pC1

  pCs F .s C 1; p C 1I p C 2I / s1

˙1

p p sX n s 1 C  2  2  Cn2 . /:

(3.8.37)

nD0

Show the following properties of h˙ : (a) h˙ .0; / D 0, (b) h .1; 0/ > 0 and hC .1; 0/ > 0, @ (c) @ h˙ .; / ¤ 0 in D. Hint: Use Exercises 3.8.21 and 3.8.22 for the two inequalities of part (b), Exercise 3.8.17 can be helpful for the first inequality. Calculate the partial derivative @ in part (c) (using Exercise 3.8.15) and assume that @ h˙ .; / D 0 for .; / 2 D. This leads to the equation   pCs F .s C 1; p C 1I p C 2I / (3.8.38) .p C 1/ s1  s1 s C F .s C 2; p C 2I p C 3I / D ˙.1  2 C  2 / 2 1 pC2 ! p p X X s s n 2 2 n1 2 s.  /  Cn . / C .1  2 C  / n Cn . / : p

nD0

nD1

Show now that the bracket on the left-hand side of (3.8.38) containing the hypergeometric functions can be reduced to .1 C /s1 . The bracket on the righthand side of (3.8.38) containing the Gegenbauer polynomials can be transformed s s 2 into .p C s/ pC1 Cp2 . /  .p C 1/ p CpC1 . / using the corresponding recurrence relations of Theorem 3.4.11. Prove the estimate s s     pCs 2 2 .1 C /s2 pCs pCs pC1 Cp . /  CpC1 . / ;  ˙ s s  1 .1  2 C  2 / 2 1 s1 1C (3.8.39)

and find the conclusion that D 1 or  D 0 which gives you the desired @ contradiction to the assumption @ h˙ .; / D 0 in D. Exercise 3.8.25. Let s D 1, p 2 N. Let D D .0; 1/ .1; 1/ and the functions h˙ W D ! R be defined as

3.8 Exercises

109

h˙ .; / D

pC1

  pCs F .s C 1; p C 1I p C 2I / s1

p p sX n s 2 ˙ 1 1 C   2  Cn2 . / nD0

D pC1 ˙ 1

p X p 1 C  2  2  n Pn . /;

(3.8.40)

nD0 1

where Pn . / D Cn2 . / is the Legendre polynomial of degree n. Show the following properties of h˙ : (a) h˙ .0; / D 0, (b) h .1; 0/ > 0 and hC .1; 0/ > 0, @ (c) @ h˙ .; / ¤ 0 in D. Hint: Note that properties (a) and (b) can be treated as in Exercise 3.8.24. Property (c) also starts out as in Exercise 3.8.24, but instead of inequalities as in (3.8.39) the equation p 1 D ˙ 1  2 C  2 .Pn . /  PnC1 . // (3.8.41) is squared and the resulting quadratic equation in  is shown to possess no realvalued solutions. Thereby, the recurrence relations of Theorem 3.4.11 can be very helpful if they are applied to the case of Legendre polynomials and their derivatives. Exercise 3.8.26. Let s 2 N, p 2 N. Let D D .0; 1/ .1; 1/ and the functions h˙ W D ! R be defined as in Exercise 3.8.24. Use the properties of Exercises 3.8.24 and 3.8.25 to prove that for all .; / 2 D hold h .; /  0 and hC .; /  0. Exercise 3.8.27. Let  2 .0; 1/, p 2 N0 and s 2 N. Use Exercise 3.8.26 to prove that the error term (3.8.27) can be estimated by   ˇ ˇ ˇ ˇ pCs ˇ ˇ s ˇ ˇ s F .s C 1; p C 1I p C 2I /; .; 1/ˇ D  pC1 ˇEpC1 .; /ˇ  ˇEpC1 s1 (3.8.42) where 2 Œ1; 1. Remark 3.8.28. One major application of the results of these exercises lies in error estimates for numerical algorithms involving truncated kernel expansions such that, e.g., the fast multipole method (see also the exercises of Chap. 4 or, e.g., Greengard and Rokhlin 1988) which goes back to Greengard (1988), Greengard and Rokhlin (1987), and Rokhlin (1985). A detailed derivation and error analysis of this kind can be found in Gutting (2008). Note that the case treated in Exercise 3.8.25 goes back to Sauter (1991). For more details the reader is referred to Gutting (2008) and the references therein.

Chapter 4

Scalar Spherical Harmonics in R3

As we have seen in the geomathematical motivation of Chap. 1, spherical functions are an essential tool for all geosciences. In this chapter we develop orthonormal function systems for scalar functions on spheres in the three-dimensional Euclidean space, namely the scalar spherical harmonics, which are then generalized to vectorial functions in Chap. 5 as well as to systems for scalar functions on spheres in the more general, q-dimensional setting in Chap. 6. The main features in each case are the addition theorem, the Funk–Hecke formula, and orthogonal invariance leading to expressions in the terms of Legendre polynomials. Scalar spherical harmonics have many fields of applications such as, e.g., geodesy and geophysics (see Sects. 1.1 or 1.2 for two examples), quantum mechanics (see, e.g, Landau and Lifshitz 2004), or numerical algorithms such as the fast multipole method of Greengard and Rokhlin (1988) (see Beatson and Greengard (1997) and the references therein for an overview or Gutting (2008) for an application in gravitational field modeling). All in all, they are essential for any analysis of spherical functions. The scalar spherical harmonics also provide the foundation for vector spherical harmonics (see Chap. 5). There is an extensive literature on spherical harmonics starting from Gauß (1801) and we only give a few recommendations: for a more mathematical point of view we refer to, e.g., Freeden et al. (1998), Freeden and Schreiner (2009), Hobson (1955), Hochstadt (1971), Lebedev (1973), Lense (1954), M¨uller (1966, 1998) and for physical aspects, in particular in quantum mechanics and in angular momentum theory, see Edmonds (1964), Jones (1985), Landau and Lifshitz (2004), Var˘salovi˘c et al. (1988) and Zare (1988). The layout of this chapter is as follows: In Sect. 4.1 we provide some basic notation for the sphere in R3 , in particular, we introduce zonal functions on the sphere. Orthogonal invariance is discussed for spaces of spherical functions in Sect. 4.2. We also lay the groundwork on orthogonal invariance for spherical vector fields which are elaborated in Chap. 5. As we will see in Sect. 4.3, we are naturally led to spherical harmonics by investigating polynomial basis systems on the sphere. We show their essential properties such as the addition theorem, orthogonality, and their role as eigenfunctions of the Beltrami operator. For their completeness in the W. Freeden and M. Gutting, Special Functions of Mathematical (Geo-)Physics, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-0348-0563-6 4, © Springer Basel 2013

113

4 Scalar Spherical Harmonics in R3

114

space of square-integrable functions on the sphere we propose and analyze two summability methods in Sect. 4.4, before the irreducibility of the space of spherical harmonics of a fixed degree is shown in Sect. 4.5. This yields the Funk–Hecke Formula. In Sect. 4.6 Green’s function on the sphere with respect to the Beltrami operator is introduced and corresponding integral theorems are established. This is another result that is required for vector spherical harmonics in Chap. 5. Moreover, we briefly present the quantum-mechanical modeling of the hydrogen atom in Sect. 4.7 as a well-known example of use of spherical harmonics in physics (see also Landau and Lifshitz 2004). Finally, the reader is encouraged to solve the exercises in Sect. 4.8 which are concerned with some important topics in approximation and numerical analysis on the sphere such as the low discrepancy method, locally supported spherical wavelets, function systems for the ball, and the fast multipole method.

4.1 Basic Notation We start by introducing some basic spherical notation. The unit sphere in R3 is defined by S2 D f 2 R3 W jj D 1g: (4.1.1) We use Greek letters for elements of S2 . The polar coordinate representation of a point x 2 R3 is given by 2 p 3 r p1  t 2 cos.'/ x.r; '; t/ D 4 r 1  t 2 sin.'/ 5 ; rt

(4.1.2)

where r D jxj 2 R0 is the distance to the origin, ' 2 Œ0; 2/ the longitude, t D cos.#/ 2 Œ1; 1 the polar distance and # 2 Œ0;  the latitude. The canonical basis in R3 is denoted by "1 ; "2 ; "3 . Another orthonormal basis is given by the moving frame consisting of the following three vectors (depending on the spherical coordinates ' and t): 2p 2 3 3  sin.'/ 1  t 2 cos.'/ p "r .'; t/ D 4 1  t 2 sin.'/ 5 ; "' .'; t/ D 4 cos.'/ 5 ; 0 t 2 3 t cos.'/ 5 "t .'; t/ D 4 t p sin.'/ : 1  t2

(4.1.3)

"' and "t are tangential vectors. Note that "r ^ "' D "t , where ^ denotes the vector product (see Fig. 4.1 for an illustration). The gradient r in R3 can be

4.1 Basic Notation

115

Fig. 4.1 The perpendicular vectors "r , "' , "t on the unit sphere S2

decomposed into a radial and an angular part, namely r D "r

1 @ C r; @r r

(4.1.4)

where the tangential operator r  is called the surface gradient and can be written in polar coordinates of (4.1.2) as follows: p 1 @ @ r  D "' p C "t 1  t 2 : 2 @' @t 1t

(4.1.5)

Another essential tangential operator is the surface curl gradient L which is defined by L F ./ D  ^ r F ./ for F 2 C.1/ .S2 /,  2 S2 . In the polar coordinates of (4.1.2) we have: p @ 1 @ : L D "' 1  t 2 C "t p 2 @t @' 1t

(4.1.6)

L F is a tangential vector field perpendicular to r  F , i.e., for  2 S2 , r F ./  L F ./ D 0:

(4.1.7)

We canonically define the surface divergence r  f ./ D

3 X

r Fi ./  "i

(4.1.8)

L Fi ./  "i ;

(4.1.9)

i D1

and the surface curl L  f ./ D

3 X i D1

where f D ŒF1 ; F2 ; F3 T 2 c.1/ .S2 /. Note that we use lower-case letters for vector fields and capital letters for scalar fields. The same convention applies to

4 Scalar Spherical Harmonics in R3

116

the corresponding function spaces such as c.1/ .S2 / for the space of continuously differentiable vector fields and C.1/ .S2 / for the space of continuously differentiable scalar fields on the sphere. The surface curl represents a scalar-valued function on the unit sphere: L  f ./ D r  .f ./ ^ /: (4.1.10) Finally, the Laplace operator  can be decomposed into D

1 2 @ @2 C 2  ; C @r 2 r @r r

(4.1.11)

where  denotes the Beltrami operator (sometimes also called Laplace–Beltrami operator) and 1 @2 @2 1 @ 2 @ .1  t C D / : 1  t 2 @' 2 @t @t 1  t 2 @' 2

 D Lt C

(4.1.12)

Note that Lt denotes the Legendre operator of (3.5.21), which is that part of the Beltrami operator depending on the polar distance t. Moreover, for F 2 C.2/ .S2 /, it holds that r  r F ./ D  F ./

and L  L F ./ D  F ./:

(4.1.13)

Definition 4.1.1 (Regular Region on the Sphere). A bounded region G  S2 is called regular, if its boundary @G is an orientable piecewise smooth Lipschitzian manifold of dimension 1. An example is a spherical cap of radius r around  2 S2 , i.e., C.; r/ D f 2 S2 W 1  r      1g. The following spherical versions of the theorems of Gauß and Stokes are wellknown (see, e.g., Freeden and Schreiner 2009 and the references therein). Theorem 4.1.2 (Surface Theorems of Gauß and Stokes). Suppose that   S2 is a regular region with continuously differentiable boundary curve @ . Let f be a .1/ tangential vector field of class ctan . /, i.e., f ./   D 0 for all  2  . Then, Z 

Z



r

Z  f ./ dS./ D

L  f ./ dS./ D

Z

  f ./ d ./;

(4.1.14)

  f ./ d ./;

(4.1.15)

@

@

where d is the arc element.

 denotes the positively oriented unit tangential vector of the boundary curve @ at  2 @ . The unit normal vector  points into the exterior of  and is perpendicular to  and , i.e.,  is perpendicular to @ at the point  2 @ , but tangential to S2 (see Fig. 4.2 for an illustration).

4.1 Basic Notation

117

Fig. 4.2 A region  on the sphere S2 and the vectors  ,  at a point  2 @

It is important to point out the assumption of the vector field f being tangential in Theorem 4.1.2. This causes an additional term in Green’s formulas involving r  , but it does not affect those for L , which is due to the fact that r    D 2, but L   D 0 for  2 S2 . The normal derivative is given by @F @

./ D   r F ./ D   L F ./

(4.1.16)

for F 2 C.1/ . /. Lemma 4.1.3. Let   S2 be a regular region with continuously differentiable boundary @ . Suppose that F , G are of class C.1/ . /. Then, Z 

G./r F ./ dS./ C

F ./r G./ dS./

G./L F ./ Z D

(4.1.17)

Z

 .F ./G.// d ./ C 2 @





Z D

Z

Z

.F ./G.// d ./; @

Z dS./ C 

F ./L G./ dS./

(4.1.18)

 .F ./G.// d ./: @

Theorem 4.1.4. Let G 2 C.2/ . /,   S2 be a regular region with a continuously differentiable boundary @ and a unit outward normal vector field . Then, we have (i) Green’s first surface identity for F 2 C.1/ . /, i.e., Z  Z      r G./  r F ./ dS./ C F ./ G./ dS./ 



Z D @

@ F ./ G./ d ./; @

(4.1.19)

4 Scalar Spherical Harmonics in R3

118

(ii) Green’s second surface identity for F 2 C.2/ . /, i.e., Z 

F ./ G./  G./ F ./ dS./ Z D

F ./ @

@ @ G./  G./ F ./ d ./: @ @

(4.1.20)

Proof. This is an immediate consequence of the surface Gauß theorem (Theorem 4.1.2). For details see, e.g., Freeden and Schreiner (2009) and the references therein. t u The aforementioned statements (Lemma 4.1.3 and Theorem 4.1.4) hold as well for the entire sphere S2 instead of a subregion  , thereby observing that the occurring boundary integrals vanish. For functions F 2 C.1/ .S2 / and tangential vector fields .1/ f 2 ctan .S2 /, this implies the following identities: Z S2

f ./ 

r F ./

f ./ 

L F ./

Z

S2

Z S2

Z dS./ D  Z

S2

dS./ D 

r  f ./ dS./ D

S2

Z S2

F ./r  f ./ dS./;

(4.1.21)

F ./L  f ./ dS./;

(4.1.22)

L  f ./ dS./ D 0:

(4.1.23)

Definition 4.1.5. Let  be a point on S2 . A function of the form G W S2 ! R, where  7! G ./ D G.  / with G W Œ1; 1 ! R, is called (-)zonal function on S2 . The application of our spherical differential operators to zonal functions yields, for F 2 C.1/ .Œ1; 1/, that r F .  / D F 0 .  /.  .  //;

(4.1.24)

L F .  / D F 0 .  /. ^ /;

(4.1.25)

while, for F 2 C.2/ .Œ1; 1/,  F .  / D 2.  /F 0 .  / C .1  .  /2 /F 00 .  /:

(4.1.26)

Theorem 4.1.6. Let G 2 L2 .Œ1; 1/. Then, for all  2 S2 , Z

Z S2

G.  / dS./ D 2

1 1

G.t/ dt:

(4.1.27)

4.2 Orthogonal Invariance

119

Proof. First, consider the case  D "3 . Note that p p  D Œ 1  t 2 cos.'/; 1  t 2 sin.'/; tT

(4.1.28)

with 1  t  1, 0  ' < 2. Therefore, "3   D t and Z

Z S2

G."3  / dS./ D

1 1

Z

2 0

ˇ ˇ ˇ @ @ ˇ ^ ˇˇ d' dt: G.t/ ˇˇ @' @t

(4.1.29)

We compute 2p 3 1  t 2 . sin.'// p @ D4 1  t 2 cos.'/ 5 ; @' 0 and find that

@ @ @'  @t 2 2

2 @ D @t

pt 2 6 p1t t 4 1t 2

cos.'/

3

7 sin.'/ 5 ;

(4.1.30)

1

ˇ ˇ p ˇ ˇ ˇ @ ˇ ˇ @ ˇ D 0, ˇ @' ˇ D 1  t 2 and ˇ @t ˇ D

p1 . 1t 2

Together with the rule

jx ^ yj D jxj jyj  .x  y/ for x; y 2 R this yields that 2

2

3

ˇ ˇ ˇ @ @ ˇ ˇ ˇ ˇ @' ^ @t ˇ D 1:

(4.1.31)

Thus, Z

Z S2

G"3 ./ dS./ D

1

1

Z

2

Z G.t/ d' dt D 2

0

1 1

G.t/ dt:

(4.1.32)

Now, let t 2 SO.3/ D fs 2 R33 W det s D 1g be a rotation with t D "3 , i.e.,  D tT "3 . Then, Z

Z S2

G.  / dS./ D Z

Z S2

D S2

G..tT "3 /  / dS./ D

S2

G."3  .t// dS./

G."3  ˛/ det tT dS.˛/ D 2

where  D tT ˛.

Z

1 1

G.t/ dt;

(4.1.33) t u

4.2 Orthogonal Invariance Systems of equations which maintain their form when the coordinate axes are subjected to an arbitrary rotation are said to be rotationally, or orthogonally, invariant. The orthogonal invariance is, of course, closely related to the group

4 Scalar Spherical Harmonics in R3

120

O.3/ of all orthogonal transformations, i.e., the group of all t 2 R33 such that ttT D tT t D i, i D .ıi;j /i;j D1;2;3 denotes the unit matrix of R33 . The set of all rotations, i.e., SO.3/ D ft 2 O.3/ W det t D 1g is a subgroup called the special orthogonal group. We briefly recapitulate some properties of these groups (see, e.g., M¨uller (1998) and Vilenkin (1968) and many others): 1. Let ;  be members of S2 . Then, there exists an orthogonal transformation t 2 O.3/ with  D t and an orthogonal transformation s 2 SO.3/ with  D s. 2. For every t 2 O.3/, t  t D   ; ;  2 S2 : (4.2.1) 3. Suppose that  2 S2 . The set O .3/ D ft 2 O.3/ W t D g is a subgroup of O.3/. Analogously, the set SO .3/ D ft 2 SO.3/ W t D g is a subgroup of SO.3/. 4. For every t 2 O.3/, we have det t D ˙1. If det t D 1, t is called a rotation, while for det t D 1, t is called a reflection . 5. Let t; t0 2 O.3/ with det t D 1, det t0 D 1. Then, t ^ t D t. ^ /; t0  ^ t0  D t0 . ^ /;

;  2 S2 ; ;  2 S2 :

(4.2.2) (4.2.3)

The following definitions will prove useful for our later considerations. Definition 4.2.1. Let F 2 L2 .S2 /, f 2 l2 .S2 /, and t 2 O.3/. For scalar and vector fields the operator Rt is defined by Rt W L2 .S2 / ! L2 .S2 /; Rt W l2 .S2 / ! l2 .S2 /;

Rt F ./ D F .t/;

(4.2.4)

Rt f ./ D tT f .t/;

(4.2.5)

respectively. Rt F and Rt f are called the t-transformed fields. Remark 4.2.2. Note that l2 .S2 / denotes the Hilbert space of square-integrable vector fields on the unit sphere S2 with the canonical scalar product. The Banach space of continuous vector fields is given by c.S2 / with the canonical norm. For further details on the notation for spherical vector fields, see Sect. 5.1. For examples illustrating how the operators Rt act on functions and vector fields, see Figs. 4.3 and 4.4, respectively. Definition 4.2.3. Let X be a subspace of L2 .S2 / or l2 .S2 /. X is called invariant with respect to orthogonal transformations or, equivalently, orthogonally invariant if, for all F 2 X and for all orthogonal transformations t 2 O.3/, the function Rt F is of class X . An orthogonally invariant space X is called reducible if there exists a proper subspace X 0  X which itself is invariant with respect to orthogonal transformations.

4.2 Orthogonal Invariance

121

Fig. 4.3 The operator Rt acting on a function

T

Fig. 4.4 The definition of the operator Rt for vector fields (note that it is necessary not only to substitute  by t, but also to transform the directions of the vectors)

Note that the expressions invariant with respect to rotations and invariant with respect to reflections are understood in analogy to the aforementioned definition. A linear, orthogonally invariant space which is not reducible is called irreducible. It should be noted that each orthogonally invariant space of dimension 1 is irreducible. Lemma 4.2.4. Let .X; h; i/ be an orthogonally invariant Hilbert subspace of the space L2 .S2 /. Let X1 be an orthogonally invariant subspace of X . Then, the orthogonal complement X1? of X1 is orthogonally invariant as well. Proof. For all F 2 X1 , F ? 2 X1? , and for all orthogonal transformations t 2 O.3/, we have Z hF; Rt F ? i D F ./F ? .t/ dS./ (4.2.6) Z

S2

D S2

F .tT /F ? ./ dS.tT / Z

D j det tT j S2

Z D S2

F .tT /F ? ./ dS./

RtT F ./F ? ./ dS./ D 0;

4 Scalar Spherical Harmonics in R3

122

since RtT F 2 X1 . This implies that Rt F ? 2 X1? and, therefore, X1? is invariant with respect to orthogonal transformations. t u Analogous results can be formulated for Hilbert spaces of square-integrable vector and tensor fields (cf. Freeden and Schreiner 2009). Lemma 4.2.4 shows that each orthogonally invariant Hilbert space can be completely decomposed into invariant parts. In view of the last result, we are particularly interested in irreducible spaces, i.e., spaces that provide us with elements that are invariant with respect to certain orthogonal transformations. The following results (see, e.g., Gervens 1989) help us to analyze the structure of such rotationally invariant functions. Lemma 4.2.5. Let F be a function of class C.S2 / with Rt F ./ D F ./ for all transformations t 2 SO.3/ and all  2 S2 . Then, F D F ."3 / D C D const:

(4.2.7)

Proof. For all  2 S2 , there exists a rotation t 2 SO.3/ with t D "3 . Consequently, for every  2 S2 , we have F ./ D Rt F ./ D F .t/ D F ."3 / D C D const. t u Theorem 4.2.6. Let  2 S2 be fixed. Furthermore, suppose F is of class C.S2 / with Rt F ./ D F ./ for all t 2 SO .3/ and for all  2 S2 . Then, F can be represented in the form F ./ D ˚.  /;  2 S2 ; (4.2.8) ˚ being a function ˚ W Œ1; 1 ! R. Proof. Without loss of generality, let  D "3 (if this were not true, we could use the function G./ D Rt0 F ./, where t0 2 O.3/ with t0 "3 D ). With  D t "3 C p 1  t 2 0 we have, by assumption, that F .t "3 C

p p 1  t 2 0 / D F .t "3 C 1  t 2 00 /;

(4.2.9)

for all points 0 ; 00 of the unit circle. Hence, F depends only on t D   "3 and is, therefore, a function of t alone, as desired. u t Lemma 4.2.7. Let  2 S2 be fixed. Let F 2 C.S2 / with Rt F ./ D .det t/F ./ for all t 2 O .3/ and all  2 S2 . Then, F D 0. Proof. Suppose that  is an element of S2 . There exists a reflection t 2 O .3/ with t D , but then—by assumption—we have F ./ D Rt F ./ D F ./, hence, F ./ D 0. t u Note that in Lemma 4.2.5 and Theorem 4.2.6, the rotations can as well be replaced by reflections, i.e., in the scalar case, we need not distinguish between rotations and reflections. In the vectorial case, however, this is not true anymore. In what follows, f is supposed to be a spherical vector field, i.e., f W S2 ! R3 . Let  be an element of S2 . In every point  ¤ ˙, we are able to introduce the

4.2 Orthogonal Invariance

123

so-called moving frame at the point , namely "1 D ;

(4.2.10)

1 .  .  //; "2 D p 1  .  /2

(4.2.11)

1 "3 D p  ^ ; 1  .  /2

(4.2.12)

such that there exist functions F1 ; F2 ; F3 W S2 ! R with f D F1 "1 C F2 "2 C F3 "3 :

(4.2.13)

For further investigations, the following lemma is helpful. Lemma 4.2.8. Let  2 S2 be fixed. Let the moving frame "i , i D 1; 2; 3, be defined as in (4.2.10)–(4.2.12). Then, for all t 2 O.3/, Rt "i D "i ;

i D 1; 2;

Rt "3 D .det t/"3 :

(4.2.14)

Proof. For t 2 O .3/, Rt "1 D tT "1t D tT t  D  D "1 :

(4.2.15)

For the tangential fields, we show the case i D 2. We have 1 tT .  .t  /t/ Rt "2 D tT "2t D p 1  .t  /2

(4.2.16)

1 D p .tT   .  tT /tT t/ 1  .  tT /2 1 D p .  .  // D "2 : 1  .  /2 The case i D 3 follows similarly.

t u

We now extend our results for rotationally invariant functions to the vector case. Lemma 4.2.9. Let f 2 c.S2 / with Rt f ./ D f ./ (or equivalently, f .t/ D tf ./) for all t 2 SO.3/ and  2 S2 . Then, there exists a constant C 2 R such that (4.2.17) f ./ D C ;  2 S2 :

4 Scalar Spherical Harmonics in R3

124

Proof. Consider the orthogonal matrix 2

1 0 t D 4 0 1 0 0

3 0 05 : 1

(4.2.18)

Then, t"3 D "3 and, by assumption, f ."3 / D tf ."3 /. Hence, in connection with (4.2.18), we have f ."3 / D C "3 , C 2 R. For  2 S2 , there exists a rotation t0 with t0 "3 D . Consequently, we have f ./ D f .t0 "3 / D t0 f ."3 / D C t0 "3 D C . t u Theorem 4.2.10. Let  2 S2 . Let f 2 c.S2 /, i.e., f is a continuous vector field on the unit sphere S2 , with Rt f ./ D f ./ for all t 2 SO .3/. Then, for  ¤ ˙, f has the representation: f ./ D ˚1 .  /"1 C ˚2 .  /"2 C ˚3 .  /"3 ;

(4.2.19)

where ˚i , i D 1; 2; 3, are functions ˚i W Œ1; 1 ! R. Proof. From Lemma 4.2.8, it follows that the functions Fi in (4.2.13) fulfill Rt Fi ./ D Fi ./;

 2 S2 ;

(4.2.20)

provided that t D . Therefore, via Theorem 4.2.6, we know that, for the functions Fi , we have Fi ./ D ˚i .  /. t u Corollary 4.2.11. Suppose that  2 S2 . Let f be of class c.S2 / with Rt f ./ D f ./ for all t 2 O .3/. Then, for  ¤ ˙, f has the representation, f ./ D ˚1 .  /"1 C ˚2 .  /"2 ;

(4.2.21)

˚i , i D 1; 2, being functions ˚i W Œ1; 1 ! R. Proof. Starting from Theorem 4.2.10, we now have to consider reflections, as well. By our assumption and Lemma 4.2.8, we get ˚3 .  / D ˚3 .  / and, therefore, ˚3 .  / D 0. t u Corollary 4.2.12. Suppose that  2 S2 . Let f be of class c.S2 / with Rt f ./ D .det t/ f ./ for all t 2 O .3/. Then, for  ¤ ˙, the field f can be represented as follows: f ./ D ˚3 .  /"3 (4.2.22) with ˚3 being a function ˚3 W Œ1; 1 ! R. Proof. Using the same reasoning as in the proof of Corollary 4.2.11, but now considering the change in sign under reflections, we end up with ˚1 .  / D ˚1 .  /; ˚2 .  / D ˚2 .  /; hence, ˚1 .  / D ˚2 .  / D 0.

(4.2.23) t u

4.3 Homogeneous Polynomials on the Unit Sphere in R3

125

4.3 Homogeneous Polynomials on the Unit Sphere in R3 In this section we consider spherical polynomials by first analyzing homogeneous polynomials on R3 , then requiring them to be harmonic and, finally, restricting these homogeneous harmonic polynomials to the unit sphere. This defines spherical harmonics and shows us how they inherit many of their properties. Such an approach to spherical harmonics is motivated by some hints in Stein and Weiss (1971). In its present form it can be found in Freeden (1979). Remark 4.3.1. Note that we present the theory of spherical harmonics in this chapter in a real framework. It can easily be carried over to the complex-valued case which is chosen in Chap. 6 for the generalized q-dimensional setting. Definition 4.3.2. A polynomial P on Rn , n 2 N, is called homogeneous of degree m 2 N if P . x/ D m P .x/ for all 2 R and all x 2 Rn . The space of all homogeneous polynomials of degree m on Rn is denoted by Homm .Rn / D

8 < :

P W P .x/ D

X Œ˛Dm

9 =

C˛ x ˛ ; x 2 Rn ; ˛ 2 Nn0 : ;

(4.3.1)

Pn Note that for the multi-index it holds that Œ˛ D i D1 ˛i (see the last part of Sect. 6.1 for a general overview on multi-indices). The set of restrictions of homogeneous polynomials to a set D  Rn is defined by Homm .D/ D fP jD W P 2 Homm .Rn /g : (4.3.2) Example 4.3.3. P .x/ D x12 C x22 C x32 2 Hom2 .R3 /, therefore, P jS2 2 Hom2 .S2 / although P jS2  1, i.e., deg.P jS2 / D 0. The index m of Homm .S2 / refers to the degree of the original polynomial on R3 . Theorem 4.3.4. The set of functions fx 7! x ˛ gŒ˛Dn is a basis of Homn .R3 / and M.n/ D dim.Homn .R3 // D .nC1/.nC2/ . 2 Proof. Homogeneous polynomials of degree n in R3 have the form X

P .x/ D

Œ˛Dn

C˛ x ˛ D

X ˛1 C˛2 C˛3 Dn

C˛1 ;˛2 ;˛3 x1˛1 x2˛2 x3˛3 ;

(4.3.3)

where ˛1 ; ˛2 ; ˛3 2 N0 . Let P .x/ D 0. Then, n X ˛1 D0

0 @

n˛ X1 ˛2 D0

1 C˛1 ;˛2 ;n˛1 ˛2 x2˛2 x3n˛1 ˛2 A x1˛1 D 0:

(4.3.4)

For x2 ; x3 2 R fixed, we have a polynomial in R with respect to x1 which is zero for all x1 2 R. Therefore,

4 Scalar Spherical Harmonics in R3

126 n˛ X1 ˛2 D0

C˛1 ;˛2 ;n˛1 ˛2 x2˛2 x3n˛1 ˛2 D 0:

(4.3.5)

Now, keep only x3 fixed and we obtain a polynomial in x2 which is zero for all x2 2 R. Thus, C˛1 ;˛2 ;n˛1 ˛2 x3n˛1 ˛2 D 0

)

C˛1 ;˛2 ;n˛1 ˛2 D 0

(4.3.6)

for all ˛ 2 N30 with Œ˛ D n. Therefore, the set fx 7! x ˛ gŒ˛Dn is a basis of Homn .R3 /. The dimension is equal to the number of monomials x ˛ with Œ˛ D n, i.e., #fx 7! ˛ x gŒ˛Dn . We have n C 1 choices for ˛1 2 f0; : : : ; ng and n C 1  ˛1 choices for ˛2 2 f0; : : : ; n  ˛1 g. In the end, there remains one choice for ˛3 D n  ˛1  ˛2 . This gives us M.n/ D dim.Homn .R3 // D

n X

.n C 1  ˛1 / D

˛1 D0

.n C 1/.n C 2/ : 2

(4.3.7) t u

Thus, we have found the desired basis properties in Homn .R3 /.

Remark 4.3.5. In (4.3.7), we have introduced the notation M.n/ D M.3I n/ D dim.Homn .R3 // in accordance with the more general q-dimensional setting in Chap. 6 which uses M.qI n/ D dim.Homn .Rq //. Formally, we can insert the gradient operator in R3 into a homogeneous polynomial Hn 2 Homn .R3 /, i.e., Hn .r/ D

X



Œ˛Dn

If Un 2 Homn .R3 / with Un .x/ D Hn .r/Un .x/ D

X X

P

C˛ CQ ˇ

Œ˛Dn ŒˇDn

ŒˇDn



@Œ˛ @x1˛1 @x2˛2 @x3˛3

:

(4.3.8)

CQ ˇ x ˇ , it holds that

@ @x1

˛1

 ˇ x1 1

@ @x2

˛2

 ˇ x2 2

@ @x3

X

C˛ CQ ˛

Œ˛Dn

ˇ

x3 3 : (4.3.9)

Now, all summands are zero except the ones with ˛ D ˇ such that Hn .r/Un .x/ D

˛3

 3  Y X @ ˛i ˛i xi D C˛ CQ ˛ ˛1 Š ˛2 Š ˛3 Š 2 R: „ ƒ‚ … @x i i D1 Œ˛Dn

˛Š

(4.3.10)

Lemma 4.3.6. The mapping h;  iHomn .R3 / W Homn .R3 /  Homn .R3 / ! R

(4.3.11)

4.3 Homogeneous Polynomials on the Unit Sphere in R3

127

given by hHn ; Un iHomn .R3 / D Hn .rx /Un .x/

(4.3.12)

3

defines an inner product on Homn .R /. The set of monomials 

 1 ˛ ˇˇ 3 x 7! p x ˇ˛ 2 N0 ; Œ˛ D n (4.3.13) ˛Š  is an orthonormal system in the Hilbert space Homn .R3 /; h;  iHomn .R3 / . Proof. If Hn ; Un 2 Homn .R3 /, then hHn ; Hn iHomn .R3 / D Hn .rx /Hn .x/ D

X

C˛2 ˛Š  0

(4.3.14)

Œ˛Dn

and the scalar product of a homogeneous P polynomial Hn of degree n with itself is hHn ; Hn iHomn .R3 / D 0 if and only if Œ˛Dn C˛2 ˛Š D 0. However, this is equivalent to C˛2 D 0 for all multi-indices ˛ 2 N30 with Œ˛ D n, i.e., with Hn  0. For the symmetry we find that X

hHn ; Un iHomn .R3 / D

C˛ CQ ˛ ˛Š D hUn; Hn iHomn .R3 / :

(4.3.15)

Œ˛Dn

Let also Vn 2 Homn .R3 / and ; 2 R. Then hHn ; Un C Vn iHomn .R3 / D

X

C˛ . CQ ˛ C D˛ /˛Š

Œ˛Dn

D hHn ; Un iHomn .R3 / C hHn ; Vn iHomn .R3 / :

(4.3.16)

For the monomials X 1 1 0 M˛.n/.x/ D p x ˛ D p ı˛;˛0 x ˛ ; ˛Š ˛Š Œ˛ 0 Dn

(4.3.17)

we obtain hM˛.n/; M˛.n/iHomn .R3 / D

2 X  1 p ı˛;˛0 ˛ 0 Š D 1; ˛Š Œ˛ 0 Dn

(4.3.18)

and for ˛ ¤ ˛ 0 ; .n/

hM˛.n/; M˛Q iHomn .R3 / D

X

1 1 p ı˛;˛0 p ı˛0 ;˛Q ˛ 0 Š D 0: ˛Š ˛Š Q Œ˛ 0 Dn

(4.3.19)

4 Scalar Spherical Harmonics in R3

128

Since the monomials form a basis of the finite dimensional space Homn .R3 / by Theorem 4.3.4 we also have completeness. t u By the fundamental theorem of Fourier analysis an orthonormal expansion of the homogeneous polynomial Hn 2 Homn .R3 / is given by X

Hn .x/ D

hHn ; M˛.n/ iHomn .R3 / M˛.n/.x/ D

Œ˛Dn

D Hn .ry /

X

1 1 Hn .ry / p y ˛ p x ˛ ˛Š ˛Š Œ˛Dn

1 X nŠ ˛ ˛ y x : nŠ ˛Š

(4.3.20)

Œ˛Dn

Thus, we have Hn .x/ D Hn .ry /

 D E 1 .x  y/n D Hn ; .x /n : Homn .R3 / nŠ nŠ

1

(4.3.21)

Definition 4.3.7. Let V ¤ ; and X be a Hilbert space on the set V . A kernel K.; / W V  V ! R

(4.3.22)

is called reproducing kernel for X if (i) K.x; / 2 X for all x 2 V , (ii) hF; K.x; /iX D F .x/ for all x 2 V and all F 2 X . If such a kernel exists in X , then X is called a reproducing kernel Hilbert space. It should be noted that a reproducing kernel is always unique (see, e.g., Aronszajn 1950; Davis 1963). Theorem 4.3.8. The mapping KHomn .R3 / W R3  R3 ! R; 1 .x  y/n nŠ  is the reproducing kernel in Homn .R3 /; h;  iHomn .R3 / . .x; y/ 7! KHomn .R3 / .x; y/ D

(4.3.23)

Proof. The second property of Definition 4.3.7 is shown in (4.3.21). The first property holds since .x /n is a homogeneous polynomial of degree n for any fixed x 2 R3 . Thus, KHomn .R3 / .x; / 2 Homn .R3 / for any fixed x 2 R3 . t u Theorem 4.3.9 (Addition Theorem for Homn .R3 /). Let fHn;j gj D1;:::;M.n/ , where M.n/ D 12 .nC1/.nC2/, be an orthonormal system in Homn .R3 /. For all x; y 2 R3 , it holds that X

M.n/ j D1

Hn;j .x/Hn;j .y/ D

1 .x  y/n D KHomn .R3 / .x; y/: nŠ

(4.3.24)

4.3 Homogeneous Polynomials on the Unit Sphere in R3

129

Proof. Since the reproducing kernel in a Hilbert space is unique, we have to show PM.n/ that j D1 Hn;j .x/Hn;j .y/ is a reproducing kernel in Homn .R3 /. Clearly, X

M.n/

X

M.n/

Hn;j .x/Hn;j 2 Homn .R3 / and

j D1

Hn;j Hn;j .y/ 2 Homn .R3 /

(4.3.25)

j D1

for any fixed x; y 2 R3 . Let Hn 2 Homn .R3 /, then X

M.n/

Hn .x/ D

hHn ; Hn;j iHomn .R3 / Hn;j .x/

(4.3.26)

j D1

and we obtain M.n/ D E X Hn ; Hn;j .x/Hn;j j D1

X

M.n/

Homn .R3 /

D

Hn;j .x/hHn ; Hn;j iHomn .R3 / D Hn .x/:

j D1

(4.3.27) PM.n/ Therefore, the kernel defined by j D1 Hn;j .x/Hn;j .y/ is the reproducing kernel in the Hilbert space Homn .R3 / with the inner product h;  iHomn .R3 / . t u

Interpolation Problem Given measurements .xj ; bj / 2 R3  R, j D 1; : : : ; M.n/, determine F 2 Homn .R3 / with the property F .xj / D bj ;

j D 1; : : : ; M.n/ D 12 .n C 1/.n C 2/:

(4.3.28)

When does a solution exist? If a solution exists, how does it look like? Definition 4.3.10. A system of M.n/ points fx1 ; : : : ; xM.n/ g  R3 is called a fundamental system relative to Homn .R3 / if the matrix A with the entries Aj;k D Hn;j .xk /, j; k D 1; : : : ; M.n/, is regular, where fHn;j g is an orthonormal system in Homn .R3 /. Lemma 4.3.11. For each n 2 N0 , there exists a fundamental system relative to the space Homn .R3 /. Proof. Let n 2 N0 be fixed. We construct the points fx1 ; : : : ; xM.n/ g inductively. Clearly there exists a point x1 2 R3 with Hn;1 .x1 / ¤ 0:

(4.3.29)

4 Scalar Spherical Harmonics in R3

130

Now, assume there exist fx1 ; : : : ; xm g, m < M.n/, such that

Hn;j .xk / j;kD1;:::;m

(4.3.30)

is regular. Furthermore, assume that for every y 2 R3 n fx1 ; : : : ; xm g the matrix 2

3 Hn;1 .x1 / : : : : : : Hn;1 .xm / Hn;1 .y/ 6 7 :: :: :: 6 7 : : : 6 7 6 7 :: :: :: 4 5 : : : Hn;mC1 .x1 / : : : : : : Hn;mC1 .xm / Hn;mC1 .y/

(4.3.31)

is singular. This means that there exists a row i 2 f1; : : : ; m C 1g such that this row can be represented as a linear combination of the other rows

Hn;j .x1 /; : : : ; Hn;j .xm /; Hn;j .y/ ; j 2 f1; : : : ; m C 1g n fi g: (4.3.32) In particular, Hn;i .y/ can be represented by a linear combination of the functions fHn;j .y/gj 2f1;:::;mC1gnfi g for all y 2 R3 . But this is a contradiction to the linear independence of fHn;1 ; : : : ; Hn;mC1 g. This inductive construction can be continued until we reach m D M.n/ D 12 .n C 1/.n C 2/. t u If the points of the interpolation problem are a fundamental system relative to Homn .R3 /, the problem can be solved uniquely. Theorem 4.3.12. Let fHn;j gj D1;:::;M.n/ be an orthonormal system in Homn .R3 / and fxk gkD1;:::;M.n/ be a fundamental system relative to Homn .R3 /. Then, each homogeneous polynomial F 2 Homn .R3 / is uniquely representable in the form (x 2 R3 ) X

M.n/

F .x/ D

X

M.n/

aj KHomn .R3 / .xj ; x/ D

j D1

j D1

aj

.xj  x/n : nŠ

(4.3.33)

Proof. It is clear that F can be written as X

M.n/

F D

bk Hn;k ;

(4.3.34)

kD1

where bk D hF; Hn;k iHomn .R3 / , k D 1; : : : ; M.n/. Let faj gj D1;:::;M.n/ be the solution of the system of linear equations X

M.n/ j D1

aj Hn;k .xj / D bk ;

k D 1; : : : ; M.n/:

(4.3.35)

4.3 Homogeneous Polynomials on the Unit Sphere in R3

131

The system is uniquely solvable since the points xk form a fundamental system (thus, the matrix is regular). Then, XX

M.n/ M.n/

F D

X

M.n/

aj Hn;k .xj /Hn;k D

kD1 j D1

aj KHomn .R3 / .xj ; /:

(4.3.36)

j D1

t u Note that the system can be very ill-conditioned.

Harmonic Polynomials Definition 4.3.13. Let D  R3 be open and connected. F 2 C.2/ .D/ is called harmonic if x F .x/ D 0 for all x 2 D. The set of all harmonic functions in C.2/ .D/ is denoted by Harm.D/. Definition 4.3.14. The set of all homogeneous harmonic polynomials on R3 with degree m 2 N0 is denoted by ˚ Harmm .R3 / D P 2 Homm .R3 / W P D 0 :

(4.3.37)

Moreover, we define for m 2 N0 ; Harm0;:::;m .R3 / D

m M

Harmi .R3 /;

(4.3.38)

Harm0;:::;i .R3 /:

(4.3.39)

i D0

Harm0;:::;1 .R3 / D

1 [ i D0

Let H 2 Homn .R3 /. There is a representation H.x/ D

n X

j

Anj .x1 ; x2 /x3 ;

(4.3.40)

j D0

where Anj 2 Homnj .R2 /. Now, let H 2 Harmn .R3 /: 0 D x H.x/ D

 n  2 n X X @ @2 j @2 j A C .x ; x /x C x Anj .x1 ; x2 /: nj 1 2 3 2 2 @x1 @x2 @x32 3 j D0 j D0 (4.3.41)

4 Scalar Spherical Harmonics in R3

132

Note that  2  @ @2 C 2 Ar .x1 ; x2 / D x;2 Ar .x1 ; x2 / D 0 @x12 @x2

for r 2 f0; 1g:

(4.3.42)

Therefore,  n2  2 n X X @ @2 j j 2 0D C 2 Anj .x1 ; x2 /x3 C j.j  1/x3 Anj .x1 ; x2 / 2 @x @x 1 2 j D0 j D2 D

n2 X j D0

j x;2 Anj .x1 ; x2 /x3

n2 X j C .j C 2/.j C 1/x3 Anj 2 .x1 ; x2 /: (4.3.43) j D0

Theorem 4.3.15. Let n 2 N0 . Let An and An1 be homogeneous polynomials of degree n and n  1 in R2 . For j D 0; : : : ; n  2, we define recursively Anj 2 .x1 ; x2 / D 

1 x;2 Anj .x1 ; x2 /: .j C 1/.j C 2/

(4.3.44)

Then, H W R3 ! R given by H.x1 ; x2 ; x3 / D

n X j D0

j

Anj .x1 ; x2 /x3

(4.3.45)

is in Harmn .R3 /. Moreover, dim.Harmn .R3 // D 2n C 1:

(4.3.46)

Proof. We already know that H 2 Harmn .R3 / from our considerations before. The degrees of freedom for H occur in the choices of An and An1 . Since Aj is of class Homj .R2 /, we know it can be represented by Aj .x/ D

X

C˛ x ˛ ;

(4.3.47)

Œ˛Dj

where ˛ 2 N20 . Thus, dim.Homj .R2 // D j C 1. This yields dim.Harmn .R3 // D dim.Homn .R2 // C dim.Homn1 .R2 //

(4.3.48)

D .n C 1/ C ..n  1/ C 1// D 2n C 1; which shows the remaining part of Theorem 4.3.15.

t u

Theorem 4.3.16. For n  2, the space Homn .R3 / can be decomposed into the orthogonal and direct sum

4.3 Homogeneous Polynomials on the Unit Sphere in R3 3 Homn .R3 / D Harmn .R3 / ˚ Harm? n .R /

133

(4.3.49)

with 3 2 3 Harm? (4.3.50) n .R / D j  j Homn2 .R / ˚ D Ln .x/ D jxj2 Hn2 .x/ W Hn2 2 Homn2 .R3 / :

Proof. (4.3.49) is just the standard decomposition of functional analysis. For the proof of (4.3.50), let n  2, Hn2 2 Homn2 .R3 / and Kn 2 Harmn .R3 / be arbitrary. Then, hj  j2 Hn2 ; Kn iHomn .R3 / D hHn2 j  j2 ; Kn iHomn .R3 / D Hn2 .rx /x Kn .x/ D 0: (4.3.51) 3 Therefore, j  j2 Hn2 2 Harm? n .R /. Because of Theorem 4.3.4 and (4.3.46) in Theorem 4.3.15 we easily calculate that 3 3 3 dim.Harm? n .R // D dim.Homn .R //  dim.Harmn .R //

D dim.Homn2 .R3 //; which gives us the desired equality of the spaces.

(4.3.52) t u

Corollary 4.3.17. Applying Theorem 4.3.16 iteratively, we obtain the decomposition bn=2c M j  j2i Harmn2i .R3 /; (4.3.53) Homn .R3 / D i D0

where

b n2 c

is the largest integer that is less than or equal to n2 .

Remark 4.3.18. Let fHn;j gj Dn;:::;n be an orthonormal system in Harmn .R3 /. This system can be completed to an orthonormal system in Homn .R3 /, i.e., fHn;j gj Dn;:::;n [ fUn;j gj D1;:::;dn2 ;

(4.3.54)

3 where fUn;j gj D1;:::;dn2 , dn2 D 12 n.n1/, is an orthonormal system in Harm? n .R /. Note that in view of the typical indexing of spherical harmonics we also use the indices n; : : : ; n for the 2n C 1 orthonormal functions of Harmn .R3 /.

Lemma 4.3.19. For Hn 2 Homn .R3 /, we have ProjHarmn .R3 / Hn .x/ D

 bn=2c X sD0

.1/s

 nŠ.2n  2s/Š jxj2s s Hn .x/: .2n/Š.n  s/ŠsŠ

(4.3.55)

4 Scalar Spherical Harmonics in R3

134

Proof. First, we prove Hobson’s formula by induction (cf. Hobson 1955) 

 bn=2c  X nŠ.2n  2s/Š jxj2s s xin .1/s .2n/Š.n  s/ŠsŠ sD0 (4.3.56) for x 2 R3 n f0g, i 2 f1; 2; 3g, n 2 N0 . To prove this, we need to remember that @ @xi

n

1 .2n/Š 1 D .1/n n jxj 2 nŠ jxj2nC1

( s xin

D

nŠ x n2s .n2s/Š i

;

n  2s;

0

;

else;

(4.3.57)

and @  2s2n1 n2s jxj D.1/.2n  2s C 1/jxj2s2.nC1/1 xinC12s xi @xi

(4.3.58)

C .n  2s/jxj2s2n1 xin2s1 : Everything else consists only of lengthy, but uneventful transformations to get (4.3.56) in the end. Because of the invariance of the operator  with respect to orthogonal transformations we find from (4.3.56) that 1 .2n/Š 1 .y  rx / D .1/n n jxj 2 nŠ jxj2nC1 n

 bn=2c X

 nŠ.2n  2s/Š 2s s jxj  .y  x/n .1/ .2n/Š.n  s/ŠsŠ sD0 (4.3.59) s

for all y 2 R3 , x 2 R3 n f0g, n 2 N0 . In more detail, we choose t 2 O.3/ such that t"i D y and set z D tx. This yields @ D "i  rx D "i  tT rz D y  rz ; @xi

(4.3.60)

z D x ; p jzj D x T tT tx D jxj; "i  x D ."i /T tT tx D y  z: Together this gives us (4.3.59). We know from Theorem 4.3.12 that each Hn 2 Homn .R3 / may be represented by M.n/ X Hn .x/ D aj .xj  x/n ; x 2 R3 ; (4.3.61) j D1

where the aj 2 R, j D 1; : : : ; M.n/, are suitable coefficients and the points xj 2 R3 , j D 1; : : : ; M.n/, form a fundamental system relative to Homn .R3 /. Therefore, we use (4.3.59) with xj as y:

4.3 Homogeneous Polynomials on the Unit Sphere in R3

Hn .rx /

135

M.n/ X 1 1 D aj .xj  rx /n jxj jxj j D1

X

M.n/

D

aj .1/n

j D1

D .1/n

(4.3.62)

.2n/Š 1 2n nŠ jxj2nC1

.2n/Š 1 2n nŠ jxj2nC1

 bn=2c X

 bn=2c X

.1/s

sD0

.1/s

sD0

 nŠ.2n  2s/Š jxj2s s .xj  x/n .2n/Š.n  s/ŠsŠ

 nŠ.2n  2s/Š jxj2s s Hn .x/: .2n/Š.n  s/ŠsŠ

For x ¤ 0, we have by Theorem 4.3.16 that Hn D Kn C j  j2 Hn2 , where the first function Kn 2 Harmn .R3 / and the second function Hn2 2 Homn2 .R3 /. This provides us with 1 .2n/Š 1 1 1 D Kn .rx / C Hn2 .rx /x D .1/n Kn .x/; jxj jxj jxj nŠ2n jxj2nC1 (4.3.63) 1 since x jxj D 0, x Kn .x/ D 0 and only the summand for s D 0 remains in (4.3.62). Finally, we obtain Hn .rx /

nŠ2n 1 jxj2nC1 Hn .rx / (4.3.64) .2n/Š jxj 0 1 bn=2c X nŠ.2n  2s/Š D@ jxj2s s A Hn .x/: .1/s .2n/Š.n  s/ŠsŠ sD0

ProjHarmn .R3 / Hn .x/ D Kn .x/ D .1/n

t u

This concludes the proof of Lemma 4.3.19. Remark 4.3.20. Additionally, we find in the course of the previous proof that

ProjHarm? (4.3.65) 3 Hn .x/ D Hn .x/  Kn .x/ n .R / 1 0 bn=2c X nŠ.2n  2s/Š jxj2s s A Hn .x/: .1/s1 D@ .2n/Š.n  s/ŠsŠ sD1 Note that the sum here starts at s D 1. Theorem 4.3.21 (Addition Theorem in Harmn .R3 /). Let fHn;j gj Dn;:::;n be an orthonormal system in Harmn .R3 / with respect to h ;  iHomn .R3 / . For every x; y 2 R3 with x D jxj, y D jyj (;  2 S2 ), we obtain n X j Dn

Hn;j .x/Hn;j .y/ D

2n nŠ n n jxj jyj Pn .  /; .2n/Š

where Pn denotes the Legendre polynomial of degree n.

(4.3.66)

4 Scalar Spherical Harmonics in R3

136

Proof. Let fKn;j gj D1;:::;M.n/ be the orthonormal system in Homn .R3 / which completes the basis from fHn;j g. Due to the uniqueness of the reproducing kernel in Homn .R3 / we know that X

M.n/

Kn;j .x/Kn;j .y/ D

j D1

.x  y/n nŠ

(4.3.67)

for x; y 2 R3 , where M.n/ D 12 .nC1/.nC2/. We project both sides to Harmn .R3 /: ProjHarmn .R3 /

 M.n/ X

 Kn;j .x/Kn;j .y/ D

j D1

n X

Hn;j .x/Hn;j .y/

(4.3.68)

j Dn

and using Lemma 4.3.19 together with x .x  y/n D n.n  1/jyj2 .x  y/n2 :  ProjHarmn .R3 /

.x  y/n nŠ

D

 D

bn=2c 1 X .1/s .2n  2s/Š.nŠ/2 jxj2s jyj2s .x  y/n2s nŠ sD0 .n  2s/Š.n  s/ŠsŠ.2n/Š

bn=2c 2n nŠ n n X .1/s .2n  2s/Š jxj jyj .  /n2s n .n  2s/Š.n  s/ŠsŠ .2n/Š 2 sD0 „ ƒ‚ …

(4.3.69)

DPn ./

due to the explicit representation of the Legendre polynomials. For details see Corollary 3.4.4 with D 12 and (by use of Lemma 2.3.3) 2N m

 .N  m C 12 /  . 12 /

D

 .2N  2m/ 2N 2m .2N  2m/Š : D N 2N 2m1  .N  m/ 2 2 .N  m/Š

(4.3.70)

This yields the addition theorem for homogeneous harmonic polynomials in R3 . u t Theorem 4.3.22. For Hm 2 Harmm .R3 / and Kn 2 Harmn .R3 /, we have hHm jS2 ; Kn jS2 iL2 .S2 / D where n D

ın;m hHm ; Kn iHomn .R3 / ; n

(4.3.71)

.2nC1/Š 42n nŠ .

Proof. We only present the ideas of the proof. By the fundamental theorem of potential theory (Theorem 6.2.11, see also, e.g., Freeden 1979 and the references therein) it holds, for all x 2 R3 with jxj < 1, that 1 Kn .x/ D 4

Z  S2

@ 1 1 @ Kn .y/  Kn .y/ jx  yj @ @y jx  yj

 dS.y/;

(4.3.72)

4.3 Homogeneous Polynomials on the Unit Sphere in R3

137

where  is the outward pointing normal to the surface S2 . Therefore, 1 Hm .rx /Kn .x/ D 4

Z 

@ 1 Kn .y/ jx  yj @  @ 1 dS.y/: (4.3.73) Kn .y/ Hm .rx / @y jx  yj S2

Hm .rx /

Now, we have Hm .rx /

.2m/Š Hm .x  y/ 1 .2m/Š Hm .y  x/ D .1/m m D m : 2mC1 jx  yj 2 mŠ jx  yj 2 mŠ jx  yj2mC1

(4.3.74)

At x D 0 we have ( Hm .rx /Kn .x/jxD0 D

0

for m ¤ n;

hHm ; Kn iHomn .R3 /

for n D m:

(4.3.75)

Therefore, 1 4

Z  S2

D

(

Hm .y/ @ @ Hm .y/ Kn .y/  Kn .y/ 2mC1 jyj @ @ jyj2mC1

0

for m ¤ n;

2m mŠ hHm ; Kn iHomn .R3 / .2m/Š

for n D m:

 dS.y/

(4.3.76)

Next, we see that on the sphere S2 with y D r, jj D 1: @ @ n @ Kn .y/ D   ry Kn .y/ D Kn .r/ D r Kn ./ D nr n1 Kn ./ (4.3.77) @ @r @r which gives us ˇ ˇ @ Kn .y/ˇˇ D nKn ./ @ y2S2

ˇ ˇ @ Hm .y/ˇˇ D mHm ./: @ y2S2

(4.3.78)

1 1 @ Hm .y/ D mHm .y/ 2mC1 C Hm .y/.2m  1/ 2mC2 ; 2mC1 @ jyj jyj jyj

(4.3.79)

and

Moreover,

and, therefore, ˇ @ Hm .y/ ˇˇ D mHm ./  .2m C 1/Hm ./ D .m C 1/Hm ./: @ jyj2mC1 ˇy2S2

(4.3.80)

4 Scalar Spherical Harmonics in R3

138

If we put these results together, this yields 1 4

Z 

 Hm .y/ @ @ Hm .y/ K dS.y/ .y/  K .y/ n n jyj2mC1 @ @ jyj2mC1 S2 Z 1 D .nHm ./Kn ./ C .m C 1/Hm ./Kn .// dS./ 4 S2 Z nCmC1 D Hm ./Kn ./ dS./: (4.3.81) 4 S2

Altogether, we find the desired result.

t u

We reformulate the addition theorem (Theorem 4.3.21). Corollary 4.3.23 (Addition Theorem in Harmn .R3 /). Let fHn;j gj Dn;:::;n be an orthonormal system in Harmn .R3 / with respect to h ;  iHomn .R3 / , i.e., the system p f n Hn;j gj Dn;:::;n is orthonormal in Harmn .R3 / with respect to h ;  iL2 .S2 / . For every x; y 2 R3 with x D jxj, y D jyj (;  2 S2 ), we find that n X 2n C 1 n n p p jxj jyj Pn .  /; n Hn;j .x/ n Hn;j .y/ D 4 j Dn

where n D of degree n.

.2nC1/Š 42n nŠ

(4.3.82)

as in Theorem 4.3.22 and Pn denotes the Legendre polynomial

Harmonic Polynomials on the Sphere In analogy to (4.3.37)–(4.3.39) we introduce harmonic polynomials on subsets of R3 , in particular the sphere S2 . Definition 4.3.24. For m 2 N0 and a set D  R3 , ˚ Harmm .D/ D P jD W P 2 Harmm .R3 / ; ˚ Harm0;:::;m .D/ D P jD W P 2 Harm0;:::;m .R3 / ; ˚ Harm0;:::;1 .D/ D P jD W P 2 Harm0;:::;1 .R3 / :

(4.3.83) (4.3.84) (4.3.85)

The elements of the space Harmn .S2 /, n 2 N0 , are called (scalar) spherical harmonics of degree n. From the comparison of h ;  iL2 .S2 / and h ;  iHomn .R3 / (see Theorem 4.3.22) it immediately follows that, for Yn 2 Harmn .S2 / and Ym 2 Harmm .S2 /,

4.3 Homogeneous Polynomials on the Unit Sphere in R3

139

Z hYn ; Ym iL2 .S2 / D

S2

Yn ./Ym ./ dS./ D 0

(4.3.86)

if n ¤ m. Theorem 4.3.25. If n 2 N0 , then dim.Harmn .R3 // D dim.Harmn .S2 // D 2n C 1:

(4.3.87)

Proof. It is clear that dim.Harmn .R3 //  dim.Harmn .S2 //:

(4.3.88)

Assume that m D dim.Harmn .S2 // < dim.Harmn .R3 // D 2n C 1. Let fHn;j g  Harmn .R3 / be a linearly independent system in Harmn .R3 / and let Yj D Hn;j jS2 , i.e., Hn;j .x/ D r n Yj ./. Suppose that F 2 C.S2 / with n X

F ./ D

aj Yj ./ D 0:

(4.3.89)

j Dn

Consider the following boundary value problem: find U 2 C.2/ .B31 / \ C.B31 / such that U D 0 in B31 ; U jS2 D F: (4.3.90) Note that B31 D fx 2 R3 W jxj < 1g denotes the open unit ball in R3 around the origin and its boundary @B31 is the unit sphere S2 . This boundary value problem is uniquely solvable by n X U.x/ D aj r n Yj ./; (4.3.91) j Dn

where x D r and we use the same coefficients aj as in (4.3.89), i.e., U.x/ D 0 for x 2 S2 . Thus, n X aj r n Yj ./ D 0 (4.3.92) j Dn

for all r < 1, or

n X

aj Hn;j .x/ D 0

(4.3.93)

j Dn

for all x 2 B31 . The left-hand side is a polynomial, i.e., n X

aj Hn;j .x/ D 0

for all x 2 R3 :

(4.3.94)

j Dn

Since the polynomials Hn;j are linearly independent, aj D 0 for all j .

t u

4 Scalar Spherical Harmonics in R3

140

Lemma 4.3.26. Any spherical harmonic Yn 2 Harmn .S2 /, n 2 N0 , is an infinitely often differentiable eigenfunction of the Beltrami operator  corresponding to the eigenvalue n.n C 1/, i.e.,  Yn ./ C n.n C 1/Yn./ D 0 for all  2 S2 :

(4.3.95)

The sequence fn.n C 1/gn2N0 is called the eigenspectrum of the Beltrami operator. Proof. Yn 2 Harmn .S2 / is the restriction of a polynomial Hn 2 Harmn .R3 / to the x sphere, i.e., x Hn .x/ D 0 for all x D r 2 R3 (r D jxj,  D jxj 2 S2 ) and  2  @ 1  2 @ C Hn .r/ D 0: C  (4.3.96) @r 2 r @r r2  Due to the homogeneity of Hn we know that Hn .x/ D r n Yn ./ with r > 0. Thus, 2 1 0 D x Hn .x/ D n.n  1/r n2 Yn ./ C nr n1 Yn ./ C 2 r n  Yn ./ r r   (4.3.97) D r n2 n2  n C 2n C  Yn ./: Since r > 0, it follows that  Yn ./ C n.n C 1/Yn ./ D 0.

t u

Definition 4.3.27. We denote a complete orthonormal system in the Hilbert space .Harmn .S2 /; h; iL2.S2 / / by fYn;j gj Dn;:::;n , i.e., (i) hYn;j ; Yn;k iL2 .S2 / D ıj;k for all j; k 2 fn; : : : ; ng. (ii) If hF; Yn;j iL2 .S2 / D 0 for all j 2 fn; : : : ; ng with F 2 Harmn .S2 /, then F D 0. We call Yn;j a spherical harmonic of degree n and order j . Remark 4.3.28. The family fYn;j gn2N0 ;j Dn;:::;n forms an L2 .S2 /-orthonormal system in Harm0;:::;1 .S2 /. Theorem 4.3.29 (Addition Theorem of Spherical Harmonics). If fYn;j gj Dn;:::;n is an L2 .S2 /-orthonormal system in Harmn .S2 /, n 2 N0 , then n X

Yn;j ./Yn;j ./ D

j Dn

2n C 1 Pn .  / 4

(4.3.98)

for all .; / 2 S2  S2 , where Pn is the Legendre polynomial of degree n. Moreover, n X  2 2n C 1 Yn;j ./ D 4 j Dn



Yn;j 2 D sup jYn;j ./j  C.S / 2S2

r

2n C 1 4

for all  2 S2 ;

for all j D n; : : : ; n:

(4.3.99)

(4.3.100)

4.3 Homogeneous Polynomials on the Unit Sphere in R3

141

Proof. This result follows immediately from the addition theorem in Harmn .R3 /, see Theorem 4.3.21, respectively Corollary 4.3.23 in combination with Theorem 4.3.22. t u Remark 4.3.30. Using the addition theorem (Theorem 4.3.29), it is easy to show that the mapping .; / 7! KHarmn .S2 / .; / D

2n C 1 Pn .  / 4

(4.3.101)

is the reproducing kernel in Harmn .S2 /, i.e., 2n C 1 4

Z S2

Yn ./Pn .  / dS./ D Yn ./:

(4.3.102)

Lemma 4.3.31. For every Yn 2 Harmn .S2 /, we have r sup jYn ./j 

2S2

2n C 1 kYn kL2 .S2 / : 4

(4.3.103)

Proof. Due to Fourier theory it is clear that by Parseval’s identity kYn k2L2 .S2 /

n X

D

jhYn ; Yn;j iL2 .S2 / j2

(4.3.104)

j Dn

and we have the representation n X

Yn D

hYn; Yn;j iL2 .S2 / Yn;j

(4.3.105)

j Dn

on the sphere S2 . Therefore, by the Cauchy–Schwarz inequality we obtain jYn ./j2 

n X

hYn ; Yn;j i2L2 .S2 /

j Dn

n X  j Dn

Yn;j ./

2

D

2n C 1 kYn k2L2 .S2 / : (4.3.106) 4

Taking the square root provides us with the desired result.

t u

While we have presented the theory of spherical harmonics in a real framework, we now introduce the complex-valued spherical harmonics which are well-known and widely used in physics. In Example 4.3.33 we offer a system of real-valued spherical harmonics of equal importance (in particular in geosciences). Example 4.3.32 (Complex-Valued Spherical Harmonics). Let n 2 N0 , j 2 Z with n  j  n. The function

4 Scalar Spherical Harmonics in R3

142 C Yn;j W S2 ! C

(4.3.107) s

C  7! Yn;j ./ D .1/j

2n C 1 .n  j /Š Pn;j .cos.#//eij' 4 .n C j /Š

is called the (complex) spherical harmonic of degree n and order j , where #; ' are the spherical coordinates of  given by (4.1.2). Pn;j W Œ1; 1 ! R;

j

t 7! Pn;j .t/ D .1  t 2 / 2

dj Pn .t/ dt j

(4.3.108)

is the associated Legendre function of degree n and order j , 0  j  n. For negative orders, we have the symmetry relation Pn;j .t/ D .1/j

.n  j /Š Pn;j .t/: .n C j /Š

(4.3.109)

These spherical harmonics are orthonormal with respect to the canonical scalar product of the space L2C .S2 / of complex-valued square-integrable functions on the unit sphere S2 . Their addition theorem reads as follows: n X j Dn

C C Yn;j ./Yn;j ./ D

2n C 1 Pn .  /: 4

(4.3.110)

For further details on this representation of spherical harmonics the reader is referred to, e.g., Edmonds (1964), Jones (1985), Var˘salovi˘c et al. (1988) and Zare (1988). Example 4.3.33 (Real Fully Normalized Spherical Harmonics). Let n 2 N0 , j 2 Z with n  j  n. The function R W S2 ! R; Yn;j

s  7!

R Yn;j ./

D

8p ˆ ˆ < 2 cos.j'/

2n C 1 .n  jj j/Š Pn;jj j .cos.#// 1 ˆ 4 .n C jj j/Š ˆ :p2 sin.j'/

(4.3.111) ; j 0

is called the (real) fully normalized spherical harmonic of degree n and order j , where #; ' are the spherical coordinates of  given by (4.1.2) and Pn;j , n 2 N0 , 0  j  n, are defined by (4.3.108). R The Yn;j are orthonormal with respect to the scalar product of L2R .S2 / (the space of real-valued, square-integrable functions on the unit sphere S2 ) and the addition R theorem holds as written in Theorem 4.3.29. The real spherical harmonics Yn;j C (cf. Fig. 4.5–4.7) and the complex spherical harmonics Yn;j (of Example 4.3.32) are related via

4.3 Homogeneous Polynomials on the Unit Sphere in R3

143

Fig. 4.5 All spherical harmonics Y3;j of degree 3 (upper row: Y3;3 to Y3;0 , lower row: Y3;1 to Y3;3 )

Fig. 4.6 Spherical harmonics Y7;0 (left), Y7;5 (middle), and Y7;7 (right). Note that if the order is j D ˙n, Yn;˙n are called sectorial spherical harmonics. The spherical harmonics Yn;0 of order 0 are called zonal spherical harmonics. All other basis functions Yn;j , i.e., j ¤ 0 and j ¤ ˙n, are called tesseral spherical harmonics

Fig. 4.7 Spherical harmonics Y10;2 (left), Y10;6 (middle), and Y10;9 (right)

8p   < 2ı0;j Y C ./ C Y C ./ ; n;j n;j 2 R  p  Yn;j ./ D j : .1/ 2 Y C ./  Y C ./ ; n;j n;j 2i for all  2 S2 , n 2 N0 , and j 2 Z with n  j  n.

j  0; j > 0;

(4.3.112)

4 Scalar Spherical Harmonics in R3

144

Note that in the case of real-valued spherical harmonics, indexing with negative integers is just one way to distinguish the two types with sine and cosine. In the literature, in particular in geodetic literature, often two different symbols are used to achieve this (see, e.g., Heiskanen and Moritz 1967; Hofmann-Wellenhof and Moritz 2005; Lense 1954; Morse and Feshbach 1953).

4.4 Closure and Completeness of Spherical Harmonics We consider Fourier expansions in terms of a set of L2 .S2 /-orthonormal spherical harmonics and use two summability methods: Abel-Poisson summability and Bernstein summability. The latter turns out to be also very helpful for vectorial and tensorial spherical harmonics (see Chap. 5 for the vectorial case and Freeden and Schreiner 2009 for the tensorial case). Moreover, it can easily be generalized to q-dimensional spherical harmonics as performed in Chap. 6. Lemma 4.4.1. For all t 2 Œ1; 1 and all h 2 .1; 1/, the following power series converge uniformly: 1 X

hn Pn .t/ D p

nD0 1 X

.2n C 1/hn Pn .t/ D

nD0

1 1 C h2  2ht

;

1  h2 3

.1 C h2  2ht/ 2

(4.4.1)

:

(4.4.2)

Proof. The right-hand side of (4.4.1) is the generating function which we have in the more general setting of Gegenbauer polynomials in Theorem 3.4.7. Relation (4.4.2) directly results from Corollary 3.4.10. t u Remark 4.4.2. Using (4.4.1) of Lemma 4.4.1, we obtain with x D jxj, y D jyj, ;  2 S2 :  1   12  1 jxj 1 X jxj n jxj2 D jyj1 1 C     2 D Pn .  /; (4.4.3) jx  yj jyj2 jyj jyj nD0 jyj provided that jxj < jyj. Moreover, we know that 1

X .1/n 1 1 D jxjn .  ry /n : jx  yj nŠ jyj nD0

(4.4.4)

By comparison we obtain the so-called Maxwell representation .  ry /n

1 Pn .  / D .1/n nŠ : jyj jyjnC1

(4.4.5)

4.4 Closure and Completeness of Spherical Harmonics

145

As y 7! jyj1 ; jyj ¤ 0; is (apart from a multiplicative constant) the fundamental solution of the Laplace equation in three dimensions, Maxwell’s representation tells us that the Legendre polynomials may be obtained by repeated differentiation of the fundamental solution in the direction of . The potential on the right-hand side may be regarded as the potential of a pole of order n with the axis  at the origin. Theorem 4.4.3 (Poisson Integral Formula). If F is continuous on S2 , then for h < 1: ˇ ˇ Z ˇ 1 ˇ 1  h2 ˇ (4.4.6) F ./ dS./  F ./ˇˇ D 0: lim sup 3 h!1 2S2 ˇ 4 S2 .1 C h2  2h.  // 2 Proof. Since jhj < 1 and kPn kC.Œ1;1/ D 1, we use Lemma 4.4.1 and find that Z

1 1

1  h2 .1 C h2  2ht/

3 2

dt D

Z 1 X .2n C 1/hn nD0

1 1

Pn .t/  1 dt D 2:

(4.4.7)

Therefore, we use Theorem 4.1.6: Z

1 1 2 2

1  h2 3

.1 C h2  2h.  // 2

S2

dS./ D 1;

 2 S2 :

(4.4.8)

Choose  2 S2 arbitrary, but fixed. Then, 1 4

Z

.1  h2 /.F ./  F .//

dS./ 3 .1 C h2  2h.  // 2 Z 1 .1  h2 /F ./ D dS./  F ./: 4 S2 .1 C h2  2h.  // 32

(4.4.9)

S2

For h 2 Œ 12 ; 1/, we split the sphere into two parts, i.e., the spherical cap of radius p 3 1  h around  2 S2 ; C.;

p p 3 3 1  h/ D f 2 S2 W 1  1  h      1g;

(4.4.10)

and its complementary set S2 n C.;

p p 3 3 1  h/ D f 2 S2 W 1      1  1  hg:

(4.4.11)

This allows us to split the integral (4.4.9) in the same way, i.e., Z

Z S2

:::

D S2 nC.;

Z ::: p 3 1h/

C C.;

p 3 1h/

::::

(4.4.12)

4 Scalar Spherical Harmonics in R3

146

For t 2 Œ1; 1 

p 3 1  h, we can estimate that

p 3 1 C h2  2ht D .1  h/2 C 2h.1  t/  2h.1  t/  2h 1  h:

(4.4.13)

This leads to 1  h2



3

.1 C h2  2ht/ 2

p 1  h2 1Ch 1h D p  2 1  h; p 3 3 3 1h .2h 1  h/ 2 .2h/ 2

(4.4.14)

which gives us for the first integral of the splitting (4.4.12) the following estimate: ˇZ ˇ ˇ ˇ

S2 nC.;

.1  h2 /.F ./  F .//

ˇ ˇ dS./ ˇˇ

(4.4.15) 3 .1 C h2  2h.  // 2 Z .1  h2 /.kF kC.S2 / C kF kC.S2 / / dS./  p 3 3 S2 nC.; 1h/ .1 C h2  2h.  // 2 3 Z 1 p 1h 1  h2 D 2 kF kC.S2 / 2 dt 3 1 .1 C h2  2ht/ 2 3 Z 1 p 1h p 2 1  h dt  4 kF kC.S2 /

p 3 1h/

1

p p 3 D 4 kF kC.S2 / 2 1  h .1  1  h C 1/ ƒ‚ … „ 2

p h!1  16 kF kC.S2 / 1  h ! 0; where the convergence is uniform with respect to  2 S2 . Since F 2 C.S2 / and S2 is compact, F is uniformly continuous, i.e., there exists a function W h ! .h/ with lim .h/ D 0;

h!1

(4.4.16)

such that jF ./  F ./j  .h/ (4.4.17) p 3 for all  2 S2 that satisfy 1  1  h      1. The upper bound .h/ is independent of . Now, we consider the second part of the integral in (4.4.12): ˇZ ˇ ˇ ˇ

C.;

p 3 1h/

.1  h2 /.F ./  F .// 3

ˇ ˇ dS./ ˇˇ

.1 C h2  2h.  // 2 Z .1  h2 /jF ./  F ./j dS./  p 3 3 C.; 1h/ .1 C h2  2h.  // 2

(4.4.18)

4.4 Closure and Completeness of Spherical Harmonics

Z  .h/

S2

147

1  h2 .1 C

h2

h!1

 2h.  //

3 2

dS./ D .h/4 ! 0:

Altogether, we find the desired convergence result.

t u

Theorem 4.4.4. Let F 2 C.S /. The series 2

1 X

n X

hn

nD0

F ^ .n; j /Yn;j ./

(4.4.19)

j Dn

with the Fourier coefficients F ^ .n; j / D hF; Yn;j iL2 .S2 / converges uniformly with respect to all  2 S2 for h 2 .0; h0 / with h0 < 1 and lim

h!1

1 X nD0

hn

n X

F ^ .n; j /Yn;j ./ D F ./;

(4.4.20)

j Dn

where this convergence is uniform. Proof. We first show uniform convergence of the series using the definition of the Fourier coefficients and the addition theorem (Theorem 4.3.29). We estimate the function values with the C.S2 /-norm and apply jPn .t/j  1 for t 2 Œ1; 1. ˇX ˇ n Z ˇ 1 n X ˇ ˇ h F ./Yn;j ./ dS./Yn;j ./ˇˇ ˇ 2 nD0

(4.4.21)

j Dn S

 kF kC.S2 /

1 X

Z n

h

nD0

 kF kC.S2 /

1 X

S2

2n C 1 jPn .  /j dS./ 4

hn .2n C 1/ < 1:

nD0

For the limit process, (4.4.2) of Lemma 4.4.1 and the addition theorem (Theorem 4.3.29) are used to obtain a representation as in Theorem 4.4.3. The dominated convergence theorem allows us to interchange the infinite series and the integral. Finally, the actual convergence is established by the Poisson integral formula (Theorem 4.4.3). t u Corollary 4.4.5. The system fYn;j gn2N0 ;nj n is closed in .C.S2 /; kkC.S2 / /, i.e., for all F 2 C.S2 / and for all " > 0, there exists a finite linear combination n N X X nD0 j Dn

dn;j Yn;j ;

(4.4.22)

4 Scalar Spherical Harmonics in R3

148

such that

n N X

X

dn;j Yn;j

F 

C.S2 /

nD0 j Dn

 ":

(4.4.23)

D C.S2 /:

(4.4.24)

In other words: spanfYn;j jn 2 N0 ; n; : : : ; ng

kkC.S2 /

Proof. Let F 2 C.S2 / and " > 0 be arbitrary. According to Theorem 4.4.4 there exists h D h."/ < 1 fixed, such that 1 n

X X

hn F ^ .n; j /Yn;j  F

nD0

C.S2 /

j Dn



" : 2

(4.4.25)

Moreover, the theorem tells us that there exists N D N."/, such that 1 N n n

X

X X X

hn F ^ .n; j /Yn;j  hn F ^ .n; j /Yn;j

nD0

j Dn

nD0

C.S2 /

j Dn



" ; 2

(4.4.26)

since the series converges uniformly for fixed h < 1 (here h D h."/). Together this gives us n N X

X " "

hn F ^ .n; j / Yn;j 2  C D "

F  „ ƒ‚ … C.S / 2 2 nD0 j Dn

(4.4.27)

Ddn;j

and, therefore, the closure of the spherical harmonics in .C.S2 /; kkC.S2 / /.

t u

Now, we consider Bernstein summability to obtain this result. The point of departure for Bernstein summability is the so-called Bernstein kernel of degree n. Definition 4.4.6. The Bernstein kernel of degree n is given by KB;n W Œ1; 1 ! R t 7! KB;n .t/ D

nC1 4



1Ct 2

n :

(4.4.28)

See Fig. 4.8 for an illustration of it. Remark 4.4.7. The name Bernstein is motivated by the fact that the kernel is proportional tothe Bernstein polynomial of degree n with the parameter  D n, i.e., B;n .t/ D n t  .1  t/n scaled to the interval Œ1; 1 (t is the polar distance

4.4 Closure and Completeness of Spherical Harmonics

149

3.5 n=10 n=20 n=40

3

2.5

2

1.5

1

0.5

0

−3

−2

−1

0

1

2

3

Fig. 4.8 Illustration of the kernel KB;n .cos #/; # 2 Œ; , for the degrees n D 10, n D 20, n D 40

between  and , i.e., t D   ). For further details we refer to Fengler et al. (2006) and Freeden and Gutting (2008). First, we mention some important properties of the Bernstein kernel that can easily be verified by the reader. Lemma 4.4.8. The Bernstein kernel (4.4.28) possesses the following properties: (i) For all t 2 Œ1; 1 and n D 0; 1; : : :, we have ^ .0/ D 2 KB;n

(ii) For all t 2 Œ1; 1, (iii) For all t 2 Œ1; 1/,

Z

1 1

KB;n .t/ dt D 1:

(4.4.29)

KB;n .t/  0:

(4.4.30)

lim KB;n .t/ D 0:

(4.4.31)

n!1

(iv) For k D 0; : : : ; n, Z 2

1 1

KB;n .t/Pk .t/ dt D

^ KB;n .k/

n .n C 1/Š nŠ k D nCkC1 D ; .n  k/Š .n C k C 1/Š k (4.4.32)

where Pk is the Legendre polynomial of degree k.

4 Scalar Spherical Harmonics in R3

150

(v) For k 2 N fixed,

^ ^ KB;n .k/ < KB;nC1 .k/:

(vi) For k 2 N0 fixed,

^ KB;n .k/

(4.4.33)

! 1 as n ! 1, i.e., ^ lim KB;n .k/ D 1:

(4.4.34)

n!1

Now, suppose that F is continuous on S2 . We use the property .i / to guarantee that Z

Z S2

KB;n .  /F ./ dS./ D F ./ C

S2

KB;n .  / .F ./  F .// dS./;

(4.4.35) where  2 S2 . We split S2 into two parts depending on a parameter 2 .0; 1/, i.e., into the spherical cap of radius around  2 S2 : C.; / D f 2 S2 W 1       1g

(4.4.36)

and the remaining sphere S2 n C.; /. The corresponding split of the integral (4.4.35) yields Z

Z S2

::: D

Z S2 nC.; /

::: C

::::

(4.4.37)

C.; /

On the one hand, we find with (ii) ˇZ ˇ ˇ ˇ

S2 nC.; /

ˇ Z ˇ KB;n .  /F ./ dS./ˇˇ  4 kF kC.S2 /

1 1

KB;n .t/ dt



nC1  2 kF kC.S2 / 1  : 2

(4.4.38)

On the other hand, F is uniformly continuous on S2 . Thus, there exists a positive function W 7! . / with lim . / D 0;

(4.4.39)

!0C

such that jF ./  F ./j  . /

(4.4.40)

for all  2 S2 with 1       1. Thus, it follows that ˇZ ˇ ˇ ˇ

C.; /

ˇ Z ˇ KB;n .  /.F ./  F .// dS./ˇˇ  . / KB;n .  / dS./ 2 S

D . /:

(4.4.41)

4.4 Closure and Completeness of Spherical Harmonics

151

Summarizing our results, we obtain the estimate ˇZ ˇ ˇ ˇ sup ˇˇ KB;n .  /F ./ dS./  F ./ˇˇ

2S2

S2

ˇZ ˇ ˇ ˇ ˇ D sup ˇ KB;n .  / .F ./  F .// dS./ˇˇ 2 2 2S

S

2S

S2 nC.; /

ˇZ ˇ  sup ˇˇ 2

ˇZ ˇ C sup ˇˇ 2 2S

ˇZ ˇ  sup ˇˇ 2 2S

Z

C.; /

S2 nC.; /

KB;n .  /F ./ dS./  F ./

S2 nC.; /

ˇ ˇ KB;n .  / dS./ˇˇ

ˇ ˇ KB;n .  / .F ./  F .// dS./ˇˇ ˇ ˇ KB;n .  /F ./ dS./ˇˇ

Z

CkF kC.S2 /

S2 nC.; /

KB;n .  / dS./ C . /

 nC1 C . /:  2kF kC.S2 / 1  2

(4.4.42)

Now, we choose D n1=2 , such that ! 0 for n ! 1, i.e.,   1

1 p 2 n

p1 n



nC1

!0

for n ! 1;

(4.4.43)

!0

for n ! 1:

(4.4.44)

This shows us that ˇZ ˇ ˇ ˇ lim sup ˇˇ KB;n .  /F ./ dS./  F ./ˇˇ D 0 n!1 2 2 2S

(4.4.45)

S

and we obtain the following result. Theorem 4.4.9. For F 2 C.S2 /, ˇZ ˇ ˇ ˇ ˇ lim sup ˇ KB;n .  /F ./ dS./  F ./ˇˇ D 0: n!1 2 2 2S

(4.4.46)

S

Now, it can be readily seen that the Legendre series of the Bernstein kernel is given by n X 2k C 1 ^ KB;n .t/ D Pk .t/ KB;n .k/ (4.4.47) 4 kD0

4 Scalar Spherical Harmonics in R3

152

for all t 2 Œ1; 1. This shows us that Z S2

KB;n .  /F ./ dS./ D

n X

C1 4

2k ^ KB;n .k/

kD0

D

n X

Z S2

Pk .  /F ./ dS./

^ KB;n .k/ ProjHarmk .F /./:

(4.4.48)

kD0

Thus, we finally have the ‘Bernstein summability’ of a Fourier series expansion in terms of spherical harmonics. Theorem 4.4.10. For F 2 C.S2 /, ˇ n ˇ k X ˇX ^ ˇ lim sup ˇˇ KB;n .k/ F ^ .k; j /Yk;j ./  F ./ˇˇ D 0: n!1 2 2S

(4.4.49)

j Dk

kD0

Theorem 4.4.10 also enables us to prove the closure of the system of spherical harmonics in the space C.S2 /. Corollary 4.4.11. The system fYk;j gk2N0 ; j Dk;:::;k is closed in C.S2 /, i.e., for any given " > 0 and each F 2 C.S2 /, there exists a linear combination N X k X

dk;j Yk;j

(4.4.50)

kD0 j Dk

such that

N X k

X

dk;j Yk;j

F 

C.S2 /

kD0 j Dk

 ":

(4.4.51)

Proof. Given F 2 C.S2 /. Then, for any given " > 0, there exists an integer N D N."/ such that ˇX ˇ k ˇ N X ˇ ^ ^ ˇ sup ˇ KB;N .k/F .k; j / Yk;j ./  F ./ˇˇ  ": ƒ‚ … 2S2 kD0 j Dk „

(4.4.52)

Ddk;j

t u

This proves Corollary 4.4.11. Since for all F 2 C.S / we have the inequality 2

kF kL2 .S2 / 

p 4 kF kC.S2 / ;

(4.4.53)

we immediately find that fYn;j gn2N0 ;nj n is also closed in .C.S2 /; kkL2 .S2 / /.

4.4 Closure and Completeness of Spherical Harmonics

153

Corollary 4.4.12. The system fYn;j gn2N0 ;nj n is closed in .L2 .S2 /; kkL2 .S2 / /. Proof. We know that C.S2 /

kkL2 .S2 /

D L2 .S2 /:

(4.4.54)

Let F 2 L .S / and " > 0. There exists G 2 C.S / such that kF  GkL2 .S2 /  2" . Due to Corollary 4.4.5 and estimate (4.4.53) there exists a finite linear combination 2

2

2

n N X X

an;j Yn;j

(4.4.55)

nD0 j Dn

corresponding to G such that n N X

X

an;j Yn;j

G  nD0 j Dn

" : 2

(4.4.56)

" " C D ": 2 2

(4.4.57)

L2 .S2 /



Combining these two results yields: n N X

X

an;j Yn;j

F  nD0 j Dn

L2 .S2 /



t u

This completes the proof of Corollary 4.4.12.

Now, we can apply Theorem 3.0.7 to the system of spherical harmonics. This gives us L2 -Fourier approximation on the of the system  sphere, i.e., completeness fYn;j gn2N0 ;nj n in the Hilbert space L2 .S2 /; k  kL2 .S2 / :

n N X X

lim F  hF; Yn;j iL2 .S2 / Yn;j

N !1 nD0 j Dn

D0

for all F 2 L2 .S2 /:

L2 .S2 /

(4.4.58) Remark 4.4.13. (i) Let X 2 fC.S2 /; Lp .S2 /gp2Œ1;1/ . If F; G 2 X with

n N X X

G  hF; Y i Y lim 2 2 n;j L .S / n;j D 0;

N !1 nD0 j Dn

(4.4.59)

X

then F D G almost everywhere on S2 . (ii) For X 2 fC.S2 /; Lp .S2 /gp2Œ1; 4 [Œ4;1/ it is not known whether the truncated 3 Fourier series converges for all F 2 X , i.e., a truncated Fourier series with L2 -Fourier coefficients might not yield an approximation in the X -topology.

4 Scalar Spherical Harmonics in R3

154

(iii) If F W S2 ! R is Lipschitz continuous, then

n N X X

hF; Yn;j iL2 .S2 / Yn;j lim F 

N !1

D 0;

(4.4.60)

C.S2 /

nD0 j Dn

i.e., the Fourier series is uniformly convergent in this case. Remark 4.4.14. Using the completeness property of the spherical harmonics system, we can prove that the spherical harmonics Yn 2 Harmn .S2 / are the only C.2/ .S2 /-eigenfunctions of the Beltrami operator.  only has the eigenvalues n.n C 1/ with n 2 N0 .

4.5 The Funk–Hecke Formula and the Irreducibility of Scalar Harmonics In this section we present a central result in the theory of spherical harmonics, i.e., the Funk–Hecke formula (see, e.g., Funk 1916; Hecke 1918; M¨uller 1998 and the literature therein). It is closely related to the irreducibility of the space of spherical harmonics which is also shown here. Lemma 4.5.1. Let Hn W R3 ! R be a homogeneous harmonic polynomial of degree n with the following properties: (i) Hn .tx/ D Hn .x/ for all orthogonal transformations t 2 SO"3 .3/, i.e., all rotations that leave "3 invariant, (ii) Hn ."3 / D 1. Then, Hn is uniquely determined and coincides with the so-called Legendre harmonic Ln of degree n, i.e., Hn .x/ D Ln .x/ D r n Pn .t/;

x D r.t "3 C

p 1  t 2 .cos '"1 C sin '"2 //; (4.5.1)

where Pn is the Legendre polynomial of degree n. Proof. We already know from Theorem 4.3.15 that Hn as a homogeneous harmonic polynomial of degree n can be written in the form Hn .x/ D

n X

Ank .x1 ; x2 /x3k ;

(4.5.2)

kD0

where the coefficients Ank .x1 ; x2 / fulfill (4.3.44) for k D 0; : : : ; n  2. Therefore, Hn is uniquely determined by the homogeneous polynomials An W .x1 ; x2 / 7! An .x1 ; x2 / and An1 W .x1 ; x2 / 7! An1 .x1 ; x2 /. Condition .i / implies that

4.5 The Funk–Hecke Formula

155

these polynomials depend only on x12 C x22 . Thus, we find with a constant C nk , 2 such that ( 0 ; n  k odd; Ank .x1 ; x2 / D (4.5.3) nk ; n  k even: C nk .x12 C x22 / 2 2

For x D "3 in (4.5.3), we get x12 C x22 D 0 and x3 D 1 such that C0 D 1. In order to determine C nk for even integers n  k, we see that 2



@2 @2 C @x12 @x22

 .x12

C

nk x22 / 2



nk D4 2

2

.x12 C x22 /

nk2 2

:

(4.5.4)

In connection with (4.3.44) we find the recursion relation  4C nk 2

nk 2

2

.x12 Cx22 /

nk2 2

D .kC2/.kC1/C nk2 .x12 Cx22 /

nk2 2

2

(4.5.5)

such that .n  k/2 C nk C .k C 2/.k C 1/C nk2 D 0; 2

2

(4.5.6)

k D 0; 2; : : : ; n  2. In other words, since C0 D 1, C nk D 1 for k D n;

(4.5.7)

2

C nk D .1/ 2

nk 2

nŠ 2kn  2 . nk /Š kŠ 2

for n  k even;

for k D 0; : : : ; n1. This shows us that Hn is uniquely determined by the conditions .i / and (ii). x It remains to prove that Ln .x/ D r n Pn .t/ (with r D jxj, t D "3  , and  D jxj ) is a homogeneous harmonic polynomial that fulfills .i / and (ii). Obviously, Ln 2 Homn .R3 / and by the addition theorem (Theorem 4.3.21) we find that Ln .x/ D jxjn Pn ."3  / D

n .2n/Š X Hn;j ."3 /Hn;j .x/ 2 Harmn .R3 /; 2n nŠ j Dn

(4.5.8)

where Hn;j , j D n; : : : ; n, is an orthonormal basis of Harmn .R3 /. Condition .i i / is also obvious, since Pn .1/ D 1. For condition .i /, consider any t 2 SO"3 .3/; Ln .tx/ D jxjn Pn ."3  t/ D jxjn Pn .tT "3  / D jxjn Pn ."3  / D Ln .x/; (4.5.9) since SO"3 .3/ is a group and, therefore, tT also leaves "3 invariant.

t u

4 Scalar Spherical Harmonics in R3

156

Corollary 4.5.2. Let Hn 2 Harmn .R3 / with the following properties: (i) Hn .tx/ D Hn .x/ for all orthogonal transformations t 2 SO .3/, i.e., all rotations that leave  2 S2 invariant, (ii) Hn ./ D 1. Then, Hn is uniquely determined by Hn .x/ D jxjn Pn .  /;

x D jxj;  2 S2 ;

(4.5.10)

where Pn is the Legendre polynomial of degree n. t u

Proof. This follows from Lemma 4.5.1 by simply rotating  to "3 .

For any function F 2 L .S /, the transformation  7! t;  2 S , produces a change in the functional values of F . Let F; G 2 L2 .S2 /, t 2 O.3/. Observing the change of coordinates  D t, we find that 2

2

2

Z

Z S2

F .t/G./ dS./ D j det tj

S2

F ./G.tT / dS./

(4.5.11)

(observe that dS.t/ D j det tj dS./). Thus, it follows that hRt F; GiL2 .S2 / D hF; RtT GiL2 .S2 / :

(4.5.12)

Theorem 4.5.3. The space Harmn .S2 / of spherical harmonics of degree n is irreducible. Proof. Assume that there exists an invariant subspace Y  Harmn .S2 / of dimension dim.Y / < dim.Harmn .S2 // D 2n C 1. Then, we would be able to show that the orthogonal complement Y ? of Y in Harmn .S2 / (with respect to h; iL2.S2 / ) is an invariant subspace (see Lemma 4.2.4). Now, because of the invariance of Y and Y ? , G and G ? being represented in terms of the L2 .S2 /-orthonormal system fYn;j g, respectively, X

dim.Y /

GD

Yn;j ."3 /Yn;j 2 Y;

(4.5.13)

j D1

G

?

dim.Harmn .S2 //

D

X

Yn;j ."3 /Yn;j 2 Y ?

(4.5.14)

j Ddim.Y /C1

satisfy G.t/ D G, G ? .t/ D G ? for all t 2 SO"3 .3/. Moreover, G; G ? do not vanish identically (note that there exist elements in Y , Y ? different from zero at "3 ). Thus, not all values of Yn;j ."3 /, j D 1; : : : ; dim.Y / or j D dim.Y / C 1; : : : ; dim.Harmn .S2 // are zero. By Lemma 4.5.1 there exists a constant a 2 Rnf0g such that G D aG ? , in contradiction to our assumption. t u

4.5 The Funk–Hecke Formula

157

The irreducibility of Harmn .S2 / leads us to simple representations of spherical harmonics (see, e.g., Freeden et al. 1998). Lemma 4.5.4. Let Zn be a member of Harmn .S2 /. There exist 2n C 1 orthogonal transformations tn ; : : : ; tn such that any Yn 2 Harmn .S2 / can be represented with real numbers an ; : : : ; an in the form n X

Yn D

aj Zn .tj /:

(4.5.15)

j Dn

Proof. Consider the set X D fZn .t/ W t 2 O.3/g:

(4.5.16)

Clearly, there exist tn ; : : : ; tn 2 O.3/ such that X D spanfZn .tn /; : : : ; Zn .tn /g:

(4.5.17)

Therefore, Zn 2 X implies Zn .t/ 2 X . From the irreducibility of Harmn .S2 / (see Theorem 4.5.3), it follows that X is an invariant non-void space of dimension dim.X / D dim.Harmn .S2 //. Moreover, the 2n C 1 linearly independent spherical harmonics Zn .tn /; : : : ; Zn .tn / form a basis. t u Lemma 4.5.5. There exist 2n C 1 points n ; : : : ; n of the unit sphere S2 such that any Yn 2 Harmn .S2 / can be represented with real numbers an ; : : : ; an in the form Yn ./ D

n X

aj Pn .j  /;

 2 S2 :

(4.5.18)

j Dn

Proof. From Lemma 4.5.4 we know that there exist 2n C 1 orthogonal transformations tn ; : : : ; tn such that any Yn 2 Harmn .S2 / admits the form Yn ./ D

n X

aj Pn ."3  tj /;

 2 S2 :

(4.5.19)

j Dn

But this means that Lemma 4.5.5 follows with j D tTj "3 ; j D n; : : : ; n.

t u

Theorem 4.5.6 (Funk–Hecke Formula). Let G 2 L1 .Œ1; 1/, n 2 N0 . Then, Z G.  /Pn .  / dS./ D 2 S2 „

Z

1 1

G.t/Pn .t/ dt Pn .  / ƒ‚ …

DG ^ .n/

for all ;  2 S2 . G ^ .n/ is called the Legendre coefficient of G.

(4.5.20)

4 Scalar Spherical Harmonics in R3

158

R Proof. Let Vn .; / D S2 G.  ˛/Pn .  ˛/ dS.˛/ for ;  2 S2 and t 2 O.3/ an orthogonal transformation. Then, Z Vn .t; t/ D Z

G.t  ˛/Pn .t  ˛/ dS.˛/

S2

D

(4.5.21)

G.t  tˇ/Pn .t  tˇ/ dS.tˇ/

S2

Z

D j det tj „ƒ‚ … D1

S2

G.  ˇ/Pn .  ˇ/ dS.ˇ/ D Vn .; /:

Furthermore, Vn .; / with fixed  2 S2 is a spherical harmonic by the addition theorem, i.e., for any  2 S2 , Z Vn .; / D

S2

G.  ˛/Pn .  ˛/ dS.˛/

Z

n X

D S2

D

G.  ˛/

j Dn

(4.5.22)

4 Yn;j .˛/Yn;j ./ dS.˛/ 2n C 1

n X

4 Yn;j ./G^ .n; j /: 2n C 1 j Dn

This means that Vn .; / is a harmonic polynomial that is invariant under orthogonal transformations that leave  invariant. Hence, by Lemma 4.5.1 or, more precisely, by Corollary 4.5.2, Vn .; / D ˛n Pn .  /: (4.5.23) Setting  D  we obtain (using Theorem 4.1.6) Z ˛n D ˛n Pn .  / D Z

1

S2

G.  ˛/Pn .  ˛/ dS.˛/

G.t/Pn .t/ dt D G ^ .n/

(4.5.24)

which corresponds to the desired Legendre coefficient.

t u

D 2

1

Corollary 4.5.7. Let G 2 L1 .Œ1; 1/. For every Yn 2 Harmn .S2 /, we have Z S2

G.  /Yn ./ dS./ D G ^ .n/Yn ./;

 2 S2 ;

where G ^ .n/ is the Legendre coefficient as defined in Theorem 4.5.6.

(4.5.25)

4.6 Green’s Function with Respect to the Beltrami Operator

159

Proof. This is a direct consequence of the Funk–Hecke formula (Theorem 4.5.6) and the fact that the Legendre polynomial Pn is the reproducing kernel in Harmn .S2 / (see Remark 4.3.30). t u

4.6 Green’s Function with Respect to the Beltrami Operator One of the most essential ingredients for our approach to vector spherical harmonics is Green’s function on the sphere S2 with respect to the Beltrami operator which we define here (cf. Freeden 1979, 1980a). Several integral theorems involving the Green function are included. It should be noted that the Green function of this section also plays an important role in the theory of tensor spherical harmonics for which we refer to Freeden and Gutting (2008) and Freeden and Schreiner (2009) and the many references therein. Definition 4.6.1. The function G. I ; / W .; / 7! G. I ; /, 1     < 1, is called Green’s function on S2 , or sphere function, with respect to the operator  , if it satisfies the following properties: (i) For every fixed  2 S2 , the function  7! G. I ; / is twice continuously differentiable on the set S2 n fg such that  G. I ; / D 

1 ; 1     < 1; 4

(4.6.1)

1 ln.1    / 4

(4.6.2)

(ii) For every  2 S2 , the function  7! G. I ; / 

is continuously differentiable on S2 . (iii) For all orthogonal transformations t 2 O.3/, G. I t; t/ D G. I ; /; (iv) For every  2 S2 , 1 4

Z S2

G. I ; / dS./ D 0:

(4.6.3)

(4.6.4)

Because of Condition (iii) we find that G. I ; /P is a zonal function and we usually write G. I   / instead of G. I ; /. Note that, for 1     < 1, ln j  j D

1 1 1 ln.2  2  / D ln.1    / C ln.2/: 2 2 2

(4.6.5)

4 Scalar Spherical Harmonics in R3

160

Lemma 4.6.2. G. I ; / is uniquely determined by its defining properties (i)–(iv). The explicit representation of the Green function, for 1     < 1, is G. I ; / D

1 1 1 ln.1    / C  ln.2/: 4 4 4

(4.6.6)

Proof. We first show uniqueness. Let D.; / be the difference between two Green functions. Then, .i / D.; / is twice continuously differentiable for all  2 S2 with 1     < 1 and  D.; / D 0, (ii) D.; / is a continuously differentiable function for every  2 S2 , (iii) For R all orthogonal transformations t 2 O.3/, D.t; t/ D D.; /, (iv) S2 D.; / dS./ D 0 for every  2 S2 . Without loss of generality, set  D "3 . Now, we know that D."3 ; / D D.t/, where @ t is one of the spherical coordinates of  D .t; '/. Thus, @' D D 0 and by .i /  and (4.1.12) (coordinate representation of  ) we find that @ @ .1  t 2 / D.t/ D 0 @t @t

(4.6.7)

for all 1  t < 1. Hence, .1  t 2 / @t@ D.t/ D C0 2 R for all 1  t < 1. D is continuously differentiable by (ii), i.e., @t@ D.t/ tends to a constant C1 2 R for t ! 1. Therefore, @ lim .1  t 2 / D.t/ D 0 D C0 (4.6.8) t !1 @t and .1  t 2 / @t@ D.t/ D 0 for all t 2 Œ1; 1. This gives us @t@ D.t/ D 0 on .1; 1/, i.e., D.t/ D C2 on .1; 1/ and by continuity on Œ1; 1. Together with .i v/, this yields Z Z 1 0D D."3 ; / dS./ D 2 D.t/ dt D 4 C2 : (4.6.9) S2

1

Thus, C2 D 0, i.e., D  0, which shows the uniqueness. Now, all we have to prove is that the explicit representation (4.6.6) fulfills the properties .i /–.i v/. Obviously, we have (iii) and, for (ii), we see that G. I ; / 

1 1 1 ln.1    / D  ln.2/ 4 4 4

(4.6.10)

is arbitrarily often differentiable. Consider now the integral Z S2

G. I ; / dS./ D 2

Z

1

1

1 .ln.1  t/ C 1  ln.2// dt D 1; 4

(4.6.11)

4.6 Green’s Function with Respect to the Beltrami Operator

161

which gives us .i v/. For .i /, we calculate by setting t D    and using (4.1.26) 4 G. I ; / D 4 G. I t/ D 2tG 0 . I t/ C .1  t 2 /G 00 . I t/ D 2t

1 1 C .1  t 2 / D 1: 1t .1  t/2

This concludes the proof for the representation (4.6.6).

(4.6.12) t u

By applying Green’s second surface theorem, i.e., (4.1.20) of Theorem 4.1.4, we see that Z  n.n C 1/ G. I ; /Yn ./ dS./ D .1  ı0;n /Yn ./: (4.6.13) S2

Therefore, we can determine the spectral representation of the Green function as the following bilinear expansion, for 1     < 1, G. I ; / D

n 1 X X nD1 kDn

D

1 Yn;k ./Yn;k ./ n.n C 1/

1 X 2n C 1 nD1

4

(4.6.14)

1 Pn .  /: n.n C 1/

Theorem 4.6.3 (Third Green’s Surface Theorem for r  ). Let  be a fixed point of S2 . Suppose that F is a continuously differentiable function on S2 . Then, F ./ D

1 4

Z S2

F ./ dS./ 

Z     r G. I   /  r F ./ dS./: (4.6.15) S2

Proof. Let  2 S2 be fixed. For each sufficiently small % > 0, Green’s first surface identity, i.e., (4.1.19) of Theorem 4.1.4, gives us with  .; %/ D S2 n C.; %/, where C.; %/ D f 2 S2 W 1  %      1g: Z

  .;%/

 F ./ G. I   / C r F ./  r G. I   / dS./ Z

D

F ./ @ .;%/

@ G. I   / d ./; @

(4.6.16)

where  is the unit normal to the circle @ .; %/ that consists of all points  2 S2 with    D 1  %.  points outward  .; %/ D S2 n C.; %/, i.e., into C.; %/. Thus,   D p ^ . ^ /: 1  .  /2

(4.6.17)

4 Scalar Spherical Harmonics in R3

162

We use the differential equation for Green’s function and obtain Z  .;%/

F ./ G. I   / dS./ D 

1 4

Z F ./ dS./:

(4.6.18)

 .;%/

Moreover, by noting that @ .; %/ D f 2 S2 W    D 1  %g, we find that Z F ./ @ .;%/

@ G. I   / d ./ @

(4.6.19)

    .1  %/ 1 .  .1  %// d ./ D F ./ p   4% 1  .1  %/2 @ .;%/ p Z 1 1  .1  %/2 D d ./ F ./ 4 @ .;%/ % p 1 2% 1  .1  %/2 p D 2 1  .1  %/2 F .% / D  F .% /; 4 % 2 Z

where the mean value theorem with % 2 f 2 S2 W 1     D %g has been used. The continuity of F yields that F .% / ! F ./ as % !  for % ! 0 such that Z lim

%!0 @ .;%/

F ./

@ G. I   / d ./ D F ./: @

(4.6.20) t u

This gives us the desired result.

Corollary 4.6.4 (Third Green’s Surface Theorem for L ). Under the assumptions of Theorem 4.6.3, F ./ D

1 4

Z S2

F ./ dS./ 

Z     L G. I   /  L F ./ dS./: (4.6.21) S2

Proof. We use Green’s surface identity for L which is analogous to (4.1.19) of Theorem 4.1.4. Note that instead of the normal vector the tangential vector  ^

 D p 1  .  /2

(4.6.22)

is required. The same reasoning as in Theorem 4.6.3 is used to prove this corollary. t u Theorem 4.6.5 (Third Green’s Surface Theorem for  ). Let  be a fixed point of S2 . Suppose that F is a twice continuously differentiable function on S2 . Then,

4.6 Green’s Function with Respect to the Beltrami Operator

F ./ D

1 4

Z

Z S2

F ./ dS./ C

S2

G. I   / F ./ dS./:

163

(4.6.23)

Proof. Theorem 4.6.5 can be seen as a consequence of Theorem 4.6.3 and relation (4.1.13). t u Remark 4.6.6. The Green function   G . /2 I ;  W .; / 7! G . /2 I    ;

(4.6.24)

where 1      1, with respect to the iterated Beltrami operator . /2 is given by the spherical convolution which is defined by Z .K F /./ D

S2

K.  /F ./ dS./

(4.6.25)

for K 2 L1 .Œ1; 1/ and F 2 L1 .S2 /. For the Green function this yields   G . /2 I    D G. I  / G. I  / ./ Z G. I   /G. I   / dS./: D

(4.6.26)

S2

The bilinear expansion of G . /2 I    can be easily calculated by 

 G . /2 I    D

(4.6.27)

Z X 1 X 1 1 1 2k C 1 2l C 1 Pk .  /Pl .  / dS./ 4 4 .k.k C 1// .l.l C 1// S2 kD1 lD1

D

Z 1 X 1 X 1 1 2k C 1 2l C 1 Pk .  /Pl .  / dS./ 4 4 .k.k C 1// .l.l C 1// S2 kD1 lD1

D

1 X 1 X 2k C 1

4

kD1 lD1

1 1 ık;l Pk .  /; .k.k C 1// .l.l C 1//

such that 1 X 1 2k C 1 G . / I    D Pk .  /: 4 .k.k C 1//2



 2

(4.6.28)

kD1

For .; / 2 S2  S2 with 1     < 1 we obtain   G . /2 I    D G. I   /:

(4.6.29)

4 Scalar Spherical Harmonics in R3

164

This differential equation  enables us to give an explicit representation of the Green function G . /2 I    , 1      1,  G . /2 I    D D

Z S2

G. I   /G. I   / dS./

1 X 2n C 1 nD1

D

4

1 Pn .  / .n.n C 1//2

8 1 ˆ ˆ ˆ 4 ˆ < 1 .1  .ln.2//2 /  4

(4.6.30)

ˆ C ln.2/ ˆ 4 ln.1 ˆ ˆ :1  4  24

;    D 1;

ln.1    / ln.1 C   /  1 ;    ¤ ˙1;  .  / /  4 L2 1 2 1 4 2

;    D 1;

where L2 denotes the dilogarithm, i.e., L2 .t/ D

1 X 1 k t ; k2

t 2 R:

(4.6.31)

kD1

For the proof the reader is referred to Freeden and Schreiner (2009). In addition, the third Green surface theorem for  (Theorem 4.6.5) admits the following reformulation. Corollary 4.6.7 (Third Green’s Surface Theorem for  ). Let  be a point of S2 . Assume that F 2 C.2/ .S2 /. Then, 1 F ./ D 4

Z

Z S2

F ./ dS./ C

S2

  G . /2 I     F ./ dS./:

(4.6.32)

The third Green surface theorems are the points of departure for a palette of methods and procedures in spherical theory. Applications in constructive approximation are equidistribution of point sets (see Cui and Freeden 1997; Freeden and Hesse 2002; Hesse et al. 2010; Hlawka 1982), (best-)approximate integration (cf. Freeden 1980a; Freeden and Reuter 1982; Freeden et al. 1998) and spline theory (cf. Freeden 1981; Freeden and Hermann 1986; Freeden et al. 1997, 1998). The essential ideas of these approaches in constructive approximation are highlighted by exercises in our book (see Sects. 4.8, 6.7 and 10.9). Furthermore, wavelet solvers for the surface gradient equation as well as the surface curl gradient equation are derivable from the third Green surface theorems (for more details the reader should consult Freeden and Gerhards 2012). Remark 4.6.8. Theorem 4.6.5 and (4.6.30) can be used to find the following integral estimate with the help of the Cauchy–Schwarz inequality for F 2 C.2/ .S2 /:

4.7 The Hydrogen Atom

165

ˇ ˇ ˇZ ˇ Z ˇ ˇ ˇ ˇ   ˇ ˇF ./ 1 ˇ F ./ dS./ˇ D ˇ G. I   / F ./ dS./ˇˇ ˇ 2 4 2 S

Z 

(4.6.33)

S

1  Z  1 2 2 2  G. I   /G. I   / dS./  F ./ dS./ : 2 S2 „S ƒ‚ … 



1 D 4

For more details we refer to Freeden and Schreiner (2009). Theorem 4.6.9 (Differential Equation for  on S2 ). Let H 2 C.S2 / with 1 4

Z S2

H./ dS./ D 0:

(4.6.34)

Let F 2 C.2/ .S2 / satisfy the Beltrami differential equation  F D H: Z

Then, F ./ D

S2

(4.6.35)

G. I   /H./ dS./;

 2 S2 ;

(4.6.36)

and F is uniquely determined with 1 4

Z S2

F ./ dS./ D 0:

(4.6.37)

Other solutions take the form FQ D F C C with a constant C 2 R. Proof. Theorem 4.6.9 is a direct consequence of Theorem 4.6.5.

t u

4.7 The Hydrogen Atom The hydrogen atom is the simplest atom we can think of. It consists of an electron with the charge e and mass m in the Coulomb potential of a nucleus with charge e and mass M where M m. The starting point of the quantum-mechanical discussion of the hydrogen atom is the Hamilton function of an electron with charge e in the Coulomb potential of a nucleus with charge e, i.e., H.p; x/ D

p2 e2  ; 2 4ε0 jxj

x  0;

(4.7.1)

4 Scalar Spherical Harmonics in R3

166

where x is the relative distance of the electron and the nucleus and is the reduced mass given by mM D m; (4.7.2) mCM since M m. Applying the substitution rules of (3.6.7) (in this case we only need pj ! i„ @x@j , j D 1; 2; 3), we obtain the Hamilton operator H D

e2 „2 x  : 2m 4ε0 jxj

(4.7.3)

Thus, the stationary Hamilton equation is (x 2 R3 ) H .x/ D E .x/

,



„2 e2 x .x/  .x/ D E .x/; 2m 4ε0 jxj

(4.7.4)

where 2 L2 .R3 / with k kL2 .R3 / D 1 and E < 0 such that we obtain bounded states. By separation of variables .x/ D U.r/F ./, x D r, r D jxj > 0 and  2 S2 . We use the representation of the Laplace operator in polar coordinates (4.1.11) and get: 

 1  e2 2 @ @2 C U.r/F ./  E U.r/F ./ D 0: C   U.r/F ./  2 2 @r r @r r 4ε0 r (4.7.5) The radial derivatives only apply to U and the Beltrami operator applies to F such that  2  „2 @ „2 U.r/  2 @  U.r/F ./  C  F ./ 2 2m @r r @r 2m r 2 e2 U.r/F ./  E U.r/F ./ D 0: (4.7.6)  4ε0 r 

„2 2m

„ Dividing by U.r/, F ./, and  2m as well as multiplying by r 2 , this is equivalent to 2

r2

U 0 .r/ 2m U 00 .r/ C 2r C 2 U.r/ U.r/ „



  F ./ e2r C Er 2 D  4ε0 F ./

(4.7.7)

for all r > 0,  2 S2 . This can only be true if both sides are equal to a constant 2 R. Thus, we obtain for the right-hand side ( 2 S2 ): 

 F ./ F ./

D

,

 F ./ D  F ./:

(4.7.8)

4.7 The Hydrogen Atom

167

This is the Beltrami differential equation which has a polynomial solution if and only if D l.l C 1/, l 2 N0 . In this case the solution is a spherical harmonic of degree l: l X F ./ D Yl ./ D am Yl;m ./; (4.7.9) mDl

i.e., the basis system of the orthonormal spherical harmonics Yl;m , m D l; : : : ; l, describes all solutions. For the left-hand side we get with D l.l C 1/ and after dividing by r 2 and multiplying with U.r/:   2mE l.l C 1/ 2 0 2m e 2 00 C U.r/ D 0; r > 0: (4.7.10)  U .r/C U .r/C r 4ε0 „2 r „2 r2 To solve this equation, we use P .r/ D rU.r/;

(4.7.11)

0

0

P .r/ D U.r/ C rU .r/; 00

0

(4.7.12)

0

00

0

00

P .r/ D U .r/ C U .r/ C rU .r/ D 2U .r/ C rU .r/;

(4.7.13)

such that we have (r > 0) 2 P 00 .r/ : U 00 .r/ C U 0 .r/ D r r Using the abbreviation aB D P 00 .r/ C

4ε0 „2 m e2

(4.7.14)

(Bohr’s atomic radius), we find that

 2 2E 4ε0 l.l C 1/  P .r/ D 0: C  aB r e 2 aB r2

(4.7.15)

Next, we introduce the Rydberg energy ER D

„2 m e4 D 2 2 2„ .4ε0 /2 2maB

(4.7.16)

and multiply (4.7.15) by aB2 such that aB2 P 00 .r/ C

 2a

B

r

C

E a2 l.l C 1/  P .r/ D 0:  B 2 ER r

(4.7.17)

Q We p introduce dimensionless coordinates by % D r=aB , P .%/ D P .r/ and  D E=ER (observe that E < 0). This yields   2 l.l C 1/ d2 Q 2 Q P .%/ D 0;  P .%/ C   d%2 % %2

% > 0:

(4.7.18)

4 Scalar Spherical Harmonics in R3

168

To solve this equation, we set (% > 0): PQ .%/ D V .%/%lC1 exp.%/:

(4.7.19)

We obtain an equation for V given by   l C1 2 d2 d V .%/   C V .%/ .1  .l C 1// D 0: V .%/ C 2 d%2 d% % %

(4.7.20)

This equation is multiplied by % and we set y D 2%, VQ .y/ D V .%/ such that d2 d y 2 VQ .y/ C .2l C 2  y/ VQ .y/ C dy dy



 1  l  1 VQ .y/ D 0: 

(4.7.21)

Comparing this equation to the differential equation of the Laguerre polynomials .˛/ Ln (see Sect. 3.7) given by y

d d2 .˛/ Ln .y/ C .˛ C 1  y/ L.˛/ .y/ C nL.˛/ n .y/ D 0; 2 dy dy n

y > 0;

(4.7.22)

we see that ˛ D 2l C 1 and that we have a polynomial solution if and only if 1  l  1 D nr ; 

nr 2 N0 :

(4.7.23)

The solution is then given by .2lC1/ VQ .y/ D C2 L.2lC1/ .y/ D C2 Lnl1 .y/ nr

(4.7.24)

with n D nr C l C 1 D 1=, n 2 N. This means that for 1= only values in N are allowed which gives the discretization of energy by 1 En 1 e2 m e4 Dn , D 2 , En D  D  ;  ER aB 4ε0 2n2 2„2 .4ε0 /2 n2 Resubstituting V into P , then P into U , and finally U into

n 2 N: (4.7.25)

, we obtain:

U.r/ D 1r P .r/ D 1r PQ .%/ D 1r V .%/%lC1 exp.%/ 1 .2lC1/ 2r /. arB /lC1 exp. nar B / D C2 Lnl1 . na B r .2lC1/

2r 2r l D C2 a1B . n2 /l Lnl1 . na /. na / exp. nar B /: B B „ ƒ‚ … CQ2

(4.7.26)

4.8 Exercises

169

n=1, l=0 n=2, l=0 n=2, l=1 n=3, l=0 n=3, l=1 n=3, l=2

0.5

0.4

0.3

0.2

0.1

0

0

5

10

15

20

25

30

2 Fig. 4.9 Radial probability densities r 2 Un;l .r/ for n D 1; 2; 3 and corresponding l

We get after normalizing k kL2 .R3 / D 1: n;l;m .x/

D C Un;l .r/Fl;m ./ (4.7.27) s       2r l .2lC1/ 2r r .n  l  1/Š 2 1 exp  Yl;m ./: D 3=2 Lnl1 .n C l/Š n2 naB naB naB a B

n 2 N is called the main quantum number, l D 0; : : : ; n  1 is called the angular momentum quantum number and m D l; : : : ; l is called the magnetic quantum number. For n D 1, we get for the energy E1 the well-known ionization energy 1 of hydrogen, i.e., 4ε E1 D 13:6 eV (see Fig. 4.9 for an illustration of the radial 0 parts).

4.8 Exercises (Low Discrepancy Method, Locally Supported Wavelets, Up Function, Anharmonic Functions for the Ball, Fast Multipole Method, Wigner Matrices, Quaternionic Generation of Spherical Harmonics) The following exercises discuss some important features in constructive approximation and numerical analysis. The essential tool is the theory of spherical harmonics of this chapter.

4 Scalar Spherical Harmonics in R3

170

Numerical Integration by Partitioning the Sphere Partitioning the sphere provides a numerical integration rule, which is useful for application whenever the high complexity of a (geo-)physical process demands simply structured integration techniques. Exercise 4.8.1. Let the sphere S2 be partitioned into a set T of N open domains Tj  S2 , j D 1; : : : ; N , such that Tj \ Tk D ;; N [

k 6D j;

Tj D S2 ;

(4.8.1) (4.8.2)

j D1

where Tj is the closure of Tj . If we choose a point j 2 Tj for j D 1; : : : ; N , then we are led to the approximate integration rule QN .F / D

N X

kTj kF .j /;

(4.8.3)

j D1

where kTj k denotes the area of Tj . Show that ˇZ ˇ N X ˇ ˇ ˇ kTj kF .j /ˇˇ  4 max sup jF ./  F ./j: ˇ 2 F ./ dS./  j D1;:::;N S

(4.8.4)

;2Tj

j D1

Exercise 4.8.2. Let F be Lipschitz-continuous with Lipschitz constant CF . Prove that ˇ ˇZ N X ˇ ˇ ˇ ˇ F ./ dS./  kT kF . / diam .Tj /; (4.8.5) j j ˇ  4 CF max ˇ S2

j D1;:::;N

j D1

where diam.Tj / is the diameter of Tj , i.e., diam.Tj / D sup j  j:

(4.8.6)

;2Tj

Spherical Low Discrepancy Method The spherical low discrepancy method provides a selection principle for the quality of nodal sets in numerical integration with respect to their equidistribution over the sphere. Definition 4.8.3. Assume that F is of class C.2/ .S2 /. Let XN D f1N ; : : : ; NN g be a system of points on S2 . Let

4.8 Exercises

171

"XN .F / D .F /  XN .F /;

(4.8.7)

where .F / is the integral mean of F on S2 , i.e., .F / D

1 4

Z S2

F ./ dS./;

(4.8.8)

and XN .F / is the arithmetic sum with respect to XN ; XN .F / D

N 1 X F .jN /: N j D1

(4.8.9)

Exercise 4.8.4. Verify that for F 2 C.2/ .S2 /; j"XN .F /j  D.XN / V .F /;

(4.8.10)

where the L2 -discrepancy D.XN / of the system of points XN is given by 11=2 N N X   X 1 D.XN / D @ 2 G . /2 I iN  jN A N i D1 j D1 0

(4.8.11)

using the Green function with respect to . /2 of Remark 4.6.6. The L2 -variance of F reads 1=2 Z  2 j F ./j dS./ : (4.8.12) V .F / D S2

Definition 4.8.5. A sequence fXN g, XN D f1N ; : : : ; NN g  S2 , is called equidistributed or an equidistribution, if .F / D lim XN .F / N !1

(4.8.13)

holds for all F 2 C.2/ .S2 /. Exercise 4.8.6. Prove that fXN g is equidistributed if and only if lim D.xN / D 0:

N !1

(4.8.14)

Note that G.. /2 /I ; ) is explicitly available in accordance with Remark 4.6.6. Exercise 4.8.7. Consider the following point sets XN  S2 . (a) Depending on a parameter 2 N the spherical coordinates of the Reuter grid (cf. Reuter 1982) are given by

4 Scalar Spherical Harmonics in R3

172 3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

1

2

3

4

5

0

6

0

1

1

0.5

0.5

0

0

−0.5

−0.5

−1 1

−1 1 1

0.5 0.5

0

0

−0.5 −1

−1

1

2

3

4

5

6

1

0.5 0.5

0

−0.5

0

−0.5 −1

−1

−0.5

Fig. 4.10 Coordinates and points of the Reuter grid of Exercise 4.8.7(a) for D 25 (left column) and the Hammersley sequence of Exercise 4.8.7(b) (right column). The number of points is N D 786 in both cases

.#0 ; '0;1 / D .0; 0/ (North Pole); (4.8.15)   i D 1; : : : ;  1; j D 1; : : : ; i ; (4.8.16) .#i ; 'i;j / D i  ; .j  12 / 2

i .# ; ' ;1 / D .; 0/ where

(South Pole);

(4.8.17)

$

1 %    cos. /cos2 .#i /

:

i D 2 arccos sin2 .#/

(4.8.18)

This leads to a system of points XN. / with N. /  2 C 4 2 . See Fig. 4.10 (left column) for an illustration. (b) For a given N 2 N define the system of points by their spherical coordinates  .#n ; 'n / D arccos.2˚2 .n/  1/; 2 n1 ; N

n D 1; : : : ; N;

(4.8.19)

4.8 Exercises

173

where we use the mapping ˚p .n/ D

s X

aj p j 1

(4.8.20)

j D0

with the coefficients aj 2 f0; : : : ; p  1g resulting from the representation n1 D

s X

aj p j

with p 2 f2; 3; : : :g:

(4.8.21)

j D0

The sequence of these point systems is called a Hammersley sequence. See Fig. 4.10 (right column) for an illustration. (c) For a given number of points N 2 N the following system of points is generated by their spherical coordinates .#n ; 'n / D .arccos.2˚3 .n/  1/; 2˚2 .n// ;

n D 1; : : : ; N;

(4.8.22)

where we use the mappings ˚p .n/ of part (b) with p D 2 and p D 3. The resulting sequence of point systems is called a Corput–Halton sequence. See Fig. 4.11 (left column) for an illustration. (d) Given N 2 N, N > 1, we define the spiral grid XN using the spherical coordinates .#1 ; '1 / D .; 0/ (South Pole); (4.8.23)     .#j ; 'j / D arccos.hj /; 'j 1 C p3:6 .1  h2j /1=2 .mod .2// ; (4.8.24) N

.#N ; 'N / D .0; 0/

(North Pole);

(4.8.25)

2 where hj D 1 C 2j and j D 2; : : : ; N  1. See Fig. 4.11 (right column) for N 1 an illustration. (e) Given N 2 N, N > 3, we introduce the improved spiral grid XN using the spherical coordinates

(South Pole); .#1 ; '1 / D .; 0/   .#j ; 'j / D arccos.hj /; 'j 1 C p3:6

2 N rj Crj 1

.#N ; 'N / D .0; 0/



 .mod .2// ;

(North Pole);

(4.8.26) (4.8.27) (4.8.28)

where /2 hj D  1 C 2k.j with k.j / D .1  N 1 q rj D 1  h2j ; r1 D 0;

1 /.j N 3

C 1/ C

1 N C1 ; 2 N 3

and j D 2; : : : ; N  1. See Fig. 4.12 (left column) for an illustration.

(4.8.29) (4.8.30)

4 Scalar Spherical Harmonics in R3

174 3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

1

2

3

4

5

0

6

1

1

0.5

0.5

0

0

−0.5

−0.5

-1 1

−1 1 0.5

1

0

1

−1 −1

4

5

6

1 0.5

0

0

−0.5

3

0.5

0.5

0

2

0

−0.5

−0.5

−0.5 −1

−1

Fig. 4.11 Coordinates and points of the Corput–Halton sequence of Exercise 4.8.7(c) (left column) and the spiral grid of Exercise 4.8.7(d) (right column). The number of points is N D 786 in both cases

(f) Depending on the parameter Ns 2 N we construct the so-called HEALPix grid consisting of N D 12Ns2 points on the sphere in three steps. At first the northern polar cap is considered:   .#j ; 'j / D arccos 1 

k2 3Ns2



   ; 2k j  2k.k  1/  12 ;

(4.8.31)

where j D 1; : : : ; Ncap D 2Ns .Ns  1/ and $r kD

j 2



q

% b j2 c

C 1:

(4.8.32)

For the second step the remaining northern hemisphere and the equator are considered:

4.8 Exercises

175

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

1

2

3

4

5

0

6

1

1

0.5

0.5

0

0

−0.5

−0.5

−1 1

−1 1

0

1

0.5

−1 −1

3

4

5

6

1 0.5

0

0

−0.5

2

0.5

0.5

0

1

0

−0.5

−0.5

−1 −1

−0.5

Fig. 4.12 Coordinates and points of the improved spiral grid of Exercise 4.8.7(e) (left column) and the HEALPix grid of Exercise 4.8.7(f) for Ns D 8 (right column). The number of points is N D 786 for the improved spiral grid and N D 768 in the HEALPix case

where j D 1; : : : ; Nbelt



   m ; 2N .j  1/.mod .4N ; // C 1  s 2 s (4.8.33) D 4Ns .Ns C 1/, m D .k  Ns C 1/.mod 2/, and

  .#Ncap Cj ; 'Ncap Cj / D arccos 43 

kD

2k 3Ns

j

j 1 4Ns

k

C Ns :

(4.8.34)

Finally, the points of the southern hemisphere are constructed by .#N j C1 ; 'N j C1 / D .  #j ; 'j /;

(4.8.35)

where j D 1; : : : ; Nsouth D Ncap C Nbelt  4Ns . See Fig. 4.12 (right column) for an illustration. Calculate and compare the discrepancy D.XN / for each of the point systems. Generate uniformly distributed random points for an additional comparison.

4 Scalar Spherical Harmonics in R3

176

Remark 4.8.8. For more details on the point systems in (a)–(c) see Freeden et al. (1998). The system of part (a) goes back to Reuter (1982). The idea of the construction using the mappings ˚p goes back to van der Corput (1935a,b). For the theory of uniform distributions in Euclidean spaces we refer to Kuipers and Niederreiter (1974) and Niederreiter (1992). For the spiral grids we refer to Rakhmanov et al. (1994) and Saff and Kuijlaars (1997), for the improvement of part (e) see Thomson (2007). The HEALPix (Hierarchical Equal Area isoLatitude Pixelization) point system is related to cosmic microwave background experiments and the reader is referred to G´orski et al. (2005) for a detailed introduction. Note that the number of points in parts (a) and (f) cannot take all values of N.

Locally Supported Wavelets on the Sphere In constructive approximation locally supported functions are nothing new, having been discussed already by Haar (1910). The primary importance of spherically locally supported Haar kernels is the generation of an entire “basis family” by means of two operations, namely dilation and rotation. In other words, an entire set of approximations is available from one locally supported “Haar mother kernel”. Exercise 4.8.9. Let fLh gh2.1;1/  L2 .Œ1; 1/ be a non-negative, Œh; 1-locally supported family. Suppose that fLh F gh2.1;1/ given by the spherical convolution Z .Lh F /./ D Lh .  /F ./ dS./ (4.8.36) S2

is an approximate identity generated by fLh gh2.1;1/ , i.e., lim kF  Lh F kC.0/ .S2 / D 0

h!1

(4.8.37)

for all F 2 C.0/ .S2 /. Show that Z

!

hQ

jF ./  Lh F ./j  2

Lh .t/ dt h

 Z C 2



1 hQ

sup jF ./  F ./j 2S2 hhQ

Lh .t/ dt

sup jF ./  F ./j(4.8.38) 2S2 Q h1

holds for every hQ 2 Œh; 1, for every F 2 C.0/ .S2 /, and for all  2 S2 . If F is additionally Lipschitz-continuous with Lipschitz-constant CF , i.e., jF ./  F ./j  CF j  j D CF

p 1    ;

(4.8.39)

4.8 Exercises

177

then kF  Lh F kC.0/ .S2 / Z

p  2 2 CF

hQ

! 1

Z

Lh .t/ dt .1  h/ 2 C

h

1 hQ



(4.8.40) !

Q 12 Lh .t/ dt .1  h/

for every hQ 2 Œh; 1. .k/

Exercise 4.8.10. The (smoothed) Haar scaling function fLh gh2.1;1/ , k 2 N0 , is .k/ .k/ defined by Lh W Œ1; 1 ! R, t 7! Lh .t/; .k/

Lh .t/ D

( D

1

.k/

.k/ 1 Bh .t/

2

where .k/ Bh .t/

R1

dt

Bh .t/;

0

if t 2 Œ1; h/;

.t h/k .1h/k

if t 2 Œh; 1:

(4.8.41)

(4.8.42)

(a) Show that the Legendre coefficients .Bh /^ .n/ D 2 .k/

Z

1 1

.k/

Bh .t/Pn .t/ dt;

n 2 N0 ;

(4.8.43)

satisfy the recursion relation for n 2 N; 2n C 1 k C 1  n .k/ ^ .k/ h.Bh /^ .n/ C .B / .n  1/; nCkC2 nCk2 h (4.8.44) where the initial values are .Bh /^ .n C 1/ D .k/

1h 6 0; D kC1   1h 1h .k/ ^ 1 : .Bh / .1/ D 2 kC1 kC2 .Bh /^ .0/ D 2 .k/

(b) Verify that

ˇ ˇ   3 ˇ .k/ ^ ˇ ˇ.Lh / .n/ˇ D O .n.1  h// 2 k

(4.8.45) (4.8.46)

(4.8.47)

for n ! 1. (c) Show that

.k/

Lh

C.0/ .Œ1;1/

.k/

D Lh .1/ D

1 kC1 ; 2 1  h

k 2 N0 :

(4.8.48)

4 Scalar Spherical Harmonics in R3

178 .k/

(d) Verify that for k  2 the kernels Lh Lipschitz-constant takes the value CL.k/ h



.k/ 0

D Lh

C.0/ .Œ1;1/

o n .k/ (e) Verify that Lh

h2.1;1/

are Lipschitz-continuous and the

0  1 k.k C 1/ .k/ D Lh .1/ D : 2 .1  h/2

(4.8.49)

is an approximate identity as introduced in (4.8.37).

Exercise 4.8.11. Suppose fhj gj 2N0  .1; 1 is a strict monotonically increasing .k/ sequence with lim hj D 1. Let fj gj 2N0 be the spherical difference wavelet j !1

given by .k/

j

.k/

.k/

D Lhj C1  Lhj ;

k 2 N0 :

(4.8.50)

Set Z   .k/ .k/ .k/ Lhj .  /F ./ dS./; Tj .F I / D Lhj F ./ D

(4.8.51)

Z   .k/ .k/ .k/ Rj .F I / D j F ./ D j .  /F ./ dS./:

(4.8.52)

S2

S2

Prove that

.k/

lim kF  TJ .F I  /kC.0/ .S2 / D 0

(4.8.53)

J !1

holds for all F 2 C.0/ .S2 /, where .k/

.k/

.k/

.k/

TJ .F I  / D TJ 1 .F I  / C RJ 1 .F I  / D TJ0 .F I  / C

J 1 X

.k/

Rj .F I  / (4.8.54)

j DJ0

for all J; J0 2 N0 and 0  J0 < J  1. Exercise 4.8.12. Show that, for hj D 1  2j , ( .k/ Lhj .t/

D

if t 2 Œ1; 1  2j /;

0 1 .k 2

C 1/2j.kC1/.t  1 C 2j /k

if Œ1  2j ; 1;

(4.8.55)

and

.k/

j .t/ D

8 ˆ 0 ˆ ˆ ˆ ˆ 1 if not mentioned otherwise (see Fig. 4.13 for an illustration varying the parameters h and ). Exercise 4.8.15. Show that for n ! 1; L^ h; .n/ Hint: Use

R1 x

Z D 2

1

1

1=2

Pn1 .t/ dt D Cn

3

Lh; .t/Pn .t/ dt D O.n 2  /: 1=2

.x/, where Cn

(4.8.63)

is the Gegenbauer polynomial.

Exercise 4.8.16. Verify that for all >  12 ; 1 X 2 2n C 1  ^ Lh; .n/ < 1: 4 nD0

(4.8.64)

Exercise 4.8.17. Show that .2/ Lh; .

Z  / D .Lh; Lh; / .  / D

S2

Lh; .  /Lh; .  / dS./

(4.8.65)

.2/

possesses the support supp.Lh; / D f 2 S2 W 2h  1      1g. Exercise 4.8.18. Prove that for n ! 1; Z 1 ^  .2/ .2/ Lh; .t/Pn .t/ dt D O.n32 /: Lh; .n/ D 2 1

(4.8.66)

4.8 Exercises

181 1.4

0.4

−2/3 −1/3 0 1 2

0.35 0.3

−0.99 0 1

1.2 1

0.25

0.8

0.2

0.6

0.15 0.4 0.1 0.2

0.05 0

0

−0.05

−0.2 −3

−2

−1

0

1

2

3

−3

−2

−1

0

1

2

3

.2/

Fig. 4.14 Iterated smoothed Haar functions Lh; .cos.#// for h D 0:2 and D 2=3; 1=3; 0; 1; 2 (left) and the up function Uph; .cos.#// for h D 0:5 and D 0:99; 0; 1 (right)

Exercise 4.8.19. Suppose that h 2 .1; 1/. Prove that the following statements hold true: .2/

(a) If > 1, then  7! Lh; .  /,  2 S2 , is of class L2 .S2 / for each  2 S2 . .2/

(b) If >  12 , then  7! Lh; .  /,  2 S2 , is of class C.0/ .S2 / for each  2 S2 . .2/

(c) If > k1 , k 2 N0 , then  7! Lh; .  /,  2 S2 , is of class C.k/ .S2 / for each 2  2 S2 (Fig. 4.14).

Exercise 4.8.20. Suppose that h 2 .1; 1/ and > 1. We set '0 D arccos.h/ and 'k D 2k '0 , hk D cos. '2k /, k 2 N. Then, the up function Uph; is defined by .2/ .2/ the infinite spherical convolution of Lh1 ; ; Lh1 ; ; : : :, i.e., .2/

1

*

.2/

Uph; D Lh1 ; Lh2 ; : : : D

k D1

.2/

Lhk ;

(4.8.67)

Verify the following properties: (a) The Legendre symbols of the up function fulfill for every n 2 N0 ; 

Uph;

^

.n/ D

1  Y

.2/

^

Lhk ;

.n/

(4.8.68)

kD1

and lim

hD!1

 ^ Uph; .n/ D 1:

Even more, the speed of convergence fulfills for n ! 1;  ^ Uph; .n/ D O.nk / for every k 2 N.

(4.8.69)

(4.8.70)

4 Scalar Spherical Harmonics in R3

182

(b) Uph; W Œ1; 1 ! R is locally supported with supp Uph; D Œh; 1. (c) The function  7! Uph; .  /,  2 S2 , is of class C.1/ .S2 / for all  2 S2 . (d) Uph; W Œ1; 1 ! R admits the uniformly convergent orthogonal expansion in terms of Legendre polynomials Uph; D

1 X 2n C 1 

4

nD0

Uph;

^

.n/Pn ;

(4.8.71)

 ^ where Uph; .0/ D 1 and, for all n 2 N0 , 1  ^ Y  ^ .2/ 0  Uph; .n/ D Lhk ; .n/  1:

(4.8.72)

kD1

(e) For all t 2 Œ1; 1, we have the estimate 0  Uph; .t/  Uph; .1/ D

1 X 2n C 1  nD0

4

Uph;

^

.n/:

Exercise 4.8.21. Show that for all > 1 and F 2 L2 .S2 /;

lim Up F  F 2 2 D 0: h!1

h;

L .S /

(4.8.73)

(4.8.74)

j

Exercise 4.8.22. The scaling function ˚h; W Œ1; 1 ! R is defined by j

˚h; D

1

*

kDj

.2/

Lhk ; ;

j 2 N:

(4.8.75)

j

(a) Show that supp ˚h; D Œhj 1 ; 1. (b) Verify the refinement equation: j C1

.2/

j

˚h; Lhj ; D ˚h; ;

j 2 N:

(4.8.76)

Exercise 4.8.23. The scale spaces Vj , j 2 N, are given by n o j Vj D ˚h; F W F 2 L2 .S2 / :

(4.8.77)

Verify that the sequence of scale spaces fVj gj 2N constitutes a multi-resolution analysis of L2 .S2 / : (a) Every Vj , j 2 N, is a linear subspace with Vj  C.1/ .S2 /  L2 .S2 /. (b) The spaces Vj are nested, i.e., Vj  Vj C1 , j 2 N.

4.8 Exercises

183

14

j=1 j=2 j=3

12

10

j=1 j=2

8

10 6

8

4

6 4

2

2 0

0 −2

−3

−2

−1

0

1

2

3

−2

−3

−2

−1

0

1

2

3

j

Fig. 4.15 The scaling functions ˚h; .cos.#// for j D 1; 2; 3, D 0:9, and h D 1 (left) and the corresponding wavelets

j h; .cos.#//

for j D 1; 2, D 0:9, and h D 1 (right)

(c) For the intersection and the union of all spaces it holds that 1 \

Vj D V1

and

j D1

1 [

kk2L .S2 /

D L2 .S2 /:

Vj

(4.8.78)

j D1

(d) There exists a family fKj gj 2N of L2 .S2 /-kernels such that (Fig. 4.15) Vj D fKj F W F 2 Vj C1 g:

(4.8.79)

Exercise 4.8.24. We now set the spherical difference wavelets j

and

j C1

j

h; D ˚h;  ˚h;

(4.8.80)

n o j Wj D h; F W F 2 L2 .S2 / :

(4.8.81)

Prove that there exist zonal functions Zj for all j 2 N that satisfy the scale relation j C1

j

h; Zj D h; :

(4.8.82)

Show that these functions Zj are given by the Legendre coefficients ^  .2/ ^ 1  Lhj ; .n/  .2/ Lhj C1 ; .n/; Zj^ .n/ D ^  .2/ 1  Lhj C1 ; .n/ where Zj^ .0/ is arbitrary.

n 2 N;

(4.8.83)

4 Scalar Spherical Harmonics in R3

184

Remark 4.8.25. Note that for numerical calculations Zj .0/ should be chosen in such a way that the kernel Zj is strongly (space) localized. This can be achieved by setting Zj .1/ D 0 such that the Legendre coefficient of order 0 should be chosen in accordance with Zj^ .0/ D 4

1 X 2n C 1 nD1

4

.1/n Zj^ .n/:

(4.8.84)

Remark 4.8.26. The up function enables a multi-resolution analysis on S2 by use of locally supported zonal kernels. The infinite smoothness of the scaling functions allows us to apply up functions as trial functions in a large class of differential equations (see Rvachev 1990) and integral equations of satellite technology (see Freeden 1999). More details about the multi-resolution analysis by spherical up functions can be found in Freeden and Schreiner (2006).

Function Systems for the Ball B3R Exercise 4.8.27. Prove that the space L2 .B3R / can be orthogonally decomposed as L2 .B3R / D Harm.B3R / ˚ Anharm.B3R /; ?

(4.8.85)

3

where Anharm.B3R / D Harm.B3R / L .BR / is the space of anharmonic functions, i.e., functions that are not harmonic in the ball of radius R in three dimensions B3R . 2

I gn;m2N0 Ij Dn;:::;n forms a Exercise 4.8.28. Show that the function system fGm;n;j 2 3 complete orthonormal system in L .BR /, where

r I Gm;n;j

D

    n 4m C 2n C 3 .0;nC1=2/  jxj2 jxj x 2 P  1 Y n;j jxj ; m R R2 R3

x 2 B3R ;

(4.8.86) .0;nC1=2/ of degree m 2 N0 (see Sect. 3.3) and the with the Jacobi polynomial Pm orthonormal spherical harmonics Yn;j , n 2 N0 , j D n; : : : ; n. II Exercise 4.8.29. Show that the function system fGm;n;j gn;m2N0 Ij Dn;:::;n also forms 2 3 a complete orthonormal system in L .BR /, where

II Gm;n;j

8q     .0;2/ jxj x < 2mC3 2 P  1 Y m n;j 3 R jxj D q R : 3 R3

.0;2/

;

x 2 B3R n f0g;

;

x D 0;

(4.8.87)

with the Jacobi polynomial Pm of degree m 2 N0 (see Sect. 3.3) and the orthonormal spherical harmonics Yn;j , n 2 N0 , j D n; : : : ; n.

4.8 Exercises

185

I Exercise 4.8.30. Verify that the system fGm;n;j gn;m2N0 Ij Dn;:::;n can be split into harmonic and anharmonic functions. II Exercise 4.8.31. Show that the functions of fGm;n;j gn;m2N0 Ij Dn;:::;n are not continuous in x D 0 for n > 0. I the radial and angular parts of Remark 4.8.32. Note that in contrast to Gm;n;j II Gm;n;j are decoupled which can be exploited numerically (cf. Michel 2010 and the references therein). II Exercise 4.8.33. Show that for the product series of the functions Gm;n;j of (4.8.86)

K.x; y/ D

1 X n 1 X X

II II K ^ .m; n/Gm;n;j .x/Gm;n;j .y/

(4.8.88)

mD0 nD0 j Dn

with x; y 2 B3R n f0g it holds that ! 1    X am .2m C 3/ .0;2/  jxj jyj 2 R  1 Pm.0;2/ 2 R  1 Pm K.x; y/ D 3 R mD0 2n C 1  x Pn jxj  bn  4 nD0 1 X

y jyj

!  ;

(4.8.89)

where Pn denotes the Legendre polynomial of degree n 2 N0 and K ^ .m; n/ D am bn with fam gm2N0 , fbn gn2N0 real sequences. I Exercise 4.8.34. Show that the product series of the functions Gm;n;j of (4.8.86)

K.x; y/ D

1 X n 1 X X

I I K ^ .m; n/Gm;n;j .x/Gm;n;j .y/;

x; y 2 B3R ;

(4.8.90)

mD0 nD0 j Dn

(a) Converges in L2 .B3R  B3R / if and only if 1 1 X X  2 n K ^ .n; m/ < 1;

(4.8.91)

mD0 nD0

(b) Converges uniformly on B3R  B3R if 1 1 X X ˇ ^ ˇ .n C m C 12 /2m ˇK .n; m/ˇ n.2m C n/ < 1: .mŠ/2 mD0 nD0

(4.8.92)

4 Scalar Spherical Harmonics in R3

186

II Exercise 4.8.35. Show that the product series of the functions Gm;n;j of (4.8.87)

K.x; y/ D

1 X n 1 X X

II II K ^ .m; n/Gm;n;j .x/Gm;n;j .y/;

x; y 2 B3R n f0g;

mD0 nD0 j Dn

(4.8.93) (a) Converges in L2 .B3R  B3R / if and only if the sequence fK ^ .n; m/gn;m2N0 fulfills 1 1 X X  2 n K ^ .n; m/ < 1;

(4.8.94)

mD0 nD0

(b) Converges uniformly on B3R  B3R if the sequence fK ^.n; m/gn;m2N0 fulfills 1 1 X X ˇ ˇ ^ ˇK .n; m/ˇ nm5 < 1:

(4.8.95)

mD0 nD0

Remark 4.8.36. The system of (4.8.86) goes back to Ballani et al. (1993) and Dufour (1977) (see also Freeden and Michel 2004; Michel 1999) and the second system of (4.8.87) is due to Tscherning (1996). Both systems can be used to construct splines and wavelets for B3R in the same way as known from the sphere S2 . These functions find many applications in modeling the Earth’s interior, e.g., in gravimetry, normal mode variations or seismic body wave tomography. For further details we refer to Michel (2010, 2012) and the references therein.

The Fast Multipole Method The fast multipole method for the Laplace equation in three dimensions (see, e.g., Greengard and Rokhlin 1988) allows the computation of the interaction of many particles which is described by the fundamental solution of the Laplace equation as a sum of the form N X j D1

aj

1 ; jxj  yj

(4.8.96)

which is evaluated in O.N / points y. The direct approach takes a numerical effort of O.N 2 / operations which is reduced by the method to O.N /. The main idea of the method consists for one part of the subdivision of the computational domain into hierarchical sets of nested cubes which are arranged in an octal tree data structure. The other part is the possibility to consider the interaction of cubes instead of each single point by summarizing all points of

4.8 Exercises

187

one cube and representing their information by a multipole expansion. Such an expansion results from the decomposition of the single pole in terms of inner and outer harmonics. For the interaction of the different cubes two different paths are taken: the direct way, i.e., all points in one cube interact with all points in another cube, and the approximate way, i.e., by translation of the expansion coefficients. The key is to apply the approximation as often as possible and on the coarsest possible level of the tree. The direct evaluation is only to be used to the closest boxes, where the approximation is not applicable. At first, we define outer and inner harmonics based on the orthonormal system of spherical harmonics defined in Example 4.3.32 (cf. Epton and Dembart 1995 or Gutting 2008). Definition 4.8.37 (Complex Outer Harmonics). Let n 2 N0 , m 2 Z, n  m  n. Then, the complex-valued function r On;m .x/ D

4 p 1 .n C m/Š.n  m/Š nC1 Yn;m 2n C 1 jxj

D .1/m



x jxj



.n  m/Š Pn;m .cos.#//eim' jxjnC1

(4.8.97) (4.8.98)

denotes the outer harmonic of degree n and order m, where .#; '/ are the spherical x and x ¤ 0. Yn;m is the complex-valued spherical harmonic coordinates of jxj of degree n and order m of Example 4.3.32 and Pn;m is the associated Legendre function of (4.3.108). Definition 4.8.38 (Complex Inner Harmonics). Let n 2 N0 , m 2 Z, n  m  n. Then, the complex-valued function r In;m .x/ D

4 1 jxjn Yn;m p 2n C 1 .n C m/Š.n  m/Š

D .1/m



x jxj



jxjn Pn;m .cos.#//eim' .n C m/Š

(4.8.99) (4.8.100)

denotes the inner harmonic of degree n and order m, where x 2 R3 . .#; '/, Yn;m , Pn;m are the same as in Definition 4.8.37. Exercise 4.8.39. Show the symmetries On;m .x/ D .1/n On;m .x/; On;m .x/ D .1/m On;m .x/;

In;m .x/ D .1/n In;m .x/; In;m .x/ D .1/m In;m .x/;

(4.8.101) (4.8.102)

where n 2 N0 , m 2 Z with n  m  n, x 2 R3 n f0g for the outer harmonics and x 2 R3 in case of the inner harmonics.

4 Scalar Spherical Harmonics in R3

188

Exercise 4.8.40. Prove that for jxj > jyj the following expansion holds: n 1 X X 1 In;m .y/On;m .x/; D jx  yj nD0 mDn

(4.8.103)

and for jx  x0 j > jy  x0 j; n 1 X X 1 D In;m .y  x0 /On;m .x  x0 /; jx  yj nD0 mDn

(4.8.104)

where the expansion center is x0 2 R3 . Exercise 4.8.41. Consider the following differential operators which are closely related to the so-called ladder operators in quantum mechanics (see, e.g., Biedenharn and Louck 1981; Edmonds 1964; Var˘salovi˘c et al. 1988): @C D

@ @ Ci ; @x1 @x2

(4.8.105)

@ D

@ @ i ; @x1 @x2

(4.8.106)

@z D

@ : @x3

(4.8.107)

Show that these operators possess the following representation in terms of spherical coordinates:   cos.#/ @ i @ @ @C D ei' sin.#/ C ; (4.8.108) C @r r @# r sin.#/ @'   cos.#/ @ i @ @ i'  ; (4.8.109) @ D e sin.#/ C @r r @# r sin.#/ @' @z D cos.#/

sin.#/ @ @  : @r r @#

(4.8.110)

Exercise 4.8.42. Prove that the complex outer harmonic of degree n 2 N0 and order m D n; : : : ; n can be written as 8 jyj. Then, the outer harmonic of degree n 2 N0 and order m 2 Z, n  m  n, at x  y can be expanded in terms of inner and outer harmonics as follows: 0

1 n X X

On;m .x  y/ D

In0 ;m0 .y/OnCn0 ;mCm0 .x/

(4.8.112)

In0 n;m0 m .y/On0 ;m0 .x/:

(4.8.113)

n0 D0 m0 Dn0 0

1 n X X

D

n0 Dn m0 Dn0

Note that by convention In;m D 0 if jmj > n which occurs in the second formulation. Exercise 4.8.44. Verify the translation theorem for inner harmonics: Let x; y 2 R3 . Then, the inner harmonic of degree n 2 N0 and order m 2 Z with n  m  n at x  y can be expanded in a finite sum of inner harmonics 0

In;m .x  y/ D

n n X X

0

.1/n In0 ;m0 .y/Inn0 ;mm0 .x/:

(4.8.114)

n0 D0 m0 Dn0

Note again that by convention In;m D 0 if jmj > n. Consider the translation of an outer harmonics expansion such as F .x/ D

n 1 X X

Fx^;O .n; m/On;m .x  x0 /; 0

(4.8.115)

nD0 mDn

which is uniformly convergent for x 2 R3 n B3r0 .x0 /, where the radius of the ball around x0 is r0 > 0. The expansion coefficients Fx^;O .n; m/ 2 C are called 0 multipole coefficients and denote the complex coefficients corresponding to the outer harmonic of degree n and order m with the center x0 in the expansion. They usually result from an expansion of a kernel like, e.g., the single pole in (4.8.104) (for more details see, e.g., Beatson and Greengard 1997; Epton and Dembart 1995; Greengard and Rokhlin 1997; Gutting 2008). Exercise 4.8.45. Show that the expansion of the function F of (4.8.115) can be shifted to 0

F .x/ D

1 n X X n0 D0 m0 Dn0

Fx^;O .n0 ; m0 /On0 ;m0 .x  x1 /; 1

(4.8.116)

4 Scalar Spherical Harmonics in R3

190

where x 2 R3 n B3r1 .x1 /  R3 n B3r0 .x0 / and the new expansion coefficients Fx^;O .n0 ; m0 / can be computed from the old ones via 1 0

Fx^;O .n0 ; m0 / 1

D

n n X X

Fx^;O .n; m/In0 n;m0 m .x0  x1 /: 0

(4.8.117)

nD0 mDn

Exercise 4.8.46. Prove that the expansion of the function F of (4.8.116) can be translated to 0

F .x/ D

1 n X X n0 D0 m0 Dn0

Fx^;I .n0 ; m0 /In0 ;m0 .x  x2 /; 2

(4.8.118)

where x 2 B3r2 .x2 / and B3r1 .x1 / \ B3r2 .x2 / D ;. The new expansion coefficients Fx^;I .n0 ; m0 / are related to (4.8.117) by 2 Fx^;I .n0 ; m0 / 2

D

n 1 X X

0

Fx^;O .n; m/.1/n Cm OnCn0 ;m0 m .x2  x1 /: 1

(4.8.119)

nD0 mDn

Remark 4.8.47. Since we are now dealing with an expansion in inner harmonics, the notation Fx^;I .n; m/ is used for the coefficient corresponding to the inner harmonic 2 of degree n and order m of the expansion of F with center x2 to distinguish the coefficients of (4.8.119) from those of (4.8.117). The coefficients Fx^;I .n; m/ are 2 called local coefficients corresponding to the inner harmonics expansion (a.k.a. local expansion) around x2 . Exercise 4.8.48. Show that the expansion of the function F of (4.8.118) can be translated to 0

F .x/ D

1 n X X n0 D0 m0 Dn0

Fx^;I .n0 ; m0 /In0 ;m0 .x  x3 /; 3

(4.8.120)

where x 2 B3r3 .x3 /  B3r2 .x2 / and the new expansion coefficients Fx^;I .n0 ; m0 / are 3 given by .n0 ; m0 / D Fx^;I 3

1 X n X nDn0

Fx^;I .n; m/Inn0 ;mm0 .x3  x2 /: 2

(4.8.121)

mDn

Remark 4.8.49. Note that we require the geometrical situation that B3r0 .x0 /  B3r1 .x1 / for Exercise 4.8.45. Moreover, we need that B3r1 .x1 / \ B3r2 .x2 / D ; in Exercise 4.8.46 as well as B3r2 .x2 / B3r3 .x3 / in Exercise 4.8.48.

4.8 Exercises

191

Remark 4.8.50. The three translation steps of Exercises 4.8.45–4.8.48 are the key ingredients of the fast multipole algorithm. Together they allow the fast interaction for points that are not located close to each other. The fast multipole method goes back to Greengard (1988), Greengard and Rokhlin (1987) and Rokhlin (1985) and has found many applications, also in geomathematics, e.g., for the solution of the oblique boundary value problem (see Gutting 2008, 2012). An overview can be found in, e.g., Beatson and Greengard (1997), Cheng et al. (1999), Greengard and Rokhlin (1997) and Gutting (2008). For further details on the translations we refer to Epton and Dembart (1995) and the references therein.

Wigner-D-Matrices Any two unit vectors ;  0 2 S2 are related by a rotation which can be described by a matrix R D R.˛; ˇ; / depending on the three Euler angles ˛, ˇ, and , i.e.,  D R 0 ;

or  0 D RT ;

(4.8.122)

where the matrix is given by R.˛; ˇ; / D 2 cos ˛ cos ˇ cos  sin ˛ sin 6 6sin ˛ cos ˇ cos C cos ˛ sin 4  sin ˇ cos

 cos ˛ cos ˇ sin  sin ˛ cos  sin ˛ cos ˇ sin C cos ˛ cos sin ˇ sin

(4.8.123) 3 cos ˛ sin ˇ 7 sin ˛ sin ˇ 7 5: cos ˇ

The Euler angles for a rotation of the coordinate system fx1 ; x2 ; x3 g ! fx10 ; x20 ; x30 g are defined as follows: (a) A rotation about the x3 -axis through an angle ˛ 2 Œ0; 2/, (b) A rotation about the new xQ 2 -axis through an angle ˇ 2 Œ0; , (c) A rotation about the new x30 -axis through an angle 2 Œ0; 2/. Alternatively, we find the same rotation as the following combination (using the same angles): (a) A rotation about the x3 -axis through an angle 2 Œ0; 2/, (b) A rotation about the initial x2 -axis through an angle ˇ 2 Œ0; , (c) A rotation about the initial x3 -axis through an angle ˛ 2 Œ0; 2/. Exercise 4.8.51. Prove that the inverse rotation is given by the Euler angles  , ˇ, ˛ and that it is also given by the Euler angles   , ˇ,   ˛.

4 Scalar Spherical Harmonics in R3

192

If we want to rotate the coordinate system in such a way that a point  whose spherical coordinates are # and ' lies on the "3 -axis afterwards, we use R D R.'; #; 0/. A spherical harmonic of degree n and order m evaluated at  0 with the spherical coordinates # 0 and ' 0 can be represented by a linear combination of spherical harmonics of the same degree at the point  with the spherical coordinates # and '; Yn;m . 0 / D

n X

n Dk;m .˛; ˇ; /Yn;k ./;

(4.8.124)

kDn

where the rotation is described in terms of the Euler angles ˛, ˇ, . Exercise 4.8.52. Let R1 , R2 be the rotations such that RT1  D "3 D RT2  0 , where ;  0 2 S2 with the corresponding spherical coordinates #; ' and # 0 ; ' 0 . Show that the rotation R with  0 D R can be described using the Euler angles ˛ D ' 0 , ˇ D # 0  # and D '. n .˛; ˇ; / 2 C of the linear combination (4.8.124) are the matrix The factors Dk;m entries of the complex Wigner rotation matrix to degree n. They can be decomposed into n n Dk;m .˛; ˇ; / D eik˛ dk;m .ˇ/eim :

(4.8.125)

This corresponds to a decomposition of the rotation into three rotations. The first n and the last of these three are obviously diagonal. The coefficients dk;m .ˇ/ 2 R that provide the matrix for the rotation around the x2 -axis have the following explicit representation (see, e.g., Edmonds 1964; Var˘salovi˘c et al. 1988) 1

n .ˇ/ D .1/nm ..n C m/Š.n  m/Š.n C k/Š.n  k/Š/ 2 dk;m

X  .1/s s

 2nmk2s  mCkC2s sin ˇ2 cos ˇ2 sŠ.n  m  s/Š.n  k  s/Š.m C k C s/Š

;

(4.8.126)

where the summation index s is extended over all integer values for which the factorials are non-negative. n Exercise 4.8.53. Show the following symmetries of the coefficients dk;m .ˇ/: n n n n .ˇ/ D .1/km dk;m .ˇ/ D .1/km dm;k .ˇ/ D dm;k .ˇ/; dk;m

(4.8.127)

n n n .ˇ/ D .1/km dk;m .ˇ/ D dm;k .ˇ/: dk;m

(4.8.128)

(See Var˘salovi˘c et al. 1988 for a complete overview of all symmetry relations.) n .ˇ/ and Exercise 4.8.54. Prove the following relation between the coefficients dk;m the Jacobi polynomials (see Sect. 3.3):

4.8 Exercises

193

 n dk;m .ˇ/ D k;m

sŠ.s C C /Š .s C /Š.s C /Š (

where k;m D

1=2        cos ˇ2 sin ˇ2 Ps. ;/ .cos.ˇ//; (4.8.129)

1 .1/

mk

;

m  k;

;

m < k;

(4.8.130)

and D jk  mj,  D jk C mj and s D n  12 . C /. Note that for the case that both k  m  0 and k C m  0 hold, we find  n .ˇ/ D .1/mk dk;m

.n C k/Š.n  k/Š .n C m/Š.n  m/Š

1=2 (4.8.131)

  kCm   km .km;kCm/ sin ˇ2  cos ˇ2 Pnk .cos.ˇ//: Remark 4.8.55. These coefficients are usually not calculated directly, but with the help of a recursion scheme (see, e.g., Biedenharn and Louck 1981; Edmonds 1964; Var˘salovi˘c et al. 1988; Zare 1988). We choose to go along with Choi et al. (1999) where a stable algorithm is developed that computes all the unitarian Wigner rotation matrices D n 2 C.2nC1/.2nC1/ for n D 1 up to a prescribed maximal degree from the rotation matrix of the coordinate system R.˛; ˇ; / or the corresponding Euler angles. The point of departure is the matrix for the first degree, i.e., n D 1, 2 1Ccos ˇ 2

ei.˛C /

sin ˇ i˛ p e 2

6 6 sin ˇ cos ˇ D 1 .˛; ˇ; / D 6  p2 ei 4 ˇ 1cos ˇ i. ˛/ p ei˛ e  sin 2 2

1cos ˇ i.˛ / e 2

3 7

7 sin ˇ i p e 7: 2 5 1Ccos ˇ i.˛C / e 2

(4.8.132)

n Note that the orders, i.e., the indices k and m of Dk;m , run from n to n. In our matrix notation k corresponds to the row index and m to the column index. Thus, 1 the upper left corner of D 1 contains the element D1;1 .

Exercise 4.8.56. Show the following recurrence relations for the computation of the matrices of higher degrees: n n 1 n1 n 1 n1 n 1 n1 D ak;m D0;0 Dk;m C bk;m D1;0 Dk1;m C bk;m D1;0 DkC1;m ; Dk;m

(4.8.133)

n n 1 n1 n 1 n1 n 1 n1 D ck;m D0;1 Dk;mC1 C dk;m D1;1 Dk1;mC1 C dk;m D1;1 DkC1;mC1 ; Dk;m (4.8.134) n n 1 n1 n 1 n1 n 1 n1 D ck;m D0;1 Dk;m1 C dk;m D1;1 Dk1;m1 C dk;m D1;1 DkC1;m1 ; Dk;m

(4.8.135)

4 Scalar Spherical Harmonics in R3

194

where (4.8.133) holds for n C 1  m  n  1, (4.8.134) for n  m  n  2 and (4.8.135) for n C 2  m  n. All three recurrence relations (4.8.133)– (4.8.135) hold for all k D n; : : : ; n. The factors ak;m , bk;m , ck;m , and dk;m are set to  n ak;m D

 n bk;m

D

.n C k/.n  k/ .n C m/.n  m/

 12

.n C k/.n C k  1/ 2.n C m/.n  m/

n ak;m D 0 for k D ˙n;

;  12

(4.8.136)

n bk;m D 0 for k D n or k D n C 1;

;

(4.8.137)  n D ck;m

 n D dk;m

2.n C k/.n  k/ .n C m/.n C m  1/ .n C k/.n C k  1/ .n C m/.n C m  1/

 12 ;

n ck;m D 0 for k D ˙n;

;

n dk;m D 0 for k D n or k D n C 1:

 12

(4.8.138)

(4.8.139) n n Note that the factors dk;m have to be distinguished from dk;m .ˇ/ of (4.8.126).

Remark 4.8.57. The recursions (4.8.134) and (4.8.135) are only applied for the cases m D n and m D n respectively. The repeated calculation of the square roots that form these factors can be avoided via a look-up table. For more details the reader is referred to Choi et al. (1999). Exercise 4.8.58. Prove the following symmetries of the matrix entries: n n Dk;m .˛; ˇ; / D .1/mk ei2k˛i2m Dk;m .˛; ˇ; /;

(4.8.140)

n n Dk;m .˛; ˇ; / D .1/mk Dm;k . ; ˇ; ˛/ n D .1/mk Dk;m .˛; ˇ; /

(4.8.141)

n . ; ˇ; ˛/; D Dm;k

and the following relations for the inverse rotation:  1 n n n D .˛; ˇ; / k;m D Dm;k .˛; ˇ; / D Dk;m . ; ˇ; ˛/:

(4.8.142)

Now, we apply these rotations to the inner and outer harmonics of Definitions 4.8.37 and 4.8.38. Their equivalents for the inner and outer harmonics, i.e.,   n;I D n;I .˛; ˇ; / D Dk;m .˛; ˇ; /

k;mDn;:::;n

  n;O .˛; ˇ; / D n;O .˛; ˇ; / D Dk;m

k;mDn;:::;n

;

(4.8.143)

;

(4.8.144)

4.8 Exercises

195

respectively, are related by p .n C m/Š.n  m/Š n;O n n n .˛; ˇ; / D Dk;m .˛; ˇ; / p .˛; ˇ; /Ok;m ; Dk;m D Dk;m .n C k/Š.n  k/Š (4.8.145) p .n C k/Š.n  k/Š n;I n n n Dk;m .˛; ˇ; / D Dk;m .˛; ˇ; / p .˛; ˇ; /Ik;m : D Dk;m .n C m/Š.n  m/Š (4.8.146) n;O n;I For the computation of the matrices Dk;m .˛; ˇ; / and Dk;m .˛; ˇ; / the factorials n n of Ok;m and Ik;m have to be incorporated in the recurrences. n W Exercise 4.8.59. Show the following recursive properties of the factors Ok;m

s

n Ok;m

.n C m/.n  m/ n1 D Ok1;m D .n C k/.n  k/ s .n C m/.n  m/ n1 D OkC1;m .n  k/.n  k  1/

s

n1 Ok;m

.n C m/.n  m/ .n C k/.n C k  1/ (4.8.147)

n and analogously for Ik;m W

s

n Ik;m

.n C k/.n  k/ n1 n1 D Ik1;m D Ik;m .n C m/.n  m/ s .n  k/.n  k  1/ n1 D IkC1;m : .n C m/.n  m/

s

.n C k/.n C k  1/ .n C m/.n  m/ (4.8.148)

Exercise 4.8.60. Calculate the initial matrices D 1;O .˛; ˇ; / and D 1;I .˛; ˇ; / corresponding to the inner and outer harmonics from (4.8.132). Exercise 4.8.61. Show the following recurrence relations which correspond to the recursions (4.8.133)–(4.8.135) of Exercise 4.8.56 for the case of the outer harmonics: n;O 1;O n1;O 1;O n1;O Dk;m DD0;0 Dk;m .1  ık;n /.1  ık;n / C D1;0 Dk1;m .1  ık;n /.1  ık;nC1 / 1;O n1;O C D1;0 DkC1;m .1  ık;n /.1  ık;n1 /;

(4.8.149)

n;O 1;O n1;O Dk;m DD0;1 Dk;mC1 .1  ık;n /.1  ık;n / 1;O n1;O C D1;1 Dk1;mC1 .1  ık;n /.1  ık;nC1 / 1;O n1;O C D1;1 DkC1;mC1 .1  ık;n /.1  ık;n1 /;

(4.8.150)

4 Scalar Spherical Harmonics in R3

196 n;O 1;O n1;O Dk;m DD0;1 Dk;m1 .1  ık;n /.1  ık;n /

1;O n1;O C D1;1 Dk1;m1 .1  ık;n /.1  ık;nC1 / 1;O n1;O C D1;1 DkC1;m1 .1  ık;n /.1  ık;n1 /;

(4.8.151)

where (4.8.149) holds for n C 1  m  n  1, (4.8.150) for n  m  n  2 and (4.8.151) for n C 2  m  n. All three recurrence relations (4.8.149)– (4.8.151) hold for all k D n; : : : ; n. Exercise 4.8.62. Show the following recurrence relations which correspond to the recursions (4.8.133)–(4.8.135) of Exercise 4.8.56 for the case of the inner harmonics: n;I 1;I n1;I Dk;m DD0;0 Dk;m

1 1;I n1;I .n C k/.n C k  1/ .n C k/.n  k/ C D1;0 Dk1;m .n C m/.n  m/ 2 .n C m/.n  m/

1 1;I n1;I .n  k/.n  k  1/ ; DkC1;m C D1;0 2 .n C m/.n  m/ n;I 1;I n1;I Dk;m D2D0;1 Dk;mC1

.n C k/.n  k/ .n  m/.n  m  1/

1;I n1;I C D1;1 Dk1;mC1

.n C k/.n C k  1/ .n  m/.n  m  1/

1;I n1;I DkC1;mC1 C D1;1 n;I 1;I n1;I Dk;m D2D0;1 Dk;m1

(4.8.152)

.n  k/.n  k  1/ ; .n  m/.n  m  1/

(4.8.153)

.n C k/.n  k/ .n C k/.n C k  1/ 1;I n1;I C D1;1 Dk1;m1 .n C m/.n C m  1/ .n C m/.n C m  1/

1;I n1;I DkC1;m1 C D1;1

.n  k/.n  k  1/ ; .n C m/.n C m  1/

(4.8.154)

where (4.8.152) holds for n C 1  m  n  1, (4.8.153) for n  m  n  2 and (4.8.154) for n C 2  m  n. All three recurrence relations (4.8.152)– (4.8.154) hold for all k D n; : : : ; n. Now, we apply these results to the translations of the fast multipole method. Suppose F is given as an expansion in terms of outer harmonics with expansion center x0 such as (4.8.115). The coefficients Fx^;O .n; m/ are to be shifted to the new expansion 0 center x1 as in Exercise 4.8.45. However, the shift is supposed to take place along the x3 -axis, i.e, the expansion needs to be rotated, shifted, and rotated back again. Exercise 4.8.63. Prove that the rotated expansion takes the form F .x/ D

n 1 X X nD0 kDn

e x^;O .n; k/On;k .R.x  x1 /  jx0  x1 j"3 /; F 0

(4.8.155)

4.8 Exercises

197

where the rotated coefficients fulfill n X

e x^;O .n; k/ D F 0

n;O Fx^;O .n; m/Dk;m .˛; ˇ; / 0

(4.8.156)

mDn

for k D n; : : : ; n. The rotation is R D R.˛; ˇ; / and for the Euler angles it holds that .˛; ˇ; / D .0; #; '/; (4.8.157) where .'; #/ denote the spherical coordinates of

x0 x1 jx0 x1 j .

Exercise 4.8.64. Show that the shift along the x3 -axis leads to the expansion 0

F .x/ D

1 n X X n0 D0 m0 Dn0

e x^;O .n0 ; m0 /On0 ;m0 .R.x  x1 // F 1

(4.8.158)

with the shifted coefficients 0

e x^;O .n0 ; m0 / D F 1

n X nDjm0 j

n0 n

e x^;O .n; m0 / jx0  x1 j F 0 .n0  n/Š

(4.8.159)

for m0 D n0 ; : : : ; n0 . Remark 4.8.65. The application of the inverse rotation gives 0

F .x/ D

1 n X X n0 D0 k 0 Dn0

Fx^;O .n0 ; k 0 /On0 ;k 0 .x  x1 /; 1

(4.8.160)

where the coefficients corresponding to degree n0 2 N0 are rotated back via 0

.n0 ; k 0 / D Fx^;O 1

n X m0 Dn0

e x^;O .n0 ; m0 /D n00 ;O0 . ; ˇ; ˛/; F k ;m 1

(4.8.161)

for k 0 D n0 ; : : : ; n0 using the Euler angles of Exercise 4.8.63. Since the rotation matrix is unitary the inverse rotation is already given by Exercise 4.8.63. Now, suppose that F is given as an expansion in outer harmonics with expansion center x1 such as (4.8.116) or (4.8.160). The multipole expansion is to be translated into an expansion in terms of inner harmonics around the new expansion center x2 as in Exercise 4.8.46. However, the shift is supposed to take place along the x3 -axis, i.e, the expansion needs to be rotated, shifted, and rotated back again. Analogously to Exercise 4.8.63 the rotated expansion takes the form

4 Scalar Spherical Harmonics in R3

198

F .x/ D

n 1 X X

e x^;O .n; k/On;k .R.x  x2 / C jx2  x1 j"3 /; F 1

(4.8.162)

nD0 kDn

where the rotated coefficients fulfill e x^;O .n; k/ D F 1

n X

n;O Fx^;O .n; m/Dk;m .˛; ˇ; / 1

(4.8.163)

mDn

for k D n; : : : ; n. The rotation is R D R.˛; ˇ; / and the Euler angles satisfy .˛; ˇ; / D .0; #; '/; where .'; #/ denote the spherical coordinates of

(4.8.164)

x2 x1 . jx2 x1 j

Exercise 4.8.66. Show that the shift along the x3 -axis leads to the expansion 0

F .x/ D

1 n X X n0 D0 m0 Dn0

e x^;I .n0 ; m0 /In0 ;m0 .R.x  x2 //; F 2

(4.8.165)

.n C n0 /Š ; jx2  x1 jnCn0 C1

(4.8.166)

with the shifted coefficients e x^;I .n0 ; m0 / D F 2

1 X nDjm0 j

e x^;O .n; m0 / F 1

for m0 D n0 ; : : : ; n0 . Remark 4.8.67. Then, the application of the inverse rotation gives 0

F .x/ D

1 n X X n0 D0 k 0 Dn0

Fx^;I .n0 ; k 0 /In0 ;k 0 .x  x2 /; 2

(4.8.167)

where the coefficients corresponding to degree n0 2 N0 are rotated back via 0

.n0 ; k 0 / Fx^;I 2

D

n X m0 Dn0

0

e x^;I .n0 ; m0 /D n0 ;I 0 . ; ˇ; ˛/; F k ;m 2

(4.8.168)

for k 0 D n0 ; : : : ; n0 using the Euler angles of (4.8.164). Finally, suppose that F is given as an inner harmonics expansion with expansion .n; m/ are to be shifted center x2 such as (4.8.118) or (4.8.167). The coefficients Fx^;I 2 to the new expansion center x3 as in Exercise 4.8.48. However, the shift is supposed to take place along the x3 -axis, i.e, the expansion needs to be rotated, shifted, and rotated back again.

4.8 Exercises

199

Exercise 4.8.68. Prove that the rotated expansion takes the form F .x/ D

n 1 X X

e x^;I .n; k/In;k .R.x  x3 / C jx3  x2 j"3 /; F 2

(4.8.169)

nD0 kDn

where the rotated coefficients fulfill n X

e x^;I .n; k/ D F 2

n;I Fx^;I .n; m/Dk;m .˛; ˇ; /; 2

(4.8.170)

mDn

for k D n; : : : ; n. The rotation is R D R.˛; ˇ; / and for the Euler angles it holds that .˛; ˇ; / D .0; #; '/; (4.8.171) where .'; #/ denote the spherical coordinates of

x3 x2 . jx3 x2 j

Exercise 4.8.69. Show that the shift along the x3 -axis leads to the expansion 0

F .x/ D

1 n X X n0 D0 m0 Dn0

e x^;I .n0 ; m0 /In0 ;m0 .R.x  x3 //; F 3

(4.8.172)

with the shifted coefficients e x^;I .n0 ; m0 / D F 3

1 X nDn

nn0

e x^;I .n; m0 / jx3  x2 j ; F 2 .n  n0 /Š 0

(4.8.173)

for m0 D n0 ; : : : ; n0 . Remark 4.8.70. The application of the inverse rotation yields 0

F .x/ D

1 n X X n0 D0 k 0 Dn0

Fx^;I .n0 ; k 0 /In0 ;k 0 .x  x3 /; 3

(4.8.174)

where the coefficients corresponding to degree n0 2 N0 are rotated back by 0

.n0 ; k 0 / D Fx^;I 3

n X m0 Dn0

0

e x^;I .n0 ; m0 /D n0 ;I 0 . ; ˇ; ˛/; F k ;m 3

(4.8.175)

for k 0 D n0 ; : : : ; n0 using the Euler angles of Exercise 4.8.68. Since the rotation matrix is unitary the inverse rotation is already given by Exercise 4.8.68. Remark 4.8.71. The rotations and the Wigner matrices can be used to accelerate the translations of the fast multipole algorithm and reduce the effort of a translation from p 4 to 3p 3 , where p denotes the truncation degree of the expansion (for more details we refer to Cheng et al. 1999; Greengard and Rokhlin 1997; Gutting 2008; White and Head-Gordon 1996). In Gutting (2008, 2012) this method is used to

4 Scalar Spherical Harmonics in R3

200

determine the Earth’s gravitational potential from known (oblique) gravity vectors on the actual Earth’s surface.

Spherical Harmonics in Quaternionic Representation Let R4 be the four-dimensional Euclidean vector space. Suppose that ei 1 D "i , i D 1; : : : ; 4, is the canonical orthonormal system in R4 . Hence, any vector a 2 R4 can be written as 3 X ai ei D a0 e0 C a; O a0 2 R; (4.8.176) aD i D0

where aO D a1 e1 C a2 e2 C a3 e3 2 R3 ;

ai 2 R; i D 1; 2; 3:

(4.8.177)

Let b 2 R4 be another vector such that a multiplication law is given by O 0 C aO ^ bO C a0 bO C ab ab D .a0 b0  aO  b/e O 0;

(4.8.178)

where aO  bO and aO ^ bO are the scalar product and the vector product in the Euclidean space R3 , respectively. Obviously, this product is not commutative. With the help of this multiplication the vector space R4 is furnished with the algebraic structure of a ring (see, e.g., G¨urlebeck and Spr¨oßig 1989), i.e., it is an algebra which is named real quaternionic algebra. In the honour of its discoverer (Hamilton 1866) it is designated by H. Exercise 4.8.72. Show that the members of H, i.e., the quaternions a 2 R4 , may be identified with matrices of R44 of the form 2

a0 6a1 aD6 4a2 a3

a1 a0 a3 a2

a2 a3 a0 a1

3 a3 a2 7 7 a1 5

(4.8.179)

a0

and the product of two quaternions corresponds to the matrix product of their matrix representations. Associated to the representation (4.8.179) the basis ei , i D 0; : : : ; 3, is given by 2

10 60 1 e0 D 6 40 0 00

3 2 00 0 1 61 0 0 07 7 ; e1 D 6 40 0 1 05 01 0 0

3 2 0 0 0 0 1 60 0 0 0 07 7 ; e2 D 6 41 0 0 0 15 1 0 0 1 0

3 2 0 00 60 0 17 7 ; e3 D 6 40 1 05 0 10

3 0 1 1 0 7 7: 0 05 0 0 (4.8.180)

4.8 Exercises

201

Remark 4.8.73. In the classical nomenclature (cf. Clifford 1878, Hamilton 1866) the basis elements are represented by 1; i; j; k. This is done in analogy to the approach known from the complex space C. The conjugate quaternions are defined by a D a0 e0 

3 X

a 2 H:

ai ei ;

(4.8.181)

i D1

From the law of multiplication (4.8.178) it immediately follows that aa D aa D

3 X

ai2 e0 ;

(4.8.182)

i D0

such p that by setting e0 D 1 (see Remark 4.8.73) the (Euclidean) norm jaj is equal to aa. The real part and the imaginary part is understood to be Re.a/ D a0 e0 D 12 .a C a/; Im.a/ D aO D

1 2 .a

(4.8.183)

 a/:

The inverse of the quaternion a 2 H n f0g is given by a1 D

(4.8.184) a . jaj2

Exercise 4.8.74. Verify that, for a; b 2 H and 2 R the following rules hold: a C b D a C b;

(4.8.185)

ab D ba;

(4.8.186)

a D a; p jaj D aa D jaj D j  aj;

(4.8.187)

j aj D j jjaj;

(4.8.189)

jabj D jajjbj;

(4.8.190)

Re.ab/ D Re.ba/;

(4.8.191)

jjaj  jbjj  ja  bj  jaj C jbj:

(4.8.188)

(4.8.192)

Exercise 4.8.75. Show that the basis quaternions satisfy the following relations: e02 D e0 ; ei2

D e0 ;

ei ej C ej ei D 0;

(4.8.193) i D 1; 2; 3; i ¤ j; i; j D 1; 2; 3;

(4.8.194) (4.8.195)

4 Scalar Spherical Harmonics in R3

202

e0 ei D ei e0 D ei ;

i D 0; 1; 2; 3;

(4.8.196)

e1 e2 D e3 ;

(4.8.197)

e2 e3 D e1 ;

(4.8.198)

e3 e1 D e2 :

(4.8.199)

Definition 4.8.76. Let R0;q stand for the real vector space Rq , q 2 N, provided with a non-degenerate symmetric bilinear form B such that, for any vectors a; b 2 R0;q ; B.a; b/ D a  b;

(4.8.200)

where a  b denotes the Euclidean inner product of a and b. In particular, B.ei ; ej / D ıi;j ;

i; j D 1; : : : ; q:

(4.8.201)

Looking at the associated quadratic form we have Q.a/ D B.a; a/ D jaj2 ;

(4.8.202)

where jaj stands for the Euclidean norm of a. Of course, Q.ei / D B.ei ; ei / D 1;

i D 1; : : : ; q:

(4.8.203)

The real Clifford algebra Cl.q/ constructed over R0;q is a real, linear associative algebra with identity e0 D "1 , of dimension 2q , containing R and R0;q as subspaces and in which for each vector a 2 R0;q ; a2 D Q.a/ D jaj2 :

(4.8.204)

such that an Henceforth we take the orthonormal basis e1 ; : : : ; eq for Pgranted q arbitrary element a 2 R0;q is representable in the form a D j D1 aj ej : A basis of Cl.q/ consists of the elements feA gAf1;:::;qg , where A is an element of the power set P .f1; : : : ; qg/ of f1; : : : ; qg such that, for A D fi1 ; : : : ; il g with 1  i1 < : : : < il  q, eA D ei1    eil and e; D e0 is the identity element. In such a way any element a 2 Cl.q/ may be written as aD

X

aA eA ;

aA 2 R:

(4.8.205)

A

Exercise 4.8.77. Verify the following statements: (a) Cl.1/ is isomorphic to C, i.e., Cl.1/ Š C. Moreover, dim.Cl.1// D 2, where e0 D e; , e1 D ef1g with e12 D 1 is a basis for Cl.1/ (in the classical Gaussian nomenclature of the complex space C, e0 D 1 and e1 D i).

4.8 Exercises

203

(b) Cl.2/ is isomorphic to H, i.e., Cl.2/ Š H. Moreover, dim.Cl.2// D 4, where e0 D e; , e1 D ef1g , e2 D ef2g , e3 D ef1;2g D e1 e2 is a basis for Cl.2/ (in the classical Hamiltonian nomenclature e0 D 1, e1 D i, e2 D j, and e3 D k). Hint: Apply (4.8.200) to the basis vectors ei , i D 0; 1; 2; 3. For more details on Clifford algebras the reader is referred to, e.g., Brackx et al. (1982), Delanghe (2001), G¨urlebeck et al. (2008) and Qian et al. (2004) and the references therein. Next, we deal with functions f defined on regions G and/or their boundaries @G in H. The so-called H-valued functions can be represented by f D

3 X

Fi ei ;

(4.8.206)

i D0

where the scalar functions Fi are real-valued. Properties such as continuity, differentiability, integrability and so on which are ascribed to f can be defined via the components Fi , i D 0; 1; 2; 3; in the standard way. In this manner, spaces of .k/ .p/ .k/ .p/ H-valued functions cH .G /, lH .G /, cH .@G /, lH .@G / are denoted accordingly. As for vector-valued functions in Chap. 5 we use lower case letters for these function spaces. 4 P3As already pointed out, any vector a 2 R corresponds to a quaternion a D i D0 ai ei 2 H. In formal analogy, the vector 

@ @ @ @ @x0 ; @x1 ; @x2 ; @x3

 (4.8.207)

corresponds to an operator @ given by @D

3 X i D0

@ e: @xi i

(4.8.208)

Remark 4.8.78. In the framework of C, the operator @ in (4.8.208) corresponds to 2 @z@ D

@ @x

@ C i @y ;

(4.8.209)

while the conjugate operator @D

@ @x0 e0



3 X i D1

@ @xi ei

(4.8.210)

can be regarded as an extension to 2 @z@ D

@ @x

@  i @y :

(4.8.211)

4 Scalar Spherical Harmonics in R3

204

Definition 4.8.79. The operator @ defined by (4.8.208) is called the Cauchy– Riemann operator, while D D @  @x@0 e0 (4.8.212) is named the Dirac operator. According to its definition the operator DD

3 X i D1

@ @xi ei

(4.8.213)

only operates on the variables x1 , x2 , x3 . Clearly, D D D ; where  is the Laplacian in R3 . The action of the Dirac operator D on an H-valued function f of the form (4.8.206) from the left and right is denoted by D f and f D, respectively. The Dirac operator D has important applications in physics, from which we mention the pre-Maxwell equations. Exercise 4.8.80. Use Exercise 4.8.75 to show that D f may be written in the form D f D r  fO C rF0 C r ^ fO;

(4.8.214)

where rF0 D

3 X i D1

r  fO D

3 X i D1

r ^ fO D



@ @xi

F0 ei ;

(4.8.215)

@ @xi

Fi ;

(4.8.216)

@ @x2 F3



@ @x3 F2



e1 C



@ @x3 F1



@ @x1 F3



e2 C



@ @x1 F2



@ @x2 F1

 e3 :

(4.8.217) In consequence, if f is H-continuously differentiable with D f D 0, we are led to the differential system r  fO D 0;

(4.8.218)

rF0 C r ^ fO D 0:

(4.8.219)

In particular, where F0 is constant, the resulting pre-Maxwell equations describe an irrotational fluid without sources. Equations (4.8.218) and (4.8.219) have been used by Moisil and Teodorescu (1931) as the point of departure for the development of the hypercomplex function theory (see also G¨urlebeck et al. 2008 for more details). Exercise 4.8.81. Let f W H ! H be given by x D .x0 ; x1 ; x2 ; x3 /T 7! f .x/ D f .x0 ; x1 ; x2 ; x3 / D x1 e1  x2 e2 :

(4.8.220)

4.8 Exercises

205

Show that the following statements are valid: (a) D f D 0, (b) D.f f / D 2x1 e1  2x2 e2 ¤ 0, (c) For x ¤ 0 we have f .x/ ¤ 0 and (in contrast to complex analysis) the function f 1 given by x1 e1 C x2 e2 (4.8.221) f 1 .x/ D x12 C x22 satisfies D f 1 .x/ D

2.x12  x22 / 4x1 x2 e0 C 2 e3 ¤ 0: .x12 C x22 /2 .x1 C x22 /2

(4.8.222)

Exercise 4.8.81 shows that the quaternionic product ff of a function f with Df D 0 does not satisfy D.ff / D 0: The trouble in Exercise 4.8.81 consists in the noncommutativity of the H-multiplication law. An idea to overcome this difficulty is due to Delanghe (1970). Now, we turn to the concept of H-holomorphic functions. Definition 4.8.82. Let f be an H-valued continuously differentiable function in a region G  H. Then f is called H-right holomorphic (or H-left holomorphic) in G , if for all x 2 G there exist quaternions wk such that f .x C h/ D f .x/ C

3 X

wk .hk  h0 ek / C o.jhj/;

h ! 0;

(4.8.223)

3 X .hk  h0 ek /wk C o.jhj/;

h ! 0;

(4.8.224)

kD1

f .x C h/ D f .x/ C

kD1

respectively, where hk are the coordinates of h. Exercise 4.8.83 (Cauchy–Riemann Differential Equations). Suppose that the Hvalued function f is continuously differentiable in G  H. Show that f is H-right holomorphic if and only if f @ D 0 in G (f is H-left holomorphic if and only if @f D 0 in G ). Remark 4.8.84. Throughout our work, we agree upon the fact that H-left holomorphic functions are just called H-holomorphic. The set of all H-holomorphic functions on G is denoted by holH .G /. The Laplace operator (in R4 ) is given by  D @@ D @@ D .2/

3 X @2 : @xi2 i D0

(4.8.225)

Obviously, if f 2 cH .G /, then the equation f D 0 follows from @f D 0 in G , i.e., each component function of an H-holomorphic function is a harmonic function.

4 Scalar Spherical Harmonics in R3

206

In this way, the theory of H-holomorphic functions refines the theory of harmonic functions (see, e.g. Brackx and Delanghe 2003 for a more general approach in terms of Clifford algebras). Assume that a real function F W G ! R mapping x D .x0 ; x1 ; x2 /T 7! F .x0 ; x1 ; x2 / with .x0 ; x1 ; x2 /T 2 G  R3 , is twice continuously differentiable and harmonic, i.e., 2 X @2 F .x/ D 0 (4.8.226) @xi2 i D0 for all x 2 G . Following our nomenclature (and considering F W G ! R with G embedded in H) we therefore have @@F D

3 2 X X @2 @2 F .x/ D F .x/ D 0 @xi2 @xi2 i D0 i D0

(4.8.227)

in G . Note that F does not depend on x3 . Identity (4.8.227) leads us to the following observation: Applying the operator @ from the left to the harmonic function F provides us with an H-holomorphic function. This strategy is used to derive Hspherical harmonics ˇ ˇ @UnC1;0 ˇ 2 ; S

ˇ ˇ @UnC1;m ˇ 2 ; S

ˇ ˇ @VnC1;m ˇ

S2

(4.8.228)

from the standard inner harmonics given by UnC1;m .x/ D r nC1 PnC1;m .cos.#// cos.m'/;

(4.8.229)

UnC1;0 .x/ D r

(4.8.230)

nC1

PnC1 .cos.#// cos.m'/;

VnC1;m .x/ D r nC1 PnC1;m .cos.#// sin.m'/;

(4.8.231)

where n 2 N0 , m D 1; : : : ; n C 1, PnC1;m are the associated Legendre functions of (4.3.108) and PnC1 is the Legendre polynomial of degree nC1. The point x 2 R3 is represented in terms of polar coordinates .r; #; '/ as introduced in (4.1.2). Exercise 4.8.85. Show that for n 2 N0 , fyn;m ; yn;0 ; zn;m gmD1;:::;nC1 given by ˇ ˇ yn;m .#; '/ D@ .UnC1;m .x// ˇ

rD1

(4.8.232)

DAn;m cos.m'/e0 C .Bn;m cos.'/ cos m'  Cn;m sin.'/ sin.m'// e1 C .Bn;m sin.'/ cos m' C Cn;m cos.'/ sin.m'// e2 ; ˇ ˇ yn;0 .#; '/ D@ .UnC1;0 .x// ˇ rD1

DAn;0 e0 C Bn;0 cos.'/e1 C Bn;0 sin.'/e2 ;

(4.8.233)

4.8 Exercises

207

ˇ ˇ zn;m .#; '/ D@ .UnC1;m .x// ˇ

(4.8.234)

rD1

DAn;m sin.m'/e0 C .Bn;m cos.'/ sin m' C Cn;m sin.'/ cos.m'// e1 C .Bn;m sin.'/ sin m'  Cn;m cos.'/ cos.m'// e2 ; with An;m

Bn;m

  ˇ 1 ˇ 2 d sin .#/ dt PnC1;m .t/ˇ D C .n C 1/ cos.#/PnC1;m .cos.#// ; t Dcos.#/ 2 (4.8.235)   ˇ 1 ˇ d sin.#/ cos.#/ dt PnC1;m .t/ˇ D  .n C 1/ sin.#/PnC1;m .cos.#// ; t Dcos.#/ 2 (4.8.236)

Cn;m D

m 1 PnC1;m .cos.#//; 2 sin.#/

(4.8.237)

represents a system of 2n C 3 H-spherical harmonics. Of course, the system of 2nC3 H-spherical harmonics developed in Exercise 4.8.85 is not closed in l2H .S2 /, since all occurring functions do not contain an e3 -coordinate different from zero. Following an approach due to G¨urlebeck et al. (2008) we consider the extended system fyn;0 ; yn;m ; zn;m ; yn;0 e3 ; yn;l e3 ; zn;l e3 gmD1;:::;nC1I lD1;:::;nC1

(4.8.238)

that consists of 4n C 6 functions. However, it is easily seen that zn;nC1 e3 D yn;nC1 ;

(4.8.239)

yn;nC1 e3 D zn;nC1 :

(4.8.240)

Therefore, we continue with the system of 4n C 4 H-spherical harmonics fyn;0 ; yn;m ; zn;m ; yn;0;3 ; yn;l;3 ; zn;l;3 gmD1;:::;nC1I lD1;:::;n

(4.8.241)

where we have introduced the abbreviations yn;0;3 D yn;0 e3 ;

yn;l;3 D yn;l e3 ;

zn;l;3 D zn;l e3 :

(4.8.242)

Exercise 4.8.86. Show that the system (4.8.241) satisfies the following properties: (a) For every n 2 N0 , the subsystem fyn;0 ; yn;m ; zn;m gmD1;:::;nC1 is an orthogonal system in the l2H .S2 /-sense, where the l2H .S2 /-norms are given by

4 Scalar Spherical Harmonics in R3

208

p .n C 1/;

kyn;0 kl2 .S2 / D H

(4.8.243) s

kyn;m kl2 .S2 / D kzn;m kl2 .S2 / D H

H

.n C 1 C m/Š  .n C 1/ 2 .n C 1  m/Š

(4.8.244)

with m D 1; : : : ; n C 1. (b) For every n 2 N0 , the subsystem fyn;0;3 ; yn;l;3 ; zn;l;3 glD1;:::;n is an orthogonal system in the l2H .S2 /-sense, where the l2H .S2 /-norms are given by kyn;0;3 kl2 .S2 / D H

p .n C 1/;

(4.8.245) s

kyn;l;3 kl2 .S2 / D kzn;l;3 kl2 .S2 / D H

H

.n C 1 C l/Š  .n C 1/ 2 .n C 1  l/Š

(4.8.246)

with l D 1; : : : ; n. (c) For every n 2 N0 and m D 1; : : : ; n C 1, l D 1; : : : ; n, we have that hyn;0 ; yn;0;3 il2 .S2 / D hyn;0 ; yn;l;3 il2 .S2 / D hyn;0 ; zn;l;3 il2 .S2 / D 0;

(4.8.247)

hyn;m ; yn;0;3 il2 .S2 / D hyn;m ; yn;l;3 il2 .S2 / D 0;

(4.8.248)

hzn;m ; yn;0;3 il2 .S2 / D hzn;m ; zn;l;3 il2 .S2 / D 0;

(4.8.249)

H

H

H

H

H

H

H

and ( hyn;m ; zn;l;3 il2 .S2 / D hzn;m ; yn;l;3 il2 .S2 / D H

H

0

; m ¤ l;

.nCmC1/Š  2 m .nmC1/Š

; m D l: (4.8.250)

In consequence, only a few functions out of the different systems are nonorthogonal. For a total orthonormalization process we first start with the normalized polynomials corresponding to (4.8.241): fyQn;0 ; yQn;m ; zQn;m ; yQn;0;3 ; yQn;l;3 ; zQn;l;3 gmD1;:::;nC1I lD1;:::;n :

(4.8.251)

Hence, from our preceeding considerations, it is clear that the two systems fyQn;0 ; yQn;m ; zQn;m gn2N0 ImD1;:::;nC1 ;

(4.8.252)

fyQn;0;3 ; yQn;l;3 ; zQn;l;3 gn2N0 I lD1;:::;n

(4.8.253)

are orthonormal systems, respectively. In addition, for every n 2 N0 , all H-spherical harmonics of the systems (4.8.252) and (4.8.253) are orthogonal, however, with the only exception.

4.8 Exercises

209

Exercise 4.8.87. Show that, for n 2 N0 and l D 1; : : : ; n, hyQn;l ; zQn;l;3 il2 .S2 / D hQzn;l ; yQn;l;3 il2 .S2 / D H

H

1 : nC1

(4.8.254)

Exercise 4.8.88. Verify that the system ˚

ON ON ON ON ON yn;0 ; yn;m ; zn;m ; yn;0;3 ; yn;l;3 ; zON n;l;3

n2N0 I mD1;:::;nC1I lD1;:::;n

(4.8.255)

with ON yn;0 D yQn;0 ;

(4.8.256)

ON D yQn;m ; yn;m

(4.8.257)

zON Qn;m ; n;m D z

(4.8.258)

ON D yQn;0;3 ; yn;0;3

(4.8.259)

1 ON yn;l;3 D p ..n C 1/yQn;l;3 C l zQn;l / ; .n C 1/2  l 2

(4.8.260)

1 ..n C 1/Qzn;l;3  l yQn;l / zON n;l;3 D p .n C 1/2  l 2

(4.8.261)

represents a fully l2H .S2 /-orthonormalized system of 4n C 4 H-spherical harmonics. The proof of Exercise 4.8.88 is straightforward by observing that the first 2n C 3 functions in (4.8.255) are already orthonormal. In the next step it should be realized that zQn;l;3 

nC1 X l ON ON ON ON yQn;l DQzn;l;3  yn;0 hyn;0 ; zQn;l;3 il2 .S2 /  yn;m hyn;m ; zQn;l;3 il2 .S2 / H H nC1 mD1



nC1 X mD1

ON zON Qn;l;3 il2 .S2 / : n;m hzn;m ; z

(4.8.262)

H

Next, we go over to the H-holomorphic inner harmonics defined by ˚ n ON n ON n ON n ON n ON n ON r yn;0 ; r yn;m ; r zn;m ; r yn;0;3 ; r yn;l;3 ; r zn;l;3 n2N

: (4.8.263)

0 I mD1;:::;nC1I lD1;:::;n

Exercise 4.8.89. Show that, for every n 2 N0 , the system of n C 1 H-holomorphic inner harmonics o n ON n ON ; r yn;2 ; r n zON (4.8.264) r n yn;0 n;2 ;3 D1;:::;b.nC1/=2cI D1;:::;bn=2c

4 Scalar Spherical Harmonics in R3

210

forms an orthogonal basis of the space of all H-holomorphic homogeneous polynomials of degree n. Exercise 4.8.90. Show that, for every n 2 N0 , the system of n C 1 H-holomorphic inner harmonics o n ON ; r n zON (4.8.265) r n yn;2C1 n;2 1;3 D0;:::;bn=2cI D1;:::;b.nC1/=2c

forms an orthogonal basis of the space of all H-holomorphic homogeneous polynomials of degree n. In order to establish orthonormality we need the following statement. Exercise 4.8.91. Let fHn;l g be an l2H .S2 /-orthonormal system of H-spherical harmonics. Prove that hHn;l ; Hk;j il2 .B3 / D H

1

1 hHn;l ; Hk;j il2 .S2 / : H nCkC3

(4.8.266)

Indeed, by virtue of Exercise 4.8.91, G¨up rlebeck et al. (2008) are able to guarantee that each of the two systems (with kn D 2n C 3 ) n n

ON ON kn r n yn;0 ; kn r n yn;2 ; kn r n zON n;2 ;3 ON kn r n yn;2C1 ; kn r n zON n;2 1;3

o

o n2N0 I D1;:::;b.nC1/=2cI D1;:::;bn=2c

n2N0 I D0;:::;bn=2cI D1;:::;b.nC1/=2c

(4.8.267) (4.8.268)

represents a closed l2H .B31 /-orthonormal system in the space l2H .B31 / \ holH .B31 / D l2H .B31 / \ ker.@/:

(4.8.269)

Remark 4.8.92. For a discussion of spherical operators and differential equations in Clifford algebras the reader is referred to, e.g., Liu and Ryan (2002) and the references therein. Homogeneous harmonic polynomials in Clifford algebras are discussed by Brackx and Delanghe (2003) and many others. In particular, the results presented in this work for quaternionic spherical harmonics can be formulated in Clifford algebras, too. In this respect it should be noted that (De Bie and Sommen 2007; De Bie et al. 2009) extended essential results known from classical spherical harmonics theory such as the Funk–Hecke formula, decomposition theorems, eigenvalue expansions to the so-called superspace using techniques of Clifford analysis.

Chapter 5

Vectorial Spherical Harmonics in R3

Various applications imply different formulations of vector spherical harmonics, putting the accent on different issues (see, e.g., Sects. 1.3 and 1.4). One important aspect in our understanding is the easy transition from scalar spherical harmonics to the vectorial ones. A simple approach is to formulate the vectorial problem in terms of Cartesian components. However, this procedure leads back to anisotropic scalar component equations, so that the physical relevance usually is difficult to realize, the mathematical formulation is lengthy, and modeling often becomes complicated. For many physically motivated applications, where the underlying differential equations are separable in spherical coordinates, it is advantageous to separate the vector fields into a normal and a tangential part (see, e.g., Sect. 1.3). Our approach to vector spherical harmonics takes this into account. These geophysically motivated aspects are guaranteed adequately within a vectorial framework, transforming scalar functions into vector fields by use of certain operators o.i / ; i D 1; 2; 3, that are defined in Sect. 5.2. The operator o.1/ separates the normal part of a vector field from the tangential part, o.2/ defines a (tangential) surface gradient field that is curl-free, while o.3/ yields a (tangential) surface curl gradient field that is divergence-free. In doing so, we are led to definitions that are independent of any particular choice of spherical harmonics and do not relate to any particular choice of a spherical coordinate system. Moreover, the rotational symmetry, i.e., the isotropy can be reflected in suitable (vectorial) manner. Further material can be found, e.g., in Blatt and Weisskopf (1952), Freeden et al. (1998), Freeden and Schreiner (2009), Gervens (1989), Hill (1954), and Morse and Feshbach (1953), for applications we refer to Backus et al. (1996), Gerhards (2011), Maier (2003), Mayer (2003) (geomagnetism), Freeden (1999), Nutz (2002), Rummel and van Gelderen (1992), Schreiner (1994), Freeden and Nutz (2011) (satellite geodesy), or Fengler (2005), Fengler and Freeden (2005), Freeden and Schreiner (2009) (Navier–Stokes equation on the sphere), Abeyratne et al. (2003), Bauch (1981), Freeden (1990), Freeden and Michel (2004, 2005), Freeden et al. (1990), Grafarend (1986), Gervens (1989), Gurtin (1972), Jaswon and Symm (1977), Lurje (1963), Knops and Payne (1971),

W. Freeden and M. Gutting, Special Functions of Mathematical (Geo-)Physics, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-0348-0563-6 5, © Springer Basel 2013

211

5 Vectorial Spherical Harmonics in R3

212

and Mochizuki (1988) (elasticity). For applications in quantum mechanics using another approach we just mention Edmonds (1964), James (1976), and Jones (1985). The layout of this chapter on vector spherical harmonics is as follows: Sect. 5.1 is concerned with the necessary notation for spherical vector fields, in particular the occurring differential operators. It also summarizes several helpful results from vector analysis. In Sect. 5.2, we introduce the vector spherical harmonics based on the properties of the operators o.i / , i D 1; 2; 3. Section 5.3 is dedicated to the Helmholtz decomposition formula for spherical vector fields by use of the Green function with respect to the Beltrami operator that has been introduced in Sect. 4.6. Section 5.4 shows the closure and completeness of vector spherical harmonics intrinsically on the sphere based on Bernstein summability. The interrelations between vector spherical harmonics and homogeneous harmonic vector polynomials are investigated in more detail in Sect. 5.5. Section 5.6 shows us that the vector spherical harmonics can be regarded as eigenfunctions of a vectorial analog of the Beltrami operator. Section 5.7 presents the formulation of the addition theorem in terms of vector spherical harmonics thereby introducing the Legendre rank-2 tensor kernels as the counterpart of the Legendre polynomials in the scalar case. In Sect. 5.8, we prove vectorial versions of the Funk–Hecke formula and deal with orthogonal invariance. Vectorial counterparts of the Legendre polynomial are investigated in detail in Sect. 5.9. For an application we go back to the Cauchy– Navier equation of Sect. 1.4 and consider vector polynomials for its solution in Sect. 5.10. We develop a system of “Cauchy–Navier polynomials” such that the vector spherical harmonics of Sect. 5.2 act as corresponding boundary values in the Cauchy–Navier boundary value problem. Finally, we provide the reader with some exercises in Sect. 5.11 that are concerned with Heisenberg’s uncertainty principle, its application to classify zonal functions, as well as the treatment of the Navier–Stokes equation on the sphere of Sect. 1.3.

5.1 Basic Notation Vector fields on the unit sphere f W S2 ! R3 can be written in the form f ./ D

3 X

Fi ./"i ;

 2 S2 ;

(5.1.1)

i D1

where the component functions Fi are given by Fi ./ D f ./  "i (as usual, vector fields will be denoted by lower case letters). First, we introduce function spaces for vector fields on the sphere. l2 .S2 / denotes the space of all square-integrable vector fields on S2 which is a Hilbert space in connection with the scalar product Z hf; gil2 .S2 / D

f ./  g./ dS./; S2

f; g 2 l2 .S2 /:

(5.1.2)

5.1 Basic Notation

213

The space c.p/ .S2 /, 0  p  1, consists of all p-times continuously differentiable vector fields on the sphere. c.S2 / D c.0/ .S2 / is the Banach space of continuous vector fields on the sphere with respect to the norm kf kc.S2 / D sup jf ./j;

f 2 c.S2 /:

(5.1.3)

2S2

We know that c.S2 /

kkl2 .S2 /

D l2 .S2 /:

(5.1.4)

As in the scalar case, there is the following norm estimate for f 2 c.S2 /: kf kl2 .S2 / 

p

4 kf kc.S2 / :

(5.1.5)

Any vector field f 2 c.S2 / can be decomposed into its normal and its tangential part: f ./ D fnor ./ C ftan ./;

(5.1.6)

where the normal and tangential field are given by  7! pnor f ./ D fnor ./ D .f ./  /;

 2 S2 ;

 7! ptan f ./ D ftan ./ D f ./  .f ./  /;

(5.1.7)  2 S2

(5.1.8)

with the projection operators pnor and ptan . For f; g 2 c.S2 /,  2 S2 , it holds that f ./  g./ D fnor ./  gnor ./ C ftan ./  gtan ./:

(5.1.9)

The spaces c.S2 / and l2 .S2 / can also be decomposed accordingly: cnor .S2 / D ff 2 c.S2 / W f D pnor f g;

(5.1.10)

ctan .S2 / D ff 2 c.S2 / W f D ptan f g;

(5.1.11)

c.S / D cnor .S / ˚ ctan .S /; 2

2

2

(5.1.12)

as well as l2nor .S2 / D cnor .S2 / l2tan .S2 / D ctan .S2 / l .S / D 2

2

l2nor .S2 /

kkl2 .S2 /

;

(5.1.13)

;

(5.1.14)

l2tan .S2 /:

(5.1.15)

kkl2 .S2 /

˚

5 Vectorial Spherical Harmonics in R3

214

From Sect. 4.1 we remember the surface gradient operator p 1 @ @ C "t 1  t 2 r  D "' p @t 1  t 2 @'

(5.1.16)

which contains the tangential derivatives of the gradient and the surface curl gradient operator (L F ./ D  ^ r F ./ for F 2 C.1/ .S2 /): p @ 1 @ : L D "' 1  t 2 C "t p 2 @t @' 1t

(5.1.17)

A simple calculation using the coordinate representation of r  (5.1.16) and L (5.1.17) as well as the definition of the surface divergence of (4.1.8) and the surface curl of (4.1.9) shows that L  r F ./ D 0

and r  L F ./ D 0:

(5.1.18)

The following results are known from surface differential geometry (see, e.g., Strubecker 1964). Lemma 5.1.1. The tangential field of f vanishes, i.e., ftan ./ D 0 for all  2 S2 , if and only if f ./   D 0 for every unit vector  that is perpendicular to . Proof. First, let ftan D 0. Then, for all  2 S2 , f ./   D .f ./  /   C .f ./  .f ./  //  

(5.1.19)

D 0 C ftan ./   D 0: On the other hand, assume now that ftan ./ D f ./  .f ./  / ¤ 0. Then,

ftan ./ jftan ./j

./ is a unit vector field that is perpendicular to . Hence, ftan ./ jfftan D jftan ./j D 0 tan ./j which is a contradiction. Thus, ftan ./ D 0. t u

Lemma 5.1.2. Let f 2 c.S2 / satisfy Z C

for every curve C  S2 . Then,

  f ./ d./ D 0

ftan ./ D 0

(5.1.20)

(5.1.21)

for all  2 S2 . Proof. Choose any point 0 2 S2 . Let 0 be a unit vector with 0  0 D 0. Then, there is a curve C  S2 which passes through 0 and has 0 as tangent vector at 0 . Let C0 be any subset of C containing 0 . Then,

5.1 Basic Notation

215

Z C0

  f ./ d./ D 0:

(5.1.22)

Now, we let the length of C0 tend to zero and find that 0  f .0 / D 0 in the limit. Then, Lemma 5.1.2 implies that ftan .0 / D 0. Since 0 has been chosen arbitrarily, we have ftan D 0. t u Lemma 5.1.3. Suppose F is of class C.1/ .S2 /. Then, (i) r F ./ D 0 holds for all  2 S2 if and only if F is constant. (ii) L F ./ D 0 holds for all  2 S2 if and only if F is constant. Proof. First, we treat the case with r  . Suppose F is continuously differentiable in an open set in R3 that contains S2 and C  S2 is any curve from 0 2 S2 to 1 2 S2 . Let  be the unit tangent vector at  on C pointing from 0 to 1 . Then, Z F .1 /  F .0 / D   r F ./ d./: (5.1.23) C

If r F ./ D 0, we obtain from (5.1.23) that F .1 / D F .0 / for all points 0 ; 1 2 S2 . If on the other hand F is constant, (5.1.23) gives us that the vector field f D R r  F fulfills C   f ./ d./ D 0 for all curves C  S2 . By Lemma 5.1.2 it holds that ftan D 0 on S2 and thus, ftan ./ D f ./  .f ./  /  D f ./ D r F ./ D 0

(5.1.24)

for all  2 S2 . Now, we consider the case with L . If L F ./ D 0, i.e.,  ^ r F ./ D 0 for all  2 S2 , then we have      ^  ^ r F ./ D   r F ./   r F ./.  / D r F ./ D 0 (5.1.25) for all  2 S2 . By part (i) of Lemma 5.1.3 it follows that F is constant. If on the other hand F is constant, then L F ./ D  ^ r F ./ D  ^ 0 D 0 for t u all  2 S2 . Lemma 5.1.4. Let f 2 c.S2 / be a tangential vector field with Z   f ./ d./ D 0

(5.1.26)

C

for every closed curve C  S2 . Then, there is a scalar field P on S2 such that f ./ D r P ./; where P 2 C.1/ .S2 / and P is unique up to a constant.

(5.1.27)

5 Vectorial Spherical Harmonics in R3

216

Proof. Take an arbitrary, but fixed 0 2 S2 . We set Z



  f ./ d./

P ./ D

(5.1.28)

0

with the integral along any curve that starts at 0 and ends at  2 S2 . Then, for 0 ; 1 2 S2 and any curve C  S2 starting at 0 and ending at 1 , we obtain Z

1

P .1 /  P .0 / D

  f ./ d./

(5.1.29)

  r P ./ d./;

(5.1.30)

0

and

Z P .1 /  P .0 / D

1 0

since the surface gradient acts like an ordinary gradient in R3 when integrating along lines on S2 . Therefore, we are able to combine the two equations to Z

1 0

  .f ./  r P .// d./ D 0

(5.1.31)

for any curve C  S2 connecting 0 and 1 . By Lemma 5.1.2 we find that f ./  r P ./ D 0;

 2 S2 :

(5.1.32)

The proof that P is continuously differentiable can be found in standard lectures on vector analysis (see, e.g., J¨anich 2004). One basic idea is to take P constant on each straight line passing through S2 in the normal direction. In order to prove uniqueness, we consider r P1 ./ D r P2 ./,  2 S2 , which implies that r .P1  P2 /./ D 0. Therefore, due to Lemma 5.1.3, P1  P2 D const. t u Theorem 5.1.5. Let f 2 c.1/ .S2 / be a tangential vector field. Then, L  f ./ D 0,  2 S2 , if and only if there is a scalar field P such that f ./ D r P ./;

 2 S2 :

(5.1.33)

P is called a potential function for f and is unique up to an additive constant. Similarly, r  f ./ D 0,  2 S2 , if and only if there is a scalar field S such that f ./ D L S./;

 2 S2 :

(5.1.34)

S is called a stream function for f and is unique up to an additive constant.

5.1 Basic Notation

217

Proof. The condition f D r  P implies that L  f D L  r  P D 0. f D L S implies that r   f D r   L S D 0. Conversely, assume that L  f ./ D 0,  2 S2 . Then, we can use the surface theorem of Stokes (Theorem 4.1.2) to obtain Z C

  f ./ d./ D 0

(5.1.35)

for every closed curve C  S2 . By Lemma 5.1.4 there exists a scalar field P such that f D r  P , where P is unique up to a constant. Suppose now that r  f ./ D 0,  2 S2 . For  2 S2 (note that f D ftan is a tangential field and that r   D 2): L  . ^ f .// D . ^ r /  . ^ f .//

(5.1.36)

D .  /.r  f .//  .  f .//.r   / D r  f ./  2.  ftan .// D r  f ./; i.e., L  . ^ f .// D 0, since r  f ./ D 0. Thus, as before, there exists a scalar field S (unique up to a constant) such that   ^ f ./ D r S./;

 2 S2 :

(5.1.37)

This is equivalent to   ^ . ^ f .// D . ^ r /S./;

 2 S2 ;

(5.1.38)

 ^ . ^ f .// D .  f .//  .  /f ./

(5.1.39)

or f D L S on S2 , because

D .  ftan .// C f ./ D f ./: t u

This yields the second part of Theorem 5.1.5.

Theorem 5.1.6. Let f 2 c.1/ .S2 / be a tangential vector field such that, for all  2 S2 ; r  f ./ D 0

and

L  f ./ D 0:

(5.1.40)

Then, f D 0 on S2 . Proof. Since L  f ./ D 0, Theorem 5.1.5 guarantees the existence of a scalar field P such that f ./ D r P ./ for all  2 S2 . Moreover, r  f ./ D 0, i.e., r  r P ./ D  P ./ D 0. Because of (4.1.19) in Theorem 4.1.4 we find that

5 Vectorial Spherical Harmonics in R3

218

Z  2 r P ./ dS./ D 0:

(5.1.41)

S2

Hence, f ./ D r P ./ D 0.

t u

5.2 Definition of Vector Spherical Harmonics Now, we define three operators mapping scalar functions to vectorial functions. We analyze the properties of these operators as well as their adjoints and use them to define vector spherical harmonics. In this form the introduction goes back to Freeden et al. (1998). Definition 5.2.1. For  2 S2 and F 2 C.0i / .S2 /, we define the operators o.i / W C.0i / .S2 / ! c.S2 /;

i D 1; 2; 3;

(5.2.1)

by .1/

o F ./ D F ./;

(5.2.2)

o F ./ D r F ./;

(5.2.3)

o F ./ D L F ./;

(5.2.4)

.2/ .3/

where we have used the abbreviation ( 0i D

0

if i D 1;

1

if i D 2; 3:

(5.2.5)

We also introduce the notation N0i , which is N0 if i D 1 and N if i D 2; 3. It is clear that o.1/ F 2 cnor .S2 / and o.2/ F; o.3/ F 2 ctan .S2 /. Remark 5.2.2. Let F 2 C.0i / .S2 / be an -zonal function, i.e., F ./ D F .  /. Then, .1/

o F ./ D F .  /;

(5.2.6)

o F ./ D r F .  / D F 0 .  /.  .  //;

(5.2.7)

o F ./ D L F .  / D F 0 .  /. ^ /:

(5.2.8)

.2/ .3/

By Green’s integral formulas we can introduce the adjoint operators.

5.2 Definition of Vector Spherical Harmonics

219

Definition 5.2.3. For f 2 c.0i / .S2 / and G 2 C.0i / .S2 / (i D 1; 2; 3), we have that ho.i /G; f il2 .S2 / D hG; O .i / f iL2 .S2 / :

(5.2.9)

Therefore, for f 2 c.0i / .S2 /,  2 S2 , we find the adjoint operators to o.1/ , o.2/ , and o.3/ : .1/

O f ./ D   pnor f ./;

(5.2.10)

O f ./ D r  ptan f ./;

(5.2.11)

O f ./ D L  ptan f ./:

(5.2.12)

.2/ .3/

Lemma 5.2.4. Let F 2 C.2/ .S2 /. .i / .j /

(i) If i ¤ j , i; j 2 f1; 2; 3g, then O o F ./ D 0,  2 S2 . ( F ./ ; i D 1; .i / .i / (ii) O o F ./ D   F ./ ; i D 2; 3: Proof. These are well-known properties of the differential operators r  and L . t u Definition 5.2.5. For any Yn 2 Harmn .S /, the vector field 2

yn.i / D o.i / Yn ;

n  0i ; i 2 f1; 2; 3g;

(5.2.13)

is called a vector spherical harmonic of degree n and type i . By harm.in / .S2 / we denote the space of all vector spherical harmonics of degree n and type i . We also set the spaces .1/

harm0 .S2 / D harm0 .S2 /;

harmn .S2 / D

3 M

harm.in / .S2 /;

n  1:

(5.2.14)

i D1 .1/

.2/

.3/

Remark 5.2.6. We have  ^ yn D 0,   yn D 0, and   yn D 0. Theorem 5.2.7. If fYn;k gn2N0 ;kDn;:::;n is an L2 .S2 /-orthonormal system of scalar spherical harmonics, then the system of .i /

yn;k ./ D q

1 .i / n

.i /

o Yn;k ./;

i D 1; 2; 3; n  0i ; k D n; : : : ; n;

(5.2.15)

with .in /

(  .i / .i /  1 D O o Yn;k L2 .S2 / D n.n C 1/

;

i D 1;

;

i D 2; 3;

(5.2.16)

5 Vectorial Spherical Harmonics in R3

220

forms an l2 .S2 /-orthonormal system of vector spherical harmonics, i.e., Z S2

.i /

.j /

yn;k ./  ym;l ./ dS./ D ın;m ık;l ıi;j :

(5.2.17)

Proof. This follows directly from the orthogonality of the scalar spherical harmont u ics and the properties of the operators o.i / , O .i / with i D 1; 2; 3. In Fig. 5.1 some orthonormal vector spherical harmonics of type 2 and of type 3 are depicted to give an impression of the different characteristic behavior.

5.3 The Helmholtz Decomposition Theorem In the following, we formulate and prove the decomposition theorem for spherical vector fields that motivates the choice of the three operators o.i / . Theorem 5.3.1 (Helmholtz Decomposition Theorem). Let f be of class c.1/ .S2 /. There exist uniquely determined functions F1 2 C.1/ .S2 / and F2 ; F3 2 C.2/ .S2 / satisfying Z S2

Fi ./ dS./ D 0;

i D 2; 3;

(5.3.1)

such that f ./ D

3 X

o.i /Fi ./ D F1 ./ C r F2 ./ C L F3 ./;

 2 S2 :

(5.3.2)

i D1

The functions Fi are given by .1/

F1 ./ D O f ./;  2 S2 ; Z G. I ; /O.2/ f ./ dS./; F2 ./ D  S2

Z F3 ./ D 

S2

G. I ; /O.3/ f ./ dS./;

(5.3.3)  2 S2 ;

(5.3.4)

 2 S2 ;

(5.3.5)

where G. I ; / is the Green function with respect to the Beltrami operator of Definition 4.6.1. Proof. Any vector field f 2 c.1/ .S2 / can be written as f D fnor C ftan with 2 fnor D pnor f 2 c.1/ nor .S /;

(5.3.6) .1/

ftan D ptan f 2 ctan .S2 /:

(5.3.7)

5.3 The Helmholtz Decomposition Theorem

221

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 0.35

0.35

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05 0 0.45

0.45 0.4

0.4

0.35

0.35

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

.i/

Fig. 5.1 Vector spherical harmonics yn;k of type i D 2 (left column) and of type i D 3 (right column), the degree n D 8 in each case, the order k D 0 in the top row, k D 4 in the middle row, and k D 8 in the bottom row .1/

Clearly, fnor ./ D o.1/ F1 ./ with F1 ./ D O f ./ for  2 S2 . For the tangential field ftan , the Stokes theorem for S2 (Theorem 4.1.2 for a closed surface, i.e., there is no boundary) yields that Z S2

L  ftan ./ dS./ D 0:

(5.3.8)

5 Vectorial Spherical Harmonics in R3

222

Thus, L  ftan is suitable as a right-hand side of the Beltrami differential equation and by Theorem 4.6.9 we find F3 2 C.2/ .S2 / such that Z

 F3 D L  ftan

and S2

F3 ./ dS./ D 0:

(5.3.9)

In other words, L  L F3 D L  ftan

or

L  .ftan  L F3 / D 0:

(5.3.10)

Note that the difference ftan  L F3 is still a tangential vector field. Therefore, we can now use Theorem 5.1.5 which gives us a scalar field F2 2 C.2/ .S2 / with ftan  L F3 D r  F2

or ftan D r  F2 C L F3 :

(5.3.11)

R R 1 If S2 F2 ./ dS./ ¤ 0, we replace F2 by FQ2 D F2  4 S2 F2 ./ dS./. Then, it holds that Z r  F2 D r  FQ2 and FQ2 ./ dS./ D 0: (5.3.12) S2

Assume that there exists another triple Gi , i D 1; 2; 3, such that .1/

.2/

.3/

.1/

.2/

f ./ D o F1 ./ C o F2 ./ C o F3 ./; .3/

f ./ D o G1 ./ C o G2 ./ C o G3 ./:

(5.3.13) (5.3.14)

Then, it follows that F1 D O .1/ f D G1 , i.e., F1 is uniquely defined. Thus, .2/

.3/

.2/

.3/

o F2 ./ C o F3 ./ D o G2 ./ C o G3 ./:

(5.3.15)

We now apply O .2/ and O .3/ which yield  F2 D  G2 ;

 F3 D  G3 :

(5.3.16)

Hence, we find uniqueness up to a constant and the normalization conditions on F2 and F3 imply that F2 D G2 and F3 D G3 . The specific representation of F2 , F3 involving integrals with Green’s function G. I ; / follows directly from Theorem 4.6.9. t u Remark 5.3.2. A vector field of the form  7! F1 ./ C r F2 ./;

 2 S2 ;

(5.3.17)

is called spheroidal, one of the form  7! L F3 ./;

 2 S2 ;

(5.3.18)

5.4 Closure and Completeness of Vector Spherical Harmonics

223

is said to be toroidal (cf. Backus 1967, 1986). Thus, the Helmholtz decomposition theorem represents the decomposition of a c.1/ -vector field into its spheroidal and its toroidal parts. It also implies an orthogonal decomposition of the space c.1/ .S2 /, namely .1/

.1/

.1/

c.1/ .S2 / D c.1/ .S2 / ˚ c.2/ .S2 / ˚ c.3/ .S2 /;

(5.3.19)

2 c.1/ .S2 / D c.1/ nor .S /; ˚  .1/ c.2/ .S2 / D f 2 c.1/ .S2 / W O .1/ f D O .3/ f D 0 ; ˚  .1/ c.3/ .S2 / D f 2 c.1/ .S2 / W O .1/ f D O .2/ f D 0 :

(5.3.20)

where .1/

(5.3.21) (5.3.22)

These definitions can also be extended to c.k/ .S2 /, k 2 N, to c.S2 / and to l2 .S2 / by a density argument, i.e., kkc.S2 / ˚ c.i / .S2 / D o.i / F W F 2 C.1/ .S2 / ;

(5.3.23)

˚ kkl2 .S2 / l2.i / .S2 / D o.i / F W F 2 C.1/ .S2 / :

(5.3.24)

Therefore, we find the orthogonal decompositions l2 .S2 / D l2nor .S2 / ˚ l2tan .S2 /;

(5.3.25)

l2tan .S2 / D l2.2/ .S2 / ˚ l2.3/ .S2 /:

(5.3.26)

5.4 Closure and Completeness of Vector Spherical Harmonics Next, we prove the closure and completeness of vector spherical harmonics intrinsically on the sphere (note that a non-intrinsic proof is included in Sect. 5.5). For our purpose here, we use vectorial variants of the scalar zonal Bernstein kernels (cf. Freeden and Gutting 2008). The vector zonal Bernstein kernel approximations can be shown to guarantee the closure property in the space of continuous spherical normal as well as tangential vector fields. In consequence, they also assure closure and completeness in the Hilbert space of (Lebesgue-)square-integrable vector fields. Essential tools are the theory of the Green function with respect to the Beltrami operator of Sect. 4.6 and the Helmholtz decomposition theorem (Theorem 5.3.1) of Sect. 5.3.

5 Vectorial Spherical Harmonics in R3

224

We start our considerations by convolving Green’s function with respect to the Beltrami operator (see Definition 4.6.1 or (4.6.6) for the explicit representation) against the Bernstein kernel (4.4.28) of Definition 4.4.6,   BGn .  / D G. I  /  KB;n . / ./ D

Z S2

G. I   /KB;n .  / dS. /: (5.4.1)

Written in terms of a Legendre series, we find the finite sum BGn .  / D

n ^ X .k/ 2k C 1 KB;n kD1

k.k C 1/

4

Pk .  /:

(5.4.2)

Note that the Bernstein kernel is a polynomial and the Green function is of class L1 .Œ1; 1 /, hence, the existence of the convolution integral as a bandlimited Legendre expansion is obvious. Next, we are interested in the Bernstein summability of Fourier expansions in terms of vector spherical harmonics. To this end, we need some preparatory material (more precisely, Lemmas 5.4.1 and 5.4.2). An essential tool of our considerations is the Green function with respect to the Beltrami operator (cf. Freeden and Gutting 2008). Lemma 5.4.1. For i 2 f1; 2; 3g, .n/

lim kFi  Fi kC.S2 / D 0;

(5.4.3)

n!1

where Fi , i D 1; 2; 3, are the functions occurring in the Helmholtz decomposition theorem (Theorem 5.3.1) .1/

F1 ./ D O f ./; Z Fi ./ D  G. I   /O.i / f ./ dS./;

(5.4.4) i D 2; 3;

(5.4.5)

S2

.n/

and Fi , i D 1; 2; 3, are given by Z .n/

F1 ./ D .n/

S2

Fi ./ D 

KB;n .  /O.1/ f ./ dS./;

(5.4.6)

Z S2

BGn .  /O.i / f ./ dS./;

i D 2; 3:

(5.4.7)

Proof. Clearly, the case i D 1 of Lemma 5.4.1 is easy to handle. It follows immediately from the scalar theory. Thus, it remains to study the cases i D 2; 3. We start from the convolution integrals

5.4 Closure and Completeness of Vector Spherical Harmonics

  .n/ Fi ./ D  BGn  O .i / f ./ D 

225

Z S2

BGn .  /O.i / f ./ dS./;

(5.4.8)

i D 2; 3. It is not difficult to see that kFi  Fi kC.S2 / D kG. I /  O .i /f  BGn  O .i / f kC.S2 / .n/

 kO .i / f kC.S2 / kG. I /  BGn kL1 .Œ1;1 / :

(5.4.9)

Since both kernels G. I / and BGn are of class L2 .Œ1; 1 / and, for all k 2 N0 , the ^ Legendre coefficients of the Bernstein kernel KB;n .k/ converge to 1 for n tending to infinity, we are able to deduce that lim kG. I /  BGn kL2 .Œ1;1 / D 0:

n!1

(5.4.10)

.n/

Obviously, this implies L1 -convergence as well as kFi  Fi kC.S2 / ! 0 for i D 1; 2; 3 and n ! 1, as required. u t Considering the o.i /-derivatives, we have to verify the following lemma. Lemma 5.4.2. For i 2 f1; 2; 3g, ˇ ˇ ˇ ˇ .i / .i / .n/ lim sup ˇo Fi ./  o Fi ./ˇ D 0:

n!1

(5.4.11)

2S2

Proof. The case i D 1 is obvious and it is not difficult to see that, for i 2 f2; 3g, .i /

.i /

.n/

ko Fi ./  o Fi ./kc.S2 / ˇ Z Z ˇ .i / .i / .i / D sup ˇˇo G. I   /O f ./ dS./o S2

2S2

ˇZ ˇ D sup ˇˇ 2 2S

S2

o G. I   /O f ./ dS./ .i /

S2

.i /

Z S2

(5.4.12) ˇ ˇ .i / BGn .  /O f ./ dS./ˇˇ

ˇ ˇ .i / .i / o BGn .  /O f ./ dS./ˇˇ;

where it is clear that the operator o.i / can be drawn inside the two integrals. This leads us to the following estimate: ˇZ ˇ Z ˇ ˇ .i / .i /  .i / .i / ˇ sup ˇ o G. I   /O f ./ dS./  o BGn .  /O f ./ dS./ˇˇ 2 2 2

2S

S

S

Z ˇ ˇˇ ˇ ˇˇ ˇ ˇ .i / .i /  sup ˇo G. I   /  o BGn .  /ˇ ˇO.i / f ./ˇ dS./ 2S2

S2

 kO .i / f kC.S2 / sup 2S2

Z ˇ ˇ ˇ .i / ˇ .i / ˇo G. I   /  o BGn .  /ˇ dS./: (5.4.13) S2

5 Vectorial Spherical Harmonics in R3

226

We have to study the convergence of the last integral. In more detail, we are interested in proving that Z ˇ ˇ ˇ .i / ˇ .i / lim sup ˇo G. I   /  o BGn .  /ˇ dS./ D 0:

n!1

2S2

(5.4.14)

S2

.i /

For that purpose, we notice that the Bernstein kernels o BGn .  /; i D 2; 3; admit the (Legendre) series expansions .2/

o BGn .  / D

n ^ X .k/ 2k C 1 KB;n kD1

.3/

o BGn .  / D

4

k.k C 1/

n ^ X .k/ 2k C 1 KB;n kD1

4

k.k C 1/

Pk0 .  / .  .  // ;

(5.4.15)

Pk0 .  / . ^ / :

(5.4.16)

Moreover, an easy calculation shows that the application of the o.i / -operators, i D 2; 3, to the Green function with respect to the Beltrami operator leads us to the identities o G. I   / D  .2/

1   .  / 1  ^ .3/ ; o G. I   / D  : 4 1     4 1     (5.4.17)

Consequently, for i D 2; 3, our integral can be expressed in the form Z ˇ ˇ ˇ ˇ .i / .i / ˇo G. I   /  o BGn .  /ˇ dS./ S2

D

(5.4.18)

ˇ .i / Z ˇ n ˇ 1 o .  / 1 X ˇ 2k C 1 ^ .i / 0 ˇ ˇ dS./  K .k/P .  /o .  / B;n k  ˇ ˇ 4 k.k C 1/ S2 4 1     kD1

ˇ Z ˇ n ˇ ˇˇ X ˇ 1 1 2k C 1 ^ ˇˇ ˇ .i / 0 ˇ dS./ D  K .  / .k/P .  / ˇˇ ˇo B;n k ˇ 4 S2 1  k.k C 1/ kD1

D

1 2

Z

1 1

ˇ ˇ n X p ˇ 1 ˇ 2k C 1 ^ 0 2 ˇ  K .k/Pk .t/ˇˇ dt: 1t ˇ 1t k.k C 1/ B;n kD1

At this point, we use the recurrence relation (3.4.56) of Theorem 3.4.11 for the Legendre polynomials, i.e., D 12 in the version for ultraspherical polynomials. This gives us the identity

5.4 Closure and Completeness of Vector Spherical Harmonics

227

Z ˇ ˇ ˇ .i / ˇ .i / ˇo G. I   /  o BGn .  /ˇ dS./

(5.4.19)

S2

1 D 2 D

1 2

Z

1

r

ˇ ˇ n X 1 C t ˇˇ PkC1 .t/  Pk1 .t/ ˇˇ ^ 1  .1  t/ KB;n .k/ ˇ dt 1t ˇ t2  1

r

ˇ ˇ n ˇ 1 X ^ 1 C t ˇˇ 1C KB;n .k/ .PkC1 .t/  Pk1 .t// ˇˇ dt: ˇ 1t 1Ct

1

Z

1

1

kD1

kD1

For the occurring sum, it follows that n X

^ KB;n .k/ .PkC1 .t/  Pk1 .t//

kD1 ^ ^ ^ ^ DKB;n .n/PnC1 .t/ C KB;n .n  1/Pn .t/  KB;n .2/P1 .t/  KB;n .1/P0 .t/

C

n1 X  ^  ^ KB;n .k  1/  KB;n .k C 1/ Pk .t/;

(5.4.20)

kD2

where a simple calculation shows that ^ ^ ^ KB;n .k  1/  KB;n .k C 1/ D KB;nC1 .k/.2k C 1/

2 : .n C 2/

(5.4.21)

We put (5.4.21) into (5.4.20) getting the following result: n X

^ KB;n .k/ .PkC1 .t/  Pk1 .t//

(5.4.22)

kD1 ^ ^ ^ .n/PnC1 .t/ C KB;n .n  1/Pn .t/  KB;n .2/ P1 .t/ D KB;n

2 X ^ KB;nC1 .k/.2k C 1/Pk .t/ nC2 n1

^  KB;n .1/P0 .t/ C

kD2

D

2 nC2

nC1 X

^ KB;nC1 .k/.2k C 1/Pk .t/  .1 C t/

kD0

 1 C t nC1 2 .n C 1/  .1 C t/: D nC2 2 Keeping this result in mind, we return to the integral (5.4.19). As a matter of fact, the identity (5.4.19) can be rewritten in the form

5 Vectorial Spherical Harmonics in R3

228

r

ˇ ˇ n ˇ 1 X ^ 1 C t ˇˇ ˇ dt K .k/ .P .t/  P .t// 1 C kC1 k1 B;n ˇ ˇ 1t 1Ct 1 kD1 ˇ  ˇ Z r 1 1 1 C t ˇˇ n C 1 1 C t n ˇˇ D ˇ dt: 2 1 1  t ˇ n C 2 2

1 2

Z

1

(5.4.23)

Clearly, as the Bernstein kernel is non-negative, we are left with the integral expression  Z ˇ Z r ˇ 1Ct 1Ct n 1nC1 1 ˇ ˇ .i / .i /  G. I   /  o BG .  / dS./ D dt ˇ ˇo n  2 n C 2 1 1  t 2 S2 D

.n C 32 / . 12 / .n C 2/

;

(5.4.24)

which follows by induction. It is well-known that the value of our integral can be estimated as follows: .n C 32 / 1 2 p < :

0, there exists an N 2 N such that N 3 X n   X X  .i / .i /  hf; yn;k il2 .S2 / yn;k  2 f 

l .S2 /

i D1 nD0i kDn

< ":

(5.5.31)

Definition 5.5.7. A second system of vector spherical harmonics is given by r .1/ yQn;k

D r

.2/ yQn;k .3/

D

n C 1 .1/ y  2n C 1 n;k n .1/ y C 2n C 1 n;k

r r

n .2/ y ; 2n C 1 n;k

(5.5.32)

n C 1 .2/ y ; 2n C 1 n;k

(5.5.33)

.3/

yQn;k D yn;k ;

(5.5.34)

where n 2 N0i and k D n; : : : ; n. .i /

Theorem 5.5.8. The l2 .S2 /-orthonormal system fyQn;k gi D1;2;3;n2N0i ;kDn;:::;n is closed and complete in l2 .S2 /, i.e., for all f 2 l2 .S2 / and " > 0 there exists N 2 N such that N 3 X n   X X  .i / .i /  hf; yQn;k il2 .S2 / yQn;k  2 2 < ": (5.5.35) f  i D1 nD0i kDn

l .S /

Remark 5.5.9. The system of vector spherical harmonics of Definition 5.5.7 does not strictly separate into radial and tangential vector spherical harmonics. However, note that  yQn1;k ./ D n.n C 1/yQn1;k ./;

n 2 N; .n  1/  k  n  1; (5.5.36)

 yQnC1;k ./ D n.n C 1/yQnC1;k ./;

n 2 N0 ; .n C 1/  k  n C 1;

.1/

.1/

.2/

.2/

(5.5.37) .3/  yQn;k ./

D n.n C

.3/ 1/yQn;k ./;

n 2 N; n  k  n:

(5.5.38)

Thus, this system is a set of eigenfunctions of the Beltrami operator which is applied componentwise. Moreover, we can show that, for n 2 N0i and k D n; : : : ; n,

5 Vectorial Spherical Harmonics in R3

236 .1/ yQn;k ./

D

.2/

yQn;k ./ D .3/

yQn;k ./ D D

  ˇ 1 ˇ rx p Y ./ ˇ ; nC1 n;k rD1 r .n C 1/.2n C 1/ ˇ 1 ˇ .rx .r n Yn;k ./// ˇ ; p rD1 n.2n C 1/  ˇ  1 1 ˇ x ^ rx p Yn;k ˇ rD1 r nC1 n.n C 1/ ˇ 1 ˇ .x ^ rx .r n Yn;k // ˇ : p rD1 n.n C 1/ 1

(5.5.39) (5.5.40)

(5.5.41)

5.6 Vectorial Beltrami Operator Next, we develop a vectorial analog of the Beltrami operator  , denoted by  , such that the vector spherical harmonics of class harmn .S2 / can be recognized as eigenfunctions of the vectorial operator  . In particular, it turns out that the operator  corresponds to the orthogonal decomposition with respect to the operators o.i / ; i D 1; 2; 3, that is to say,  f 2 c.i / .S2 / for all i D 1; 2; 3, provided .2/ that f 2 c.i / .S2 / (see Remark 5.3.2 for the definition of these spaces). Our construction is based on the componentwise application of the (scalar) Beltrami operator  . The point of departure is the following convention: If f 2 c.2/ .S2 / is of the form f ./ D

3 X

"i Fi ./;

 2 S2 ;

(5.6.1)

i D1

then  f is understood to be  f ./ D

3 X

"i  Fi ./;

 2 S2 :

(5.6.2)

i D1

Observing this setting, we are able to deduce the following identities. Lemma 5.6.1. The following statements are true. (i) Let F W S2 ! R be sufficiently smooth. Then,  o.1/ F D o.1/ .  2/F C 2o.2/ F;

(5.6.3)

 o.2/ F D 2o.1/  F C o.2/  F;

(5.6.4)

 .3/



 o F D o  F: .3/

(5.6.5)

5.6 Vectorial Beltrami Operator

237

(ii) Let f W S2 ! R3 be a sufficiently smooth vector field. Then, O .1/  f D .  2/O .1/ f C 2O .2/ f;

(5.6.6)

O .2/  f D 2 O .1/ f C  O .2/ f;

(5.6.7)





O  f D O .3/

.3/

f:

(5.6.8)

Proof. We verify only the first formula of part .i /, since the other assertions follow by similar arguments. Assume that the function under consideration is a spherical harmonic Yn of class Harmn .S2 /, n 2 N0 . Then, it follows from (5.5.25) that  1  .1/ n n oQ n r Yn ./jrD1 C oQ .2/ (5.6.9) n r Yn ./jrD1 2n C 1 .n C 1/.n C 2/ .1/ n n.n  1/ .2/ n .oQ n r Yn .//jrD1  .oQ r Yn .//jrD1 D 2n C 1 2n C 1 n

 o Yn ./ D  .1/

holds for all  2 S2 , since the first summand is of degree n C 1 and the second summand is of degree n  1 (see Lemma 5.5.3). Using (5.5.22) and (5.5.23), we obtain  o Yn ./ D  .1/



.n C 1/2 .n C 2/ .1/ .n C 1/.n C 2/ .2/ o Yn ./ C o Yn ./ 2n C 1 2n C 1 .n  1/n2 .1/ n.n  1/ .2/ o Yn ./  o Yn ./ 2n C 1  2n C 1  .1/

.2/

D .n.n C 1/  2/o Yn ./ C 2o Yn ./ .1/

.2/

D o .n.n C 1/  2/Yn ./ C 2o Yn ./ D o .  2/Yn ./ C 2o Yn ./: .1/

.2/

(5.6.10)

Thus, our formula is true for every spherical harmonic. The completeness of the system of vector spherical harmonics in l2 .S2 / then implies the validity for all sufficiently smooth functions. Part .i i / follows from the adjointness of the operators o.i / and O .i / and the self-adjointness of  . t u Lemma 5.6.1 helps us to find a vectorial Beltrami operator by observing the identities . C 2/o.1/Yn D n.n C 1/o.1/ Yn C 2o.2/ Yn ;  o.2/ Yn D 2n.n C 1/o.1/Yn  n.n C 1/o.2/ Yn ;  .3/

 o Yn D n.n C 1/o Yn : .3/

(5.6.11) (5.6.12) (5.6.13)

5 Vectorial Spherical Harmonics in R3

238

In consequence, we have pnor . C 2/o.1/ Yn D n.n C 1/o.1/Yn ;  .2/

(5.6.14)

ptan  o Yn D n.n C 1/o Yn ;

(5.6.15)

ptan  o.3/ Yn D n.n C 1/o.3/Yn :

(5.6.16)

.2/

In other words, (5.6.14)–(5.6.16) motivate the introduction of the vectorial Beltrami operator in the following way. Definition 5.6.2. The operator  W c.2/ .S2 / ! c.S2 / given by  D pnor . C 2/pnor C ptan  ptan

(5.6.17)

is called the vectorial Beltrami operator, where the application of  on vector fields is understood in the sense of (5.6.2). As an immediate consequence of Lemma 5.6.1, we obtain the following result. Lemma 5.6.3. If F W S2 ! R and f W S2 ! R3 are sufficiently smooth, then, for i 2 f1; 2; 3g,  o.i /F D o.i /  F;

(5.6.18)

O .i /  f D  O .i / f:

(5.6.19)

Lemma 5.6.3 motivates us to characterize the spectrum of the operator  . Theorem 5.6.4. Any vector spherical harmonic yn 2 harmn .S2 / of degree n 2 N0 is an infinitely often differentiable eigenfunction of the vectorial Beltrami operator  with respect to the eigenvalue . /^ .n/ D n.n C 1/. Conversely, any infinitely often differentiable eigenfunction of  is a vector spherical harmonic. Remark 5.6.5. The vector spherical harmonics can be seen to be the eigenfunctions of an operator  that can be introduced (without projection operators) only by the use of differentiation processes in R3 (see Gervens 1989). More explicitly, the following theorem is valid. Theorem 5.6.6. Let  be given by  f ./ D  f ./  2. ^ r / ^ f ./  2f ./;

f 2 c.2/ .S2 /:

(5.6.20)

Then, for n 2 N0 ,  yn C n.n C 1/yn D 0;

yn 2 harmn .S2 /:

(5.6.21)

5.7 Vectorial Addition Theorem

239

5.7 Vectorial Addition Theorem Our interest is the formulation of a vectorial analog of the addition theorem for scalar spherical harmonics (Theorem 4.3.29) of Sect. 4.3. This vectorial addition theorem assures the existence of a reproducing kernel of tensorial structure which is a basic tool, e.g., in the theory of vectorial zonal functions (cf. Freeden et al. (1998) and Freeden and Schreiner (2009) and the references therein). In addition, this addition theorem offers a better insight into orthogonal invariance within the theory of vector spherical harmonics. .i / Let fyn;j gi D1;2;3I j Dn;:::;n be an l2 .S2 /-orthonormal basis of harmn .S2 /, n 2 N0 , as defined by (5.2.15) in Theorem 5.2.7, corresponding to an L2 .S2 /-orthonormal .i / .i / basis fYn;j gj Dn;:::;n of Harmn .S2 /, i.e., yn;j D .n /1=2 o.i /Yn;j , n 2 N0i . Then, the announced vectorial analog of the addition theorem deals with the question of determining the expression n X 2n C 1 .i;k/ .i / .k/ pn .; / D yn;j ./ ˝ yn;j ./; 4 j Dn

;  2 S2 ;

(5.7.1)

where ˝ denotes the dyadic tensor product of two vectors. Note that rank-2 tensor fields are denoted by lower case bold letters. Furthermore, we require the following bits of notation for tensor fields: The identity tensor in R33 , i.e., iD

3 X

"i ˝ "i ;

(5.7.2)

i D1

is projected onto the tangential components at a point on the unit sphere. This defines the surface identity tensor field itan ./ D i   ˝ ;

 2 S2 :

(5.7.3)

The surface rotation tensor field is given by jtan ./ D  ^ i D

3 X . ^ "i / ˝ "i ;

 2 S2 :

(5.7.4)

i D1

Our purpose is to explain how a vectorial counterpart of tensorial nature for the Legendre polynomial comes into play. For that purpose, we first extend in canonical way the definition of the o.i /-operators defined by (5.2.2)–(5.2.4) in Definition 5.2.1 to vector fields.

5 Vectorial Spherical Harmonics in R3

240

Suppose that f W S2 ! R3 is a sufficiently smooth vector field with the representation 3 X f ./ D Fk ./"k ; Fk ./ D f ./  "k ;  2 S2 : (5.7.5) kD1

Then, we set .i /

o f ./ D

3 X

.i /

.o Fk .// ˝ "k D

kD1

3 X  .i /  o .f ./  "k / ˝ "k ;

(5.7.6)

kD1

where i D 1; 2; 3. Thus, o.i / maps scalar functions to vector fields and vector fields to rank-2 tensor fields, respectively. In accordance with this nomenclature, (5.7.1) can be expressed as follows: n X

1=2 yn;j ./ ˝ yn;j ./ D ..in / /1=2 ..k/ n / .i /

.k/

j Dn

n X

.i /

o Yn;j ./ ˝ o.k/  Yn;j ./

j Dn n X

1=2 D ..in / /1=2 ..k/ o o.k/ n /  .i /

Yn;j ./Yn;j ./

j Dn 1=2 D ..in / /1=2 ..k/ n /

2n C 1 .i / .k/ o o Pn .  /: 4

(5.7.7)

.i;k/

In other words, pn W S2  S2 ! R33 , n 2 N, is given in terms of the onedimensional Legendre polynomial by .i / 1=2 1=2 ..k/ o o.k/ p.i;k/ n .; / D .n / n /  Pn .  /; .i /

.1;1/

;  2 S2 ;

(5.7.8)

.1/ .1/

and p0 .; / D o o P0 . / D  ˝. Altogether, this leads us to the following variant of the addition theorem for vector spherical harmonics. .i /

Theorem 5.7.1 (Addition Theorem for harmn .S2 /). Let fyn;j gi D1;2;3Ij Dn;:::;n be an l2 .S2 /-orthonormal basis of the space harmn .S2 /, n 2 N0 , as given by (5.2.15) in Theorem 5.2.7. Then, n X j Dn

.i /

.k/

yn;j ./ ˝ yn;j ./ D

2n C 1 .i;k/ pn .; / 4

1=2 D ..in / /1=2 ..k/ n /

(5.7.9) 2n C 1 .i / .k/ o o Pn .  / 4

holds for ;  2 S2 , n 2 N, and i; k 2 f1; 2; 3g. For .i; k/ D .1; 1/, we have

5.7 Vectorial Addition Theorem n X

241

.1/

.1/

yn;j ./ ˝ yn;j ./ D

j Dn

2n C 1 .1;1/ pn .; /; 4

(5.7.10)

for n 2 N0 , i.e., also for n D 0. .i;k/

Definition 5.7.2. The kernel pn

W S2  S2 ! R33 , i; k 2 f1; 2; 3g, given by

.i / 1=2 1=2 p.i;k/ ..k/ o o.k/ n .; / D .n / n /  Pn . / D .i /

n X 4 .i / .k/ y ./˝yn;j ./ 2n C 1 j Dn n;j

(5.7.11) is called the Legendre rank-2 tensor kernel of type .i; k/ and degree n 2 N0i;k with respect to the dual system of operators o.i /; O .i / , i 2 f1; 2; 3g. Note that we use the notation ( N0 ; .i; k/ D .1; 1/; N0i;k D (5.7.12) N ; else: The kernel pn D

3 3 X X

p.i;k/ n

(5.7.13)

i D1 kD1

is called the Legendre rank-2 tensor kernel of degree n 2 N with respect to the dual .1;1/ system of operators o.i / ; O .i /; i D 1; 2; 3, and p0 D p0 . .i;k/

The main problem to be solved is the evaluation of pn As auxiliary results we verify the following identities.

as introduced by (5.7.8).

Lemma 5.7.3. For ;  2 S2 , .2/

(5.7.14)

.3/

(5.7.15)

o .  .  // D itan ./  .  .  // ˝ ; o .  .  // D jtan ./  . ^ / ˝ ; .2/

(5.7.16)

.3/

(5.7.17)

o . ^ / D jtan ./   ˝  ^ ; o . ^ / D .  /itan ./  .  .  // ˝ :

Proof. We prove the first and third formulas. The second and fourth formulas follow by similar arguments, since we know that L D  ^ r . First, we get

5 Vectorial Spherical Harmonics in R3

242

.2/

.2/

o .  .  // D o

3 X

..  "l /  .  /.  "l //"l

(5.7.18)

lD1

D

3 X

."l  .  "l // ˝ "l  ..  "l /.  .  // ˝ "l /

lD1

D itan ./  .  .  // ˝ : Furthermore, we have .2/

.2/

o . ^ / D o

3 X

.. ^ /  "l /"l

(5.7.19)

lD1 .2/

D o

3 X

.."l ^ /  /"l

lD1

D

3 X

."l ^   .."l ^ /  // ˝ "l

lD1

D

3 X

. ^ "l / ˝ "l  .. ^ /  "l / ˝ "l

lD1

D jtan ./   ˝  ^ ; t u

which yields the third formula.

If F W Œ1; 1 ! R is sufficiently smooth, then, for ;  2 S2 , we obtain from Lemma 5.7.3: .1/

o o.1/  F .  / D F .  / ˝ ;

(5.7.20)

0 o o.2/  F .  / D F .  / ˝ .  .  //;

(5.7.21)

0 o o.3/  F .  / D F .  / ˝ . ^ /:

(5.7.22)

.1/ .1/

.2/ .1/

.3/ .1/

Similar results hold for o o F .  / and o o F .  /. Treating the tangential operators, we find, for ;  2 S2 ,  0 o o.2/  F .  /Dr ˝ .F .  /.  .  /// .2/

(5.7.23)

D.r F 0 .  // ˝ . .  //CF 0 .  /r ˝ . .  // DF 00 .  /.  .  // ˝ .  .  // CF 0 .  /.itan ./  .  .  // ˝ /

5.7 Vectorial Addition Theorem

243

and  0 o o.3/  F .  / D r ˝ .F .  / ^ / .2/

(5.7.24)

D .r F 0 .  // ˝ . ^ / C F 0 .  /r ˝ . ^ / D F 00 .  /.  .  // ˝ . ^ / CF 0 .  / .jtan ./   ˝ . ^ // : Similar calculations yield the formulas 00 o o.2/  F .  / D F .  /. ^ / ˝ .  .  // .3/

(5.7.25)

CF 0 .  /.jtan ./  . ^ / ˝ / and 00 o o.3/  F .  / D F .  /. ^ / ˝ . ^ / .3/

(5.7.26)

CF 0 .  /..  /itan ./  .  .  // ˝ /: Applying these results to the scalar Legendre polynomial Pn , we obtain the .i;k/ following representation for the Legendre rank-2 tensor kernel pn of type .i; k/ and degree n 2 N0i;k with respect to the dual system of operators o.i /, O .i / , i; k 2 f1; 2; 3g. Theorem 5.7.4. Suppose that n 2 N0i;k , i; k 2 f1; 2; 3g. Then, the identities p.1;1/ .; / D Pn .  / ˝ ; n

(5.7.27)

1 p.1;2/ Pn0 .  / ˝ .  .  //; .; / D p n n.n C 1/

(5.7.28)

1 Pn0 .  / ˝  ^ ; .; / D p p.1;3/ n n.n C 1/

(5.7.29)

1 .; / D p p.2;1/ Pn0 .  /.  .  // ˝ ; n n.n C 1/

(5.7.30)

1 p.3;1/ Pn0 .  / ^  ˝ ; .; / D p n n.n C 1/

(5.7.31)

.; / D p.2;2/ n

1 .P 00 .  /.  .  // ˝ .  .  // n.n C 1/ n C Pn0 .  /.itan ./  .  .  // ˝ //;

(5.7.32)

5 Vectorial Spherical Harmonics in R3

244

p.2;3/ .; / D n

1 .P 00 .  /.  .  // ˝  ^  n.n C 1/ n C Pn0 .  /.jtan ./   ˝  ^ //;

.; / D p.3;2/ n

1 .P 00 .  / ^  ˝ .  .  // n.n C 1/ n C Pn0 .  /.jtan ./   ^  ˝ //;

.; / D p.3;3/ n

(5.7.33)

(5.7.34)

1 .P 00 .  / ^  ˝  ^  n.n C 1/ n C Pn0 .  /..  /itan ./  .  .  // ˝ //

(5.7.35)

hold for all .; / 2 S2  S2 . The case  D  in the previous theorem is of particular interest. Observing that Pn .1/ D 1, Pn0 .1/ D 12 n.n C 1/, and Pn00 .1/ D 18 n.n C 1/.n.n C 1/  2/ (use D 12 in (3.4.7) and Theorem 3.4.5 in Sect. 3.4), we obtain the following corollary. Corollary 5.7.5. For n 2 N0i;k , i; k 2 f1; 2; 3g, and all  2 S2 , p.1;1/ .; / D  ˝ ; n / .i;1/ p.1;i n .; / D pn .; / D 0;

(5.7.36) i D 2; 3;

1 itan ./; 2 1 p.2;3/ .; / D p.3;2/ .; / D  jtan ./: n n 2 .; / D p.3;3/ .; / D p.2;2/ n n

(5.7.37) (5.7.38) (5.7.39)

It follows readily that trace ..  .  // ˝ .  .  /// D .  /.1  .  /2 /

(5.7.40)

and trace .. ^ / ˝ . ^ // D .1  .  /2 /:

(5.7.41)

Hence, we get the following identities from Theorem 5.7.4. Lemma 5.7.6. For n 2 N0i;k , i; k 2 f1; 2; 3g, and all ;  2 S2 , we have .; // D Pn .  /.  /; trace .p.1;1/ n

(5.7.42)

.; // D trace .p.2;1/ .; // trace .p.1;2/ n n 1 Pn0 .  /.1  .  /2 /; D p n.n C 1/

(5.7.43)

5.7 Vectorial Addition Theorem

trace .p.2;2/ .  // D n

245

1 .P 00 .  /.1  .  /2 /.  / C 2Pn0 .  //; n.n C 1/ n (5.7.44)

.; // D Pn .  /; trace .p.3;3/ n

(5.7.45)

.; // D trace .p.3;1/ .; // D trace .p.2;3/ .; // trace .p.1;3/ n n n .; // D 0: D trace .p.3;2/ n

(5.7.46)

Taking  D  in Lemma 5.7.6, we get in connection with Theorem 5.7.1 the following result. .i /

Lemma 5.7.7. Let n 2 N0i and i 2 f1; 2; 3g. If yn;j ; j D n; : : : ; n, forms an l2 .S2 /-orthonormal basis of harm.in / .S2 /, then n ˇ n ˇ X X 2n C 1 ˇ .i / ˇ2 .i / .i / : ./ D trace yn;j ./ ˝ yn;j ./ D ˇyn;j ˇ 4 j Dn j Dn

(5.7.47)

.i /

Every vector spherical harmonic yn 2 harm.in / .S2 / of degree n and type i can be represented by its orthogonal expansion yn.i / D

n X

.i /

an;jyn;j

(5.7.48)

j Dn

with

.i /

an;j D hyn.i / ; yn;j il2 .S2 / ;

j D n; : : : ; n:

(5.7.49)

The application of the Cauchy–Schwarz inequality in combination with Lemma 5.7.7 yields the estimate .i /

jyn ./j2 

n  X j Dn

2 an;j

n  X j Dn

n  2n C 1 X 2 2n C 1 .i / 2 .i / jyn;j ./j2 D an;j D kyn kl2 .S2 / 4 4 j Dn

(5.7.50) for all  2 S2 . Thus, we finally obtain the following lemma. .i /

Lemma 5.7.8. Suppose yn is a member of harm.in / .S2 /, n 2 N0i , i 2 f1; 2; 3g. Then, r kyn.i / kc.S2 /



2n C 1 .i / kyn kl2 .S2 / : 4

In particular,

r .i / kyn;j kc.S2 /



2n C 1 : 4

(5.7.51)

(5.7.52)

5 Vectorial Spherical Harmonics in R3

246

5.8 Vectorial Funk–Hecke Formulas Next, we deal with generalizations of the Funk–Hecke formula (Theorem 4.5.6 of Sect. 4.5) to the vectorial case. It turns out that we find two variants: (i) Let g.; / W S2 ! R3 be a vector field which is invariant with respect to all orthogonal transformations t 2 SO.3/ leaving  2 S2 fixed. Determine the integral Z S2

g.; /  yn.i / ./ dS./;

(5.8.1)

.i /

where yn 2 harm.in / .S2 /, n 2 N0i . (ii) Let G 2 L1 .Œ1; 1 /;  2 S2 fixed. Determine the integral Z S2

G.  /yn.i / ./ dS./;

(5.8.2)

.i /

where yn 2 harm.in / .S2 /, n 2 N0i . Remark 5.8.1. Note that the integral (5.8.1) is scalar-valued, while (5.8.2) is vectorvalued. This difference causes completely different ways of establishing the Funk– Hecke formulas. The first variant uses certain properties of invariant vector fields, while the second one is based on the Cartesian representation of vector spherical harmonics. We remember the representation theory needed for our studies on the Funk–Hecke formula (see Sects. 4.2 and 4.5). Now, assume that f; g are of class l2 .S2 /; t 2 SO.3/. Then, it follows that hRt f; gil2 .S2 / D hf; RtT gil2 .S2 / :

(5.8.3)

But this means that the adjoint operator of Rt is given by RtT . Let F 2 C.1/ .S2 / and t 2 SO.3/. Then, we find with (4.2.5) of Definition 4.2.1 that   .1/ o Rt F ./ D F .t/ D tT tF .t/ D Rt o.1/ F ./

(5.8.4)

and by the chain rule   .2/ o Rt F ./ D r F .t/ D tT .r  F /.t/ D Rt o.2/ F ./:

(5.8.5)

An analogous result holds for o.3/ . Together with (5.8.3), for F 2 C.1/ .S2 / and any t 2 SO.3/, we obtain that, for all i 2 f1; 2; 3g and  2 S2 , .i /

o Rt F ./ D Rt o.i / F ./

(5.8.6)

5.8 Vectorial Funk–Hecke Formulas

247

and for f 2 c.1/ .S2 /; .i /

O Rt f ./ D Rt O .i / f ./:

(5.8.7)

Therefore, we remember in connection with the results of Sect. 4.2 and Theorem 4.5.3 that the space l2.i / .S2 / is SO.3/-invariant for all i 2 f1; 2; 3g. Another coordinate-free classification of vector spherical harmonics can be given by looking at the following system of partial differential equations:  .  f .// C n.n C 1/f ./ D 0;

n  0;

(5.8.8)

r .r

 f .// C n.n C 1/f ./ D 0;

n  1;

(5.8.9)

L .r  .f ./ ^ // C n.n C 1/f ./ D 0;

n  1:

(5.8.10)

.1/

.1/

Solutions yn of (5.8.8) fulfill ^yn ./ D 0 and, consequently, there exists a scalar .1/ function F such that yn ./ D F ./. In connection with (5.8.8), this leads to . F ./ C n.n C 1/F .// D 0;

(5.8.11)

which means that F is a spherical harmonic of degree n, i.e., solutions of (5.8.8) are .1/ of the form yn ./ D Yn ./. .2/ .2/ .2/ For solutions yn of (5.8.9), we get  yn ./ D 0 and r . ^yn .// D 0, such that there exists a scalar function G with yn ./ D r G./. Together with (5.8.9), this leads to r . G./ C n.n C 1/G.// D 0: (5.8.12) .2/

Consequently, we have  G./ C n.n C 1/G./ D const:

(5.8.13)

This means that, up to a constant, G is a spherical harmonic of degree n, i.e., .2/ solutions of (5.8.9) are of the form yn ./ D r Yn ./. Analogously, solutions yn of (5.8.10) fulfill both  yn ./ D 0 and r yn ./, .3/

.3/

.3/

which means that there exists a scalar function H such that yn ./ D L H./. Consequently, L . H./ C n.n C 1/H.// D 0 (5.8.14) .3/

and, therefore, yn ./ D L Yn ./. As we have seen, the solutions of the differential Eqs. (5.8.8)–(5.8.10) are the vector spherical harmonics (as defined by (5.2.15) in Definition 5.2.5). This observation has immediate consequences for the spaces harm.in / .S2 / of vector spherical harmonics. In fact, they can be regarded as “the smallest” orthogonally invariant spaces. .3/

Theorem 5.8.2. The spaces harm.in / .S2 / of vector spherical harmonics are orthogonally invariant and irreducible.

5 Vectorial Spherical Harmonics in R3

248

Proof. The orthogonal invariance is a direct consequence of the invariant differential operators of (5.8.8)–(5.8.10). To be more concrete, suppose that there exists an orthogonally invariant subspace of harm.in / .S2 /. The application of the operators O .i / to the respective elements would—because of Definition 5.2.5—generate an orthogonally invariant subspace in the space of scalar spherical harmonics. This, however, is a contradiction to the irreducibility of the spaces Harmn .S2 /. t u Theorem 5.8.2 shows that vector spherical harmonics have the same significance for spherical vector fields as the spherical harmonics in the theory of scalar spherical fields. Further consequences for vector spherical harmonics can be found, e.g., in Freeden and Schreiner (2009). Assume that F 2 L2 .S2 / with Rt F D F for all t 2 SO .3/. Then, we already know that there exists a function FQ 2 L2 .Œ1; 1 / such that F ./ D FQ . /;  2 S2 . Furthermore, we have shown in Lemma 4.5.1 that if F is, in addition, a spherical harmonic of degree n, i.e., F 2 Harmn .S2 /, there exists a constant C 2 R such that F ./ D C Pn .  /;

 2 S2 :

(5.8.15)

A generalization of these results to the vectorial case can be written down as follows by using (5.8.6) or (5.8.7), respectively: (i) If f 2 c.1/ .S2 / satisfies Rt f D f for all t 2 SO .3/,  fixed, then there exist functions Fi 2 C.Œ1; 1 /, i D 1; 2; 3, such that .i /

O f ./ D Fi .  /;

 2 S2 :

.i /

(5.8.16) .i /

.i /

(ii) Let i 2 f1; 2; 3g and yn 2 harm.in / .S2 /, n 2 N0i , with Rt yn D yn for all rotations t 2 SO .3/,  fixed. There exists a constant C 2 R such that .i /

yn.i / ./ D C o Pn .  /;

 2 S2 :

(5.8.17)

Let  2 S2 be fixed. Assume that g.; / 2 c.1/ .S2 / is a spherical vector field with Rt g.; / D g.; /,  2 S2 , for all t 2 SO .3/. It follows from the considerations .i / above that, for i 2 f1; 2; 3g, the functions O g.; / D Gi .  / depend only on the inner product   . Thus, we may define in analogy to Theorem 4.5.6 ^

.O g/ .n/ D 2 .i /

Z

1 1

Gi .t/Pn .t/ dt: .i /

(5.8.18)

It follows from (5.2.9) and Corollary 4.5.7 that yn D o.i / Yn 2 harm.in / .S2 /, n 2 N0i , satisfies

5.8 Vectorial Funk–Hecke Formulas

249

Z

Z S2

g.; /  yn.i / ./ dS./ D

.i /

S2

O g.; /Yn ./ dS./

(5.8.19)

D .O .i / g/^ .n/Yn ./: This leads us to the first variant of the vectorial Funk–Hecke formula. Theorem 5.8.3 (Funk–Hecke Formula). Let  2 S2 be fixed. Assume that g.; / 2 c.1/ .S2 / satisfies Rt g.; / D g.; / (5.8.20) .i /

for all t 2 SO .3/ and all  2 S2 . Then, for i 2 f1; 2; 3g and yn 2 harm.in / .S2 /; n  0i , Z g.; /  yn.i / ./ dS./ D ..in / /1 .O .i / g/^ .n/O.i / yn.i / ./; (5.8.21) S2

where .O .i / g/^ .n/ is defined by (5.8.18). By virtue of the addition theorem for vector spherical harmonics, we immediately obtain the following consequence. Corollary 5.8.4. Let  2 S2 be fixed, g.; / 2 c.1/ .S2 /. Assume that Rt g.; / D g.; /

(5.8.22)

for all t 2 SO .3/ and all  2 S2 . Then, for all 2 S2 , i 2 f1; 2; 3g, and n 2 N0i , Z

/ .i / 1 .i / ^ g.; /T p.i;i n .; / dS./ D .n / .O g/ .n/ o Pn .  /: .i /

S2

(5.8.23)

We now come to the second variant of the vectorial Funk–Hecke formula as announced in (5.8.2). The basic ideas to handle this problem are the representation of vector spherical harmonics by means of restrictions of homogeneous harmonic vector polynomials and the componentwise application of the (scalar) Funk–Hecke formula. Let Yn 2 Harmn .S2 / be a spherical harmonic. Then, we know from Lemma 5.5.3 that the Cartesian components of the spherical vector field  7! oQ .in / r n Yn ./jrD1 ;

 2 S2 ; i 2 f1; 2; 3g; n 2 N0i ;

(5.8.24)

are (scalar) spherical harmonics of degree deg.i / .n/ (which is n  1, n, or n C 1 depending on the type i , see Lemma 5.5.3). Thus, we get, for G 2 L1 .Œ1; 1 / and  2 S2 , Z S2

G.  /oQ .in / r n Yn ./jrD1 dS./ D G ^ .deg.i / .n//oQ .in / r n Yn ./jrD1 :

(5.8.25)

5 Vectorial Spherical Harmonics in R3

250

We know from the formulas (5.5.25)–(5.5.27) how a vector spherical harmonic is expressible by restrictions of homogeneous harmonic vector polynomials. Combining these results, we get the second variant of the vectorial Funk–Hecke formula. Theorem 5.8.5 (Funk–Hecke Formula). Let G be of class L1 .Œ1; 1 /. Assume that Yn 2 Harmn .S2 /. Then, for all  2 S2 and for all n 2 N, Z S2

Z

S2

Z

S2

^ ^ .2/ G.  /o Yn ./ dS./ D G.1;1/ .n/o.1/  Yn ./ C G.1;2/ .n/o Yn ./; (5.8.26) .1/

^ ^ .2/ G.  /o Yn ./ dS./ D G.2;1/ .n/o.1/  Yn ./ C G.2;2/ .n/o Yn ./; (5.8.27) .2/

^ G.  /o Yn ./ dS./ D G.3;3/ .n/o.3/  Yn ./; .3/

(5.8.28)

^ where the coefficients G.i;j / .n/ are given by

 1  .n C 1/G ^ .n C 1/ C nG ^ .n  1/ ; 2n C 1  1  ^ ^ G.1;2/ .n/ D G .n  1/  G ^ .n C 1/ ; 2n C 1  n.n C 1/  ^ ^ G .n  1/  G ^ .n C 1/ ; G.2;1/ .n/ D 2n C 1  1  ^ ^ nG .n C 1/ C .n C 1/G ^ .n  1/ ; G.2;2/ .n/ D 2n C 1 ^ G.1;1/ .n/ D

^ G.3;3/ .n/ D G ^ .n/:

(5.8.29) (5.8.30) (5.8.31) (5.8.32) (5.8.33)

.i /

Remark 5.8.6. For n D 0, we know that o Yn ./ D 0, i D 2; 3, but we can find the following analog to Theorem 5.8.5: Z

G.  /o Y0 ./ dS./ D G ^ .1/o.1/  Y0 ./: .1/

S2

(5.8.34)

^ In terms of the coefficients G.i;j / .n/, we set ^ .0/ D G ^ .1/; G.1;1/ ^ G.1;2/ .0/

D

^ G.2;1/ .0/

(5.8.35) D

^ G.2;2/ .0/

D

^ G.3;3/ .0/

D 0:

Note that the space l2.3/ .S2 / is invariant with respect to the defined integral operator, while l2.1/ .S2 / and l2.2/ .S2 / are not. However, it is clear that l2.1/ .S2 / ˚ l2.2/ .S2 / is an invariant subspace of l2 .S2 /.

5.9 Vectorial Counterparts of the Legendre Polynomial

251

5.9 Vectorial Counterparts of the Legendre Polynomial In this section, our purpose is to extend the operators O .i / to rank-2 tensor fields. For sufficiently smooth fields f W S2 ! R3 ˝ R3 of the form 3 3 X X

f./ D

Fj k ./"j ˝ "k ;

(5.9.1)

j D1 kD1

we let .i / O f./

D

3 X

.i / O

X 3

kD1

Fj k ./" "k : j

(5.9.2)

j D1 .i;i /

According to (5.9.2), it is clear that the Legendre rank-2 tensor kernel pn .; / of type .i; i / and degree n 2 N0i is the reproducing kernel of harm.in / .S2 / in the following sense: (i) For all  2 S2 , .i /

/ .i / 2 O p.i;i n .; / 2 harmn .S /:

(5.9.3)

(ii) For every f 2 harm.in / .S2 / and all  2 S2 , D E .i / .i / / O f ./ D O p.i;i n .; /; f 2

l .S2 /

:

(5.9.4)

Moreover, let a 2 R3 and  2 S2 be fixed. For n 2 N0i;k , the vector field p.i;k/ n .; /  a D

n X 4 .k/ .i / .y ./  a/yn;j 2n C 1 j Dn n;j

(5.9.5)

is a vector spherical harmonic of degree n and type i . Thus, we obtain from (5.7.50) and Lemma 5.7.7 2 jp.i;k/ n .; /  aj 

2n C 1 4



4 2n C 1

2 X n

.k/

.yn;j ./  a/2  jaj2

(5.9.6)

j Dn

for n 2 N0i;k and all ;  2 S2 . This gives us the following result. Lemma 5.9.1. Let i; k; l 2 f1; 2; 3g, n 2 N0i;k . For all ;  2 S2 , l jp.i;k/ n .; /  " j  1

(5.9.7)

5 Vectorial Spherical Harmonics in R3

252 .i;k/

and the modulus of pn

.; / is

jp.i;k/ n .; /j D

3 X

l 2 jp.i;k/ n .; /" j

1=2



p 3;

(5.9.8)

lD1

where for rank-2 tensors t the modulus is defined by jtj2 D

P3

i;j D1 jtij j

2

.

Lemma 5.9.1 generalizes the estimate jPn .t/j  1, t 2 Œ1; 1 , of the scalar Legendre polynomial. Remark 5.9.2. The addition theorem enables us to represent a vector-valued function on the sphere S2 by use of the Legendre tensors. More explicitly, suppose that f is of class l2 .S2 / with f D

3 X

f .i / ;

f .i / 2 l2.i / .S2 /:

(5.9.9)

i D1

Then, it follows that f ./ D

1 X 3 X n Z X 2 i D1 nD0i mDn S

D

1 X 3 X n Z X i D1 nD0i mDn

D

1 Z 3 X X i D1 nD0i

D

(5.9.10)

.i / .i / .yn;m ./ ˝ yn;m .//f ./ dS./

S2

2n C 1 .i;i / pn .; /f ./ dS./ 4

S2

2n C 1 .i;i / pn .; /f .i / ./ dS./: 4

1 Z 3 X X i D1 nD0i

S2

.i / .i / yn;m ./  f ./ dS./ yn;m ./

The expansion of vector fields in terms of the Legendre tensors can be regarded as a natural extension of the scalar Fourier theory to the vectorial case. In order to motivate an alternative approach based on Legendre vectors, we write the representation of a vector-valued function on the sphere S2 in the following way: f ./ D

1 X 3 X n Z X 2 i D1 nD0i mDn S

D

1 X 3 X n Z X 2 i D1 nD0i mDn S

.i / .i / f .i / ./  yn;m ./ dS./ yn;m ./

(5.9.11)

f .i / ./..in / /1=2 o.i / Yn;m ./ dS./..in / /1=2 o Yn;m ./ .i /

5.9 Vectorial Counterparts of the Legendre Polynomial

D

1 3 X X

..in / /1=2

i D1 nD0i

D

1 3 X X

..in / /1=2

i D1 nD0i

Z S2

Z S2

253

O.i / f .i / ./..in / /1=2 o

.i /

n X

Yn;m ./Yn;m ./ dS./

mDn

C1 Pn .  / dS./: 4

.i / 2n

O.i / f .i / ./..in / /1=2 o

This leads us to the following definition. .i /

Definition 5.9.3. The Legendre vector field pn W S2  S2 ! R3 ; i 2 f1; 2; 3g, of type i and degree n 2 N0i with respect to the dual system of operators o.i /; O .i / ; i 2 f1; 2; 3g, is given by 1=2 .i /  pn.i / .; / D .in / o Pn .  /;

;  2 S2 :

(5.9.12)

The Legendre vector field pn W S2  S2 ! R of degree n with respect to the dual system of operators o.i / ; O .i /; i D 1; 2; 3, is defined by pn .; / D

3 X

pn.i / .; /;

n 2 N;

.1/

p0 .; / D p0 .; /:

(5.9.13)

i D1

Following our considerations given above, every vector-valued function f 2 l2 .S2 / admits an expansion f ./ D

Z 1 3 X X 2n C 1 .i / 1=2 .n / pn.i / .; /O.i / f ./ dS./: 2 4 S i D1 nD0

(5.9.14)

i

Obviously, the Legendre vectors fulfill an addition theorem, which reads as follows: Theorem 5.9.4. Let the system fYn;m gmDn;:::;n be an L2 .S2 /-orthonormal basis .i / of Harmn .S2 / and let yn;m D o.i / Yn;m be the corresponding vector spherical harmonics. Then, for i 2 f1; 2; 3g, n 2 N0i , and all ;  2 S2 , n X mDn

.i / yn;m ./Yn;m ./ D

2n C 1 .i / pn .; /: 4

(5.9.15)

The Legendre polynomials and the corresponding Legendre vector and rank-2 tensor functions are related in the following way: .i;k/

Lemma 5.9.5. Let Pn be the Legendre polynomials of degree n 2 N, pn be the .i / corresponding Legendre tensors of degree n 2 N and type .i; k/ and pn be the corresponding Legendre vector of degree n 2 N and type i . Then, for all ;  2 S2 ,

5 Vectorial Spherical Harmonics in R3

254

1=2 Pn .  / D ..in / /1=2 ..k/ O O.i / p.i;k/ n / n .; /; .k/

and

Pn .  / D ..in / /1=2 O pn.i / .; /: .i /

(5.9.16)

(5.9.17)

.1/

Furthermore, with 0 D 1 we have .1/

.1/

P0 .  / D O O.1/ p.1;1/ .; / D O pn.1/ .; /: n

(5.9.18)

Remark 5.9.6. The whole approach to vector spherical harmonics can be extended in a canonical way to the theory of tensor spherical harmonics which are also generated from the scalar ones by application of certain operators mapping scalar functions to tensor fields. In the tensorial setup it is also possible to determine complete orthonormal systems, characterize tensor spherical harmonics as eigenfunctions, and find a tensorial addition theorem as well as a tensorial Funk–Hecke formula. We refer to Freeden et al. (1998), Freeden and Schreiner (2009) and the many references therein.

5.10 Application to Elastic Fields Now, we come back to the example of Sect. 1.4, where we derived the Cauchy– Navier equation of linear elasticity. Starting with the homogeneous vector polynomials in R3 , we develop vector polynomials that solve the Cauchy–Navier equation (1.4.19), i.e., }u D u C

C r.r  u/ D 0; 

(5.10.1)

where and  denote the two Lam´e parameters. Let navn .R3 / be the class of homogeneous vector polynomials of degree n satisfying Navier’s equations in R3 :

C 3 3 ; navn .R / D u 2 homn .R / W }u D u C r.r  u/ D 0;  D  (5.10.2) where a vector polynomial hn 2 homn .R3 / if and only if hn "i 2 Homn .R3 / for i D 1; 2; 3. Note that hn .˛x/ D ˛ n hn .x/ for all ˛ 2 R, x 2 R3 , and hn 2 homn .R3 /. Remark 5.10.1. If  D 0, (5.10.2) leads back to the space harmn .R3 / of vectorial harmonic polynomials (well-known from Sect. 5.5 or, e.g., Freeden and Schreiner 2009).

5.10 Application to Elastic Fields

255

Every vector field u 2 navn .R3 / can be written in the form u.x/ D

n X

j

cnj .x1 ; x2 / x3 ;

x 2 R3 ; x D Œx1 ; x2 ; x3 T ;

(5.10.3)

j D0

where the cnj W R2 ! R3 denote homogeneous vector polynomials of degree n  j . It can be readily seen that }u allows the following representation: }x u.x/ D x u.x/ C rx .rx  u.x// Da

(5.10.4)

2

@ @ u.x/ C bx u.x/ C cx ; 2 @x @x3 3

x 2 R3 ;

where we have used the matrix operators a; bx ; cx given by 2 3 10 0 a D 40 1 0 5 ; 0 0 1C 2 3 0 0  @x@1 6 7 0  @x@2 5 ; bx D 4 0  @x@ 1  @x@2 0 2 @2 @2  @x@ 1 @x@2 .1 C / @x 2 C @x22 1 6 @2 @2  @x@ 2 @x@1 C .1 C / @x cx D 6 2 @x12 4 2 0 0

(5.10.5)

(5.10.6) 3

0 0 @2 @x12

C

@2 : @x22

7 7: 5

(5.10.7)

Observing the fact that X @u j .x1 ; x2 ; x3 / D .j C 1/cnj 1 .x1 ; x2 /x3 ; @x3 j D0

(5.10.8)

X @2 u j .x1 ; x2 ; x3 / D .j C 2/.j C 1/cnj 2 .x1 ; x2 /x3 ; 2 @x3 j D0

(5.10.9)

n1

n2

we get from (5.10.3) the recursion relation .j C 2/.j C 1/acnj 2 .x/ C .j C 1/bx cnj 1 .x/ C cx cnj .x/ D 0; (5.10.10) x D Œx1 ; x2 T , j D 0; : : : ; n2. Since the matrix a is regular (note that  ¤ 1), all polynomials cj are determined, provided that the homogeneous vector polynomials cn (of degree n) and cn1 (of degree n  1) are known. For cn 2 homn .R2 / there are 3.n C 1/ possible choices and for cn1 2 homn1 .R2 / there are 3n. Summarizing our results, we obtain the following theorem.

5 Vectorial Spherical Harmonics in R3

256

Theorem 5.10.2. Let cn , cn1 W R2 ! R3 be homogeneous polynomials of degree n, n  1, respectively. For j D 0; : : : ; n  2, we define recursively acnj 2 .x1 ; x2 / D 

  1 .j C 1/bx cnj 1 .x1 ; x2 / C cx cnj .x1 ; x2 / : .j C 2/.j C 1/ (5.10.11)

Then, un W R3 ! R3 given by un .x1 ; x2 ; x3 / D

n X

j

cnj .x1 ; x2 /x3

(5.10.12)

j D0

is a homogeneous polynomial of degree n in R3 satisfying the Navier equation }x un .x/ D 0; x 2 R3 . Moreover, the number of linearly independent homogeneous polynomials is equal to the total number of coefficients of cn and cn1 , i.e., dim.navn .R3 // D 3.2n C 1/:

(5.10.13)

Remark 5.10.3. We know (see, e.g., Freeden and Schreiner 2009 and the references therein) that homogeneous harmonic polynomials of different degrees are orthogonal (in the l2 .S2 /-sense). This fact, however, is not true for the spaces navn .R3 / with  ¤ 0, as the following example shows. The vector fields 3 2 2 3 x1 x2 0 7 6 u0 .x/ D 415 ; u2 .x/ D 4 2.C3/ .x12 C x22 C x32 /5 (5.10.14) 0 0 are elements of nav0 .R3 / and nav2 .R3 /, respectively. But it follows by an easy calculation that Z 2 ¤ 0: (5.10.15) u0 ./  u2 ./ dS./ D  2  C3 S Nevertheless, we are able to prove the following result. Theorem 5.10.4. Let un 2 navn .R3 /; um 2 navm .R3 /. Then, Z hun; um il2 .S2 / D

S2

un ./  um ./ dS./ D 0

(5.10.16)

if jn  mj ¤ 2 and n ¤ m. Proof. The proof of Theorem 5.10.4 can be found, e.g., in Freeden and Schreiner (2009). t u Next, we are interested in giving explicit representations of homogeneous polynomials of degree n which solve the Navier equation in R3 . This can be done, for

5.10 Application to Elastic Fields

257

example, by using the recursion formula (5.10.10). But we are also able to use known information about scalar homogeneous harmonic polynomials. We start with a preparatory lemma which follows straightforwardly. Lemma 5.10.5. Let Hn 2 Harmn .R3 /, n 2 N0 . Then, (i) x .Hn .x/x/ D 2rx Hn .x/, m m2 (ii) x .jxj Hn .x/; m  2;  2 Hn .x// D m.m C 2n C 1/jxj (iii) x x rx Hn .x/ D 2.2n C 1/rx Hn .x/. We are now interested in the following lemma that can be obtained by elementary calculations. Lemma 5.10.6. Let Hn W R3 ! R be a homogeneous harmonic polynomial of degree n. Then, the following identities are valid: (i) For all x 2 R3 ; }x .rx Hn .x// D 0. (ii) For all x 2 R3 ; }x .x ^ rx Hn .x// D 0. (iii) For all x 2 R3 ; }x .xHn .x/ C ˛n jxj2 rx Hn .x// D 0, where ˛n D 

.3 C n/ C .5 C n/ : 2 .n C .3n C 1//

(5.10.17)

(iv) For all x 2 R3 , k D 1; 2; 3, }x .Hn .x/"k C ˇn jxj2 rx rx  .Hn .x/"k // D 0, where C : (5.10.18) ˇn D  .2 C 6/ n  2  4   (v) For all x 2 R3 , k D 1; 2; 3, }x Hn .x/"k C n ."k  rx Hn .x//x D 0, where

n D 

C : .n C 2/ C .n C 4/

(5.10.19)

Lemma 5.10.6 enables us to develop three important systems of polynomial solutions of the Navier equation. Lemma 5.10.7. Let fHn;j gj Dn;:::;n be a linearly independent system of scalar homogeneous harmonic polynomials of degree n. The functions wn;j;k W R3 ! R3 with k D 1; 2; 3 defined by   wn;j;k .x/ D Hn;j .x/"k C ˇn jxj2 rx rx  .Hn;j .x/"k / ;

x 2 R3 ;

(5.10.20)

form a set of 3.2nC1/ linearly independent elements of navn .R3 /, where ˇn is given by (5.10.18). Lemma 5.10.8. Let fHn;j gj Dn;:::;n be a linearly independent system of scalar homogeneous harmonic polynomials of degree n. The functions vn;j;k W R3 ! R3 with k D 1; 2; 3 defined by   vn;j;k .x/ D Hn;j .x/"k C n "k  rx Hn;j .x/ x;

x 2 R3 ;

(5.10.21)

5 Vectorial Spherical Harmonics in R3

258

form a set of 3.2nC1/ linearly independent elements of navn .R3 /, where n is given by (5.10.19). Remark 5.10.9. The system (5.10.20) can be found in Lurje (1963), while the system (5.10.21) has been discussed in Bauch (1981). Unfortunately both systems do not show orthogonal invariance, i.e., tT vn;j;k .t/ (resp. tT wn;j;k .t/) generally is not a member of the span of the system fvn;j;k g (resp. fwn;j;k g). A polynomial system showing this property will be listed now. Lemma 5.10.10. Let fHk;j gkDn1;n;nC1;j Dk;:::;k be a linearly independent system .i / of scalar homogeneous harmonic polynomials. Then, the functions un;j W R3 ! R3 with i D 1; 2; 3 defined by .1/

un;j .x/ DHn1;j .x/x C ˛n1 jxj2 rx Hn1;j .x/; .2/

un;j .x/ Drx HnC1;j .x/; .3/

un;j .x/ Dx ^ rx Hn;j .x/;

n 2 N; j D n C 1; : : : ; n  1; (5.10.22)

n 2 N0 ; j D n  1; : : : ; n C 1; n 2 N; j D n; : : : ; n;

(5.10.23) (5.10.24)

form a set of 3.2nC1/ linearly independent elements of navn .R3 /, where ˛n is given by (5.10.17). .2/

.3/

The functions un;j ; un;j are characterized by the following properties: .2/

.2/

(5.10.25)

.3/

(5.10.26)

rx  un;j .x/ D 0; rx ^ un;j .x/ D 0; .3/

x  un;j .x/ D 0;

rx  un;j .x/ D 0: .2/

From a physical point of view this means that un;j is a poloidal field (i.e., a vector .3/

field free of dilatation and torsion), while un;j is a toroidal field. Only the functions .1/

un;j are responsible for a change of the volume. Remark 5.10.11. There is a very interesting relation between the systems fwn;j;k g, .i / fvn;j;k g, k D 1; 2; 3, introduced by Lemmas 5.10.7 and 5.10.8 and the system fun;j g, P3 i D 1; 2; 3, of Lemma 5.10.10. Replacing Hn1;j by kD1 "k  rHn;j (note that "k  rHn;j is a homogeneous harmonic polynomial of degree n  1 due to a result .1/ in Freeden and Schreiner (2009)) in the representation of un;j (see (5.10.22)), we obtain a field zn;j defined as follows: zn;j .x/ D

X 3 kD1

X 3  k  k   2 "  rx Hn;j .x/ x C ˛n1 x rx "  rx Hn;j .x/ : kD1

(5.10.27)

5.10 Application to Elastic Fields

259

It is clear that zn;j satisfies the Navier equation. Moreover, it is easy to see that

n D ˇn =˛n1 . But this shows that 3 X

vn;j;k D

kD1

3 X

wn;j;k 

kD1

ˇn zn;j ; ˛n1

n 2 N0 ; j D n; : : : ; n:

(5.10.28)

Assuming that the scalar system fHn;j gn2N0 ;j Dn;:::;n forms an orthonormal system of homogeneous harmonic polynomials with respect to L2 .S2 /, the following orthogonal relations can be guaranteed: Z Z Z

.i /

S2

S2

S2

.k/

un;j ./  un;l ./ dS./ D 0 if i ¤ k or j ¤ l; .i /

.i /

.3/

.i /

(5.10.29)

un;j ./  um;k ./ dS./ D 0 if n ¤ m or j ¤ k; i D 1; 2; 3; (5.10.30) un;j ./  um;k ./ dS./ D 0 if i D 1; 2:

(5.10.31)

This shows us the following lemma. Lemma 5.10.12. The space navn .R3 /, n > 0, defined by (5.10.2) can be decom.i / posed into three subspaces navn .R3 /, i D 1; 2; 3, given by ˚ .1/  3 nav.1/ n .R / D span un;j W j D n C 1; : : : ; n  1 ; ˚ .2/  3 nav.2/ n .R / D span un;j W j D n  1; : : : ; n C 1 ; ˚ .3/  3 nav.3/ n .R / D span un;j W j D n; : : : ; n ;

(5.10.32) (5.10.33) (5.10.34)

such that 3 .2/ 3 .3/ 3 navn .R3 / D nav.1/ n .R / ˚ navn .R / ˚ navn .R /:

(5.10.35)

˚ .2/  .2/ nav0 .R3 / D nav0 .R3 / D span u0;j W j D 1; 0; 1 :

(5.10.36)

For n D 0,

Moreover, we have the following dimensions:   3 dn.1/ D dim nav.1/ n .R / D 2n  1; dn.3/

  3 dn.2/ D dim nav.2/ n .R / D 2n C 3; (5.10.37)     .2/ 3 D dim nav.3/ dim nav0 .R3 / D d0 D 3: n .R / D 2n C 1;

5 Vectorial Spherical Harmonics in R3

260 .i /

As mentioned above, the spaces navn .R3 /; i D 1; 2; 3, are orthogonally invariant .i / in the sense that un 2 navn .R3 / is equivalent to tT un .t/ 2 nav.in / .R3 /;

(5.10.38)

i D 1; 2; 3, for every orthogonal transformation t 2 O.3/. Thus, we have found a decomposition of navn .R3 / into three invariant subspaces. .i / .i / .i / Next, assume that wn is a member of navn .R3 /. Consider the space hn of all .i / linear combinations of functions wn .t/, where t is an orthogonal transformation: ˚  h.in / D span w.in / .t/ W t 2 O.3/ : .i /

(5.10.39)

.i /

Then, it is clear that 0 < dim.hn /  dn . Moreover, it can be shown that there .i / exists no orthogonal invariant subspace in navn .R3 /. Thus, it follows immediately .i / .i / 3 that navn .R / D hn . This leads us to the following lemma. .i /

.i /

.i /

Lemma 5.10.13. Let wn be of class navn .R3 /. There exist dn orthogonal .i / .i / .i / transformations tj ; j D 1; : : : ; dn , such that any element un 2 navn .R3 / can be written in the form .i /

u.in /

D

dn X

.i /

cj tTj w.in / .tj /;

(5.10.40)

j D1 .i /

where cj are real numbers. .i /

Finally, we formulate the addition theorem for the system fun;j g developed in Lemma 5.10.10. By separation of radial and angular components we first obtain after some simple calculations for x D jxj: .1/

.1/

.1/

.1/

.2/

un;j .x/ D n .jxj/yn1;j ./ C ın .jxj/yn1;j ./;

n 2 N; j D n C 1; : : : ; n  1;

(5.10.41) .2/ un;j .x/

D

.2/ .1/ .2/ .2/

n .jxj/ynC1;j ./ C ın .jxj/ynC1;j ./;

n 2 N0 ; j D n  1; : : : ; n C 1;

(5.10.42) .3/

.3/

.3/

un;j .x/ D n .jxj/yn;j ./;

n 2 N; j D n; : : : ; n;

(5.10.43)

where we have used the abbreviations

n.1/ .jxj/ D jxjn .1 C .n  1/˛n1 /; p ın.1/ .jxj/ D jxjn ˛n1 .n  1/n;

n.2/ .jxj/

D jxj .1 C n/; n

(5.10.44) (5.10.45) (5.10.46)

5.10 Application to Elastic Fields

261

p ın.2/ .jxj/ D jxjn .n C 1/.n C 2/; p

n.3/ .jxj/ D jxjn n.n C 1/:

(5.10.47) (5.10.48)

Remembering the addition theorem for vector spherical harmonics (see Theorem 5.7.1), we obtain the following theorem that uses the Legendre rank-2 tensor kernels of Definition 5.7.2 whose explicit representation is given in Theorem 5.7.4. Theorem 5.10.14. For x; y 2 R3 ; x D r; y D %; r D jxj; % D jyj, n1 X

.1/

.1/

un;j .x/˝un;j .y/

(5.10.49)

j DnC1 .1;1/

.1;2/

D n.1/ .r/ n.1/ .%/pn1 .; / C n.1/ .r/ın.1/ .%/pn1 .; / .2;1/

.2;2/

C ın.1/ .r/ n.1/ .%/pn1 .; / C ın.1/ .r/ın.1/ .%/pn1 .; /; nC1 X

.2/

.2/

un;j .x/˝un;j .y/

(5.10.50)

j Dn1 .1;1/

.1;2/

D n.2/ .r/ n.2/ .%/pnC1 .; /; C n.2/ .r/ın.2/ .%/pnC1 .; / .2;1/

.2;2/

C ın.2/ .r/ n.2/ .%/pnC1 .; / C ın.2/ .r/ın.2/ .%/pnC1 .; /; n X

.3/

.3/

un;j .x/˝un;j .y/ D n.3/ .r/ n.3/ .%/p.3;3/ .; /; n

(5.10.51)

j Dn

where (5.10.49) holds for n  2, (5.10.50) for n 2 N, and (5.10.51) for n 2 N0 . .2/ .2/ .1;1/ 1 Note that u1;0 .x/ ˝ u1;0 .y/ D 4 x ˝ y D jxjjyj p0 .; /. 4 In particular, we find the following result for the traces if x D y (see also Corollary 5.7.5 and Lemma 5.7.6). Lemma 5.10.15. If x 2 R3 ; r D jxj; x D r, then n1 X ˇ ˇ .1/   ˇu .x/ˇ2 D r 2n .1 C .n  1/˛n1 /2 C ˛ 2 n.n  1/ 2n  1 ; n1 n;j 4 j DnC1

(5.10.52) nC1 2 X ˇ ˇ .2/ ˇu .x/ˇ2 D r 2n .n C 1/.2n C 3/ ; n;j 4 j Dn1

(5.10.53)

n X ˇ ˇ .3/ ˇu .x/ˇ2 D r 2n n.n C 1/.2n C 1/ ; n;j 4 j Dn

(5.10.54)

where (5.10.52) as well as (5.10.54) hold for n 2 N and (5.10.53) holds for n 2 N0 .

5 Vectorial Spherical Harmonics in R3

262

From our considerations given above it is clear that there are different ways of computing linearly independent systems of homogeneous polynomial solutions to the Navier equations. Of course, the recursion procedure of Theorem 5.10.2 can be used to derive an algorithm quite analogously to the method used for scalar homogeneous polynomials. Next, we are interested in determining elastic potentials corresponding to vector spherical harmonics as boundary values. .i /

Lemma 5.10.16. Let vn;j ; R3 ! R3 ; i D 1; 2; 3, n 2 N0i , j D n; : : : ; n, be defined by   .1/ vn;j .x/ D Hn;j .x/x C ˛n jxj2  1 rx Hn;j .x/;

(5.10.55)

1  .2/ .1/ vn;j .x/ D .n.n C 1// 2 rx Hn;j .x/  nvn;j .x/ ;

(5.10.56)

1

vn;j .x/ D .n.n C 1// 2 x ^ rx Hn;j .x/; .3/

(5.10.57)

where ˛n D 

n C 2 C 3 ; 2.n. C 2/ C 1/

Hn;j .x/ D jxjn Yn;j ./;

x D jxj;  2 S2 : (5.10.58)

.i / Then, vn;j satisfies the .i / .i / values vn;j jS2 D yn;j .

Cauchy–Navier equation

.i / }vn;j .x/

D 0 in B31 with boundary

Proof. It is not hard to see that .1/

}x vn;j .x/ D2rx Hn;j .x/ C .3 C n/rx Hn;j .x/

(5.10.59)

C ˛n ..6 C 4.n  1//rx Hn;j .x/ C 2nrx Hn;j .x// D 0; 1  .2/ .1/ }x vn;j .x/ D.n.n C 1// 2 }x rx Hn;j .x/  n}x vn;j .x/ D 0;

(5.10.60)

1

}x vn;j .x/ D.n.n C 1// 2 }x .x ^ rx Hn;j .x// D 2rx ^ rx Hn;j .x/ D 0: (5.10.61) .3/

Using the polar coordinates x D r; r D jxj;  2 S2 , we obtain after simple calculations for n 2 N, j D n; : : : ; n, .1/

.1/

.2/

(5.10.62)

.2/

.1/

.2/

(5.10.63)

.3/

.3/

vn;j .x/ D n.1/ .r/yn;j ./ C n.1/ .r/yn;j ./; vn;j .x/ D n.2/ .r/yn;j ./ C n.2/ .r/yn;j ./; vn;j .x/ D n.3/ .r/yn;j ./;

(5.10.64)

5.10 Application to Elastic Fields .1/

.1/

263

.1/

and v0;0 .x/ D 0 .r/y0;0 ./ D r p1 D 4

p1 x, 4

where

   n.1/ .r/ D r n1 r 2 C n˛n r 2  1 ; n.2/ .r/ D .n.n C n.3/ .r/

1 1// 2 n.1

(5.10.65)

  C n˛n /r n1 1  r 2 ;

(5.10.66)

Dr ; n

(5.10.67)

1   n.1/ .r/ D ˛n .n.n C 1//C 2 r n1 r 2  1 ;    n.2/ .r/ D r n1 1  n˛n r 2  1 : .i /

(5.10.68) (5.10.69)

.i /

This shows us that vn;j jS2 D yn;j , as required.

t u

It should be mentioned that .1/

.1/

vn;j D unC1;j  ˛n rHn;j ;

n 2 N0 ; j D n; : : : ; n;

1  .2/ .2/ .1/  vn;j D .n.n C 1// 2 un1;j  nvn;j ; 1

vn;j D .n.n C 1// 2 un;j ; .3/

.3/

(5.10.70)

n 2 N; j D n; : : : ; n;

n 2 N; j D n; : : : ; n:

(5.10.71) (5.10.72)

.i /

.i /

Thus, the polynomial solution vn;j , i D 1; 2, corresponding to yn;j on S2 is not homogeneous. Remark 5.10.17. Observe that, under the assumption 3 C 2 > 0;  > 0, it follows that C 1 3 C 2 1 D D C > : (5.10.73)  3 3 3 Since  > 0, it is not difficult to deduce that, for all n  3, j˛n j D

11C 1 n C 3 C 2 D 2 n C 2n C 1 21C

3 n 2n n

C C

2 n 1 n



12C 21C

2 n 1 n

 1;

(5.10.74)

while, for all n  1, because of  > 0; j˛n j D

3 3  1 1 C n2 1 2n  2: C  C 2nC1 2nC1 2 1 C n 2 2 n C 2n C 1 1 C n

(5.10.75)

Thus, the sequence .˛n /n is uniformly bounded with respect to . .i /IR

Remark 5.10.18. Let us denote by vn;j W B3R ! R3 , i D 1; 2; 3, n 2 N0i , j D n; : : : ; n, the vector fields   jxj jxj .1/IR .1/ .2/ yn;j ./ C n.1/ yn;j ./; vn;j .x/ D n.1/ (5.10.76) R R

5 Vectorial Spherical Harmonics in R3

264

 jxj .1/ .2/ .2/ jxj yn;j ./ C n yn;j ./; D R R  jxj .3/IR .3/ yn;j ./; vn;j .x/ D n.3/ R 

.2/IR vn;j .x/

n.2/

.i /

(5.10.77) (5.10.78)

.i /

.i /IR

where x D jxj; jxj  R and n ; n are given by (5.10.65)–(5.10.69). Then, vn;j is the unique solution of the boundary-value problem     .i /IR vn;j 2 c B3R \ c.2/ B3R ;

.i /IR

˙vn;j D 0

.i /IR

in B3R ;

(5.10.79)

.i /

corresponding the boundary values vn;j jS2 D yn;j . R

For further details on polynomial systems for the Cauchy–Navier equation in spherical geometry the reader is referred to Freeden and Michel (2004), Freeden and Schreiner (2009) and the references therein.

5.11 Exercises (Uncertainty Principle, Classification of Zonal Functions, Coupling Integrals and Navier–Stokes Equation) In the following, we present some exercises reflecting the framework of vector spherical harmonics.

Uncertainty Principle The uncertainty principle opens various perspectives in classifying special functions on the sphere S2 . Basic quantities are the expectation value and the variance in the space and frequency (more precisely, momentum) domain. A more detailed discussion of the following uncertainty relation can be found in Freeden and Schreiner (2009). Definition 5.11.1. Let F 2 C.1/ .S2 / satisfy kF kL2 .S2 / D 1. Then, the expectation value and the variance in the space domain, respectively, are given by .1/ gFo

o.1/

F

Z   o.1/ F ./ F ./ dS./; D

(5.11.1)

S2

D

Z  S2

o.1/

o.1/  gF F ./

!1=2

2

2 F ./

dS./

;

(5.11.2)

5.11 Exercises

265

Fig. 5.2 Localization in a .1/ spherical cap, oF is the .1/ projection of gFo onto the sphere S2 and is the center of the spherical cap .1/ C D f 2 S2 W 1    oF  .1/ 1  jgFo jg. The boundary of the cap @C is a circle of .1/ radius Fo

(1)

ηFo C

(1)

(1) σFo

goF

1

while the expectation value and the variance in the frequency domain, respectively, are given by .3/

Z   o.3/ F ./ F ./ dS./;

gFo

D

.3/ Fo

!1=2 2 Z  2 .3/ o.3/ o  gF F ./ F ./ dS./ D :

(5.11.3)

S2

(5.11.4)

S2

The geometric interpretation of the localization in space domain is provided by Fig. 5.2. If we consider a zonal function F to be a spherical “window function”, the spherical cap C in Fig. 5.2 determines the window.  .1/ 2  .1/ 2  .1/ 2 D 1  gFo and 0  Fo  1. Exercise 5.11.2. Show that Fo Exercise 5.11.3. Prove that for F 2 C.1/ .S2 /; Z 2 n 1 X  .3/ 2 X o F D n.n C 1/ F ./Yn;k ./ dS./ : nD0 kDn

(5.11.5)

S2

Exercise 5.11.4. Prove that for F 2 C.1/ .S2 / and a constant vector a 2 R3 ; Z

  F ./ .  a/ ^ o.3/ F ./ dS./ D S2

Z F ./.o.3/ ^ ..  a/F ./// dS./: S2

(5.11.6)

Exercise 5.11.5. Verify that for all F 2 C.1/ .S2 / with kF kL2 .S2 / D 1;  .1/ 2  .3/ 2 Fo Fo  gF2 ;

(5.11.7)

5 Vectorial Spherical Harmonics in R3

266

Table 5.1 The uncertainty principle and its consequences space localization

ideal space localization

no space localization frequency localization

ideal frequency localization

no frequency localization kernel type

Legendre kernel

where gF D

bandlimited

locally supported

Z ˇ ˇˇ  ˇ .1/ ˇ ˇ .1/ ˇ o  gFo F ./ˇ ˇo.3/F ./ ˇ dS./:

Dirac kernel

(5.11.8)

S2

Exercise 5.11.6. Use Exercise 5.11.5 to guarantee the uncertainty principle  .1/ 2  .3/ 2  .1/ 2 Fo Fo  gFo

(5.11.9)

for all F 2 C.1/ .S2 / with kF kL2 .S2 / D 1. .1/

.3/

Remark 5.11.7. The meaning of Fo and Fo is a measure for space localization and frequency (momentum) localization, respectively. The uncertainty relation measures the trade off between the “spread in space and frequency”. It states that sharp localization in space and frequency is mutually exclusive (cf. Table 5.1). The reason for the validity of the uncertainty principle (5.11.9) is that the operators o.1/ and o.3/ do not commute. Thus, o.1/ and o.3/ cannot be sharply defined simultaneously in the sense of (5.11.9). Extremal members in the space/frequency relation are the polynomials (i.e., spherical harmonics) and the Dirac functional. Remark 5.11.8. The uncertainty principle was first derived by Heisenberg (1927) and later concretized by Weyl (1931) in the nomenclature of standard derivations. An overview within the context of time-frequency analysis is due to Benedetto (1994) and Cohen (1995) (see also the references therein).

Classification of Zonal Functions The uncertainty principle allows a quantitative classification in the form of a specific hierarchy of the space/frequency localization of zonal kernel functions

5.11 Exercises

267 2

10

1

10

10

0

0

10

−1

10

−2

10

0

1

2

10

10

10

−2

−1

10

10

10

0

Fig. 5.3 Uncertainty classification of the normalized Gaussian kernel (left) and Abel-Poisson .1/ .3/ kernel (right). Presented are o in blue, o in red, and the product of both in green as functions of (left) or h (right) in a double logarithmic setting

 7! K.  / D

1 X 2n C 1

4

nD0

K ^ .n/Pn .  /:

(5.11.10)

Exercise 5.11.9. The Gaussian kernel G , > 0, is given by

 7! G .  "3 / D e 2 .1" / ; 3

 2 S2 :

(5.11.11)

Note that G takes its name from the bell-shaped Gaussian probability density ˛2 2



function, since e 2 .1cos ˛/ e 2

for small ˛  0.

(a) Show that  7! GQ .  "3 / D . /G .  "3 /;

 2 S2 ;

(5.11.12)

satisfies kGQ kL2 .Œ1;1 / D 1, where 1

. / D p 4



1 .1  e2 / 2

 12 :

(5.11.13)

(b) Verify that the best value of the uncertainty principle is 1 by realizing (see Fig 5.3 (left)) that .1/ tGoQ

 

.1/

GoQ



.3/

GoQ



2

."3 /

."3 /

2

Z ˇ .1/ ˇ ˇ o ˇ D ˇgGQ ."3 / ˇ D 2

1 1

tjGQ .t/j2 dt D coth. /  1 ;

(5.11.14)

2  D 1  coth. /  1 ;

(5.11.15)

D

(5.11.16)

2

coth. /  12 :

5 Vectorial Spherical Harmonics in R3

268

In other words,  lim

!1

.1/

GoQ



."3 /

2  .3/ 2  .1/ 2 GoQ ."3 / D 1 D lim gGoQ ."3 / :

!1



(5.11.17)

Exercise 5.11.10. The Abel-Poisson kernel Qh , 0 < h < 1, (see also Theorem 4.4.3) is given by  7! Qh .  "3 / D

1 1  h2 : 2 4 .1 C h  2h.  "3 //3=2

(5.11.18)

Show that QQ h D kQh k1 Q satisfies L2 .Œ1;1 / h r .1/ oQQ h



.3/ oQQ h

D

3 > 1; 2

(5.11.19)

where (see Fig 5.3 (right)) .1/

o.1/ Q Qh

D

o Q Q

h .1/

(5.11.20)

o jgQ Q j h

and

.3/

.3/

h

h

o oQQ D Q Q :

(5.11.21)

Exercise 5.11.11. Use a recursion relation for the Legendre polynomial to justify 

o.1/ K. /

2

D1

1 1 X .n C 1/K ^.n/K ^ .n C 1/ 2 nD0

!2 ;

(5.11.22)

for K 2 C.2/ .Œ1; 1 /. Equation (5.11.22) can be heuristically interpreted in the following way: If K ^ .n/ K ^ .nC1/ for many successive integers n, then the support of (5.11.10) in space domain is small. Exercise 5.11.12. Let the Shannon kernel ˚% , % > 0, be given by  7! ˚% .  "3 / D

X 2n C 1 Pn .  "3 /; 4 1

 2 S2 :

(5.11.23)

n%

Show that ˚Q % D k˚% k1 ˚ satisfies L2 .Œ1;1 / % 0  .1/ 2 ˚oQ D1@ %

1 k˚% k2

12 1 X 2n C 2 A D 2b% c C 1 ; 4 .b%1 c C 1/2 nD0

b%1 c1

(5.11.24)

5.11 Exercises

269 2

2

10

1

10

0

10

−1

10−1

10

1

10

0

10

10

−2

10

10

−2

−1

0

10

10

10−2 −2 10

−1

0

10

10

Fig. 5.4 Uncertainty classification of the normalized smoothed Haar kernel BQh with k D 1 (left) .1/ .3/ and k D 3 (right). Depicted are o in blue, o in red, and the product of both in green as functions of h in a double logarithmic setting .k/

such that (see Fig. 5.5 (left)) .1/ o˚Q %

p 2b%1 c C 1 D b%1 c

(5.11.25)

and 

.3/ ˚oQ %

2

1

b% c X 2n C 1 1 4 n.n C 1/ D b%1 c.b%1 c C 2/; D 1 2 .b% c C 1/ nD0 4 2

r .3/

o˚Q D %

(5.11.26) .b%1 c C 1/2  1 : 2

(5.11.27)

Note that the Shannon kernel formally converges to the Dirac kernel as % ! 0C. .k/

Exercise 5.11.13. Let the smoothed Haar kernel Bh , h 2 .0; 1/, k 2 N0 , (see also (4.8.42) in Exercise 4.8.10) be given by (  7!

.k/ Bh .

" /D 3

0 .t h/k .1h/k

;

  "3 2 Œ1; h/;

;

  "3 2 Œh; 1 :

(5.11.28)

.k/ .k/ .k/ Verify that BQh D kBh k1 B satisfies (see Fig. 5.4) L2 .Œ1;1 / h

2  .1  h/.h C 4k C 3/ o.1/  Q .k/ D ; Bh .2k C 2/2 p 1 .1/ .1  h/.h C 4k C 3/; oQ .k/ D Bh 1 C h C 2k

(5.11.29) (5.11.30)

5 Vectorial Spherical Harmonics in R3

270 102

101 100 100

10−1 0 10

101

102

102

101

100

Fig. 5.5 Uncertainty classification of the normalized Shannon kernel (left) and Bernstein kernel .1/ .3/ (right). Presented are o in blue, o in red, and the product of both in green as functions of 1 % in a double logarithmic setting

and  .3/  oQ .k/

Bh .

2 "3 /

Z s

o.3/

 Q .k/ D Bh

1

D 2 1

.k/ .k/ BQh .t/Lt BQ h .t/ dt D

k.h C 2k/ ; .1  h/.2k  1/

k.h C 2k/ ; .1  h/.2k  1/

(5.11.31)

(5.11.32)

where Lt denotes the Legendre operator of (3.5.21). Exercise 5.11.14. Consider the Bernstein kernel KB;% (see also Definition 4.4.6)  7! KB;% .  "3 / D

nC1 4



1 C   "3 2

b%1 c ;

(5.11.33)

^ whose Legendre coefficients KB;% .k/, k 2 N0 , are given by (4.4.32) in Lemma 4.4.8 1 by setting n D b% c.

(a) Show that kKB;% kL2 .S2 / D p b% (b) Prove that for KQ B;% D

1 cC1

.

.2b%1 cC1/ kKB;% k1 K it L2 .S2 / B;% 2

holds that (see Fig. 5.5 (right))

 .1/ 2 2b%1 c C 1 KoQ D ; B;% .b%1 c C 1/2 p 2b%1 c C 1 o.1/ ; KQ D B;% b%1 c

(5.11.34) (5.11.35)

5.11 Exercises

271

Table 5.2 Three types of kernels: bandlimited, locally supported, and globally supported/nonbandlimited Legendre kernel K ^ .n/ D ın;k , n 2 N0 , k 2 N0 fixed

Zonal kernels General case

Dirac kernel K ^ .n/ D 1, n 2 N0

Bandlimited K ^ .n/ D 0, n > N

Locally supported K.  / D 0, 1     > ı

e.g., Shannon K ^ .n/ D 1, n  N

e.g., Haar K.  / D 1, 1      ı

 .3/ 2 KoQ D B;%

.3/

o K Q

B;%

b%1 c 2 ;

q

D

and .1/

.3/

oKQ oKQ B;%

B;%

D

(5.11.36)

b%1 c ; 2

q

1C

(5.11.37)

1 ; 2b%1 c

(5.11.38)

which converges to the best value 1 for % ! 0C. The varieties of the intensity of localization on S2 can also be illustrated by discussing the expansion coefficients K ^ .n/ in more detail: By choosing K ^ .n/ D ın;k we obtain a Legendre kernel of degree k, i.e., we are in starting position of the scheme (Table 5.2). By formally taking K ^ .n/ D 1 for all integers n we are led to the Dirac kernel. Bandlimited kernels have the property K ^ .n/ D 0 for all n  N , N 2 N0 . Non-bandlimited kernels satisfy K ^ .n/ 6D 0 for an infinite number of integers n 2 N0 . Assuming the condition lim K ^ .n/ D 0;

n!1

(5.11.39)

we come to the conclusion that the slower the sequence fK ^.n/gn2N0 converges to zero, the lower is the frequency localization, and the higher is the space localization. Altogether, Table 5.1 gives a qualitative illustration of the consequences of the uncertainty principle to the theory of zonal functions. More details can be found in Freeden and Schreiner (2009). La´ın Fern´andez and Prestin (2003) discuss the optimal tradeoff by the Gaussian kernel in detail and for the Bernstein kernel see Fengler et al. (2006).

Wigner Symbols The next few pages are dedicated to Wigner symbols. They play an important role in quantum mechanics, and are involved in the coupling of two angular

5 Vectorial Spherical Harmonics in R3

272

momenta. In the following, we concentrate on the relations that are needed for operating on coupling terms and integrals. For a detailed introduction to Wigner3j symbols we refer to Edmonds (1964) and for more general results in the context of spherical tensors to Zare (1988). A detailed description for the computation of Wigner-3j symbols can be found in Zare (1988). Special emphasis on the fast and stable computation of Wigner-3j is given in Schulten and Gordon (1975, 1976). First, the Wigner-3j symbol is defined as follows (see Edmonds 1964). Definition 5.11.15 (Wigner-3j Symbols). Suppose that j1 ; j2 ; j3 2 N0 and that m1 ; m2 ; m3 2 Z with j1  m1  j1 , j2  m2  j2 , j3  m3  j3 . The Wigner-3j symbol is defined by 

j1 j2 j3 m1 m2 m3





.2j3 C 1/.j1 C j2  j3 /Š.j1  m1 /Š .j1 C j2 C j3 C 1/Š.j1  j2 C j3 /Š.j1 C j2 C j3 /Š 1 .j2  m2 /Š.j3  m3 /Š.j3 C m3 /Š 1=2  .1/m3 j1 Cj2 p .j1 C m1 /Š.j2 C m2 /Š 2j3 C 1

Dım1 Cm2 ;m3



1 X .1/sCj1 m1 sD0

.j1 C m1 C s/Š.j2 C j  m1  s/Š ; sŠ.j1  m1 /Š.j3 C m3  s/Š.j2  j3 C m1 C s/Š

(5.11.40) where the sum runs over all s for which the faculty expressions in the denominator are not negative. The definition can be extended in that sense, that the Wigner-3j symbol vanishes if any of the orders jmi j > ji , i D 1; 2; 3. Exercise 5.11.16 (3j -Symmetries). Assume that j1 ; j2 ; j3 2 N0 and m1 ; m2 ; m3 2 Z with j1  m1  j1 , j2  m2  j2 , j3  m3  j3 . (a) Show that the value of the Wigner-3j symbol is invariant with respect to an even permutation of columns, i.e.,    j2 j3 j1 j3 j1 j2 j1 j2 j3 D D : (5.11.41) m1 m2 m3 m2 m3 m1 m3 m1 m2 (b) Prove that, for an odd permutation, we have the multiplication with the factor .1/j1 Cj2 Cj3 , i.e.,     j1 j2 j3 j2 j1 j3 j1 j3 j2 j3 j2 j1 D D D : .1/j1 Cj2 Cj3 m1 m2 m3 m2 m1 m3 m1 m3 m2 m3 m2 m1 (5.11.42) Exercise 5.11.17 (Triangle Relation). Suppose that j1 ; j2 ; j3 2 N0 , m1 ; m2 ; m3 2 Z with j1  m1  j1 , j2  m2  j2 , j3  m3  j3 . Prove that the Wigner-3j symbol given by

5.11 Exercises

273



j1 j2 j3 m1 m2 m3

(5.11.43)

vanishes, if the following two conditions are not satisfied: m1 C m2 C m3 D 0; jj1  j2 j  j3  j1 C j2 :

(5.11.44) (5.11.45)

In special cases, the rather cumbersome sum in Definition 5.11.15 can be reduced to trivial expressions. Exercise 5.11.18 (One Vanishing Argument). Let j1 2 N0 and m1 2 Z, where j1  m1  j1 . Show that then the following identity is valid: 

j1 j1 1 D .1/j1 m1 .j1 .j1 C 1/.2j1 C 1//1=2 m1 : m1 m1 0

(5.11.46)

Exercise 5.11.19 (Two Vanishing Arguments). Assume that j1 ; j3 2 N0 and that m1 ; m3 2 Z with j1  m1  j1 and j3  m3  j3 . Prove that the following identity holds: j1 0 j3 D .1/j1 m3 .2j3 C 1/1=2 ım1 ;m3 ıj1 ;j3 : m1 0 m3



(5.11.47)

Exercise 5.11.20 (Three Vanishing Arguments). Suppose that j1 ; j2 ; j3 2 N0 and J D j1 C j2 C j3 . Prove that we have the following identity: 8 0 ˆ ˆ  <  1=2 j1 j2 j3 Cj3 j2 /Š.j2 Cj3 j1 /Š D .1/J=2 .j1 Cj2 j3 /Š.j1.J C1/Š ˆ 0 0 0 ˆ : .J=2/Š  .J=2j1 /Š.J=2j 2 /Š.J=2j3 /Š

if J is odd,

if J is even. (5.11.48)

Remark 5.11.21. For further details on Wigner-3j symbols and their properties we refer to Brink and Satchler (1968), Edmonds (1964), Messiah (1990), Var˘salovi˘c et al. (1988), and Zare (1988). Now, the Wigner-6j symbol is considered which occurs in coupling three or more angular momenta. A detailed introduction can be found in any standard textbook on quantum mechanics (see, e.g., Edmonds 1964; Messiah 1990; Rose 1957; Shore and Menzel 1968; Var˘salovi˘c et al. 1988). For its definition we introduce so-called metric tensors. Definition 5.11.22 (Metric Tensors). Assume that j 2 N0 and m; m0 2 Z. Then, the quantity

5 Vectorial Spherical Harmonics in R3

274



j D .1/j Cm ım;m0 m m0

(5.11.49)

is called a metric tensor. Definition 5.11.23 (Wigner-6j Symbols). Let j1 ; j2 ; j3 ; j4 ; j5 ; j6 2 N0 and m1 ; m2 ; m3 ; m4 ; m5 ; m6 , m01 ; m02 ; m03 ; m04 ; m05 ; m06 2 Z. Then, the expression

j1 j2 j3 j4 j5 j6

(5.11.50)

   6 X X  X j1 j2 j3 j1 j5 j6 j4 j2 j6 j4 j5 j3 m1 m2 m3 m01 m5 m06 m04 m02 m6 m4 m05 m03 0

D

i D1 mi 2Z mi 2Z





j1 m1 m01



j2 m2 m02



j3 m3 m03



j4 m4 m04



j5 m5 m05



j6 ; m6 m06

is called a Wigner-6j symbol. In view of Exercise 5.11.16 it is interesting to note that the Wigner-6j symbol also possesses appealing symmetry properties. Exercise 5.11.24 (6j -Symmetries). Prove that the value of the 6j -symbol is invariant (a) To the interchange of any two columns, (b) To the interchange of the upper and lower arguments in each of any two columns. Exercise 5.11.25 (One Vanishing Argument). For j1 ; j2 ; j4 ; j5 ; j6 2 N0 , show that the following identity holds true:

j1 j2 0 D .1/j1 Cj4 Cj6 ..2j1 C 1/.2j4 C 1//1=2 ıj1 ;j2 ıj4 ;j5 : j4 j5 j6

(5.11.51)

Exercise 5.11.26 (One Argument is Equal to 1). Assume that j1 ; j2 ; j3 2 N and J D j1 C j2 C j3 . Show that the following identities are valid:

2 .j2 .j2 C 1/ C j3 .j3 C 1/  j1 .j1 C 1// j1 j2 j3 p ; D .1/J C1 p 1 j3 j2 2j2 .2j2 C 1/.2j2 C 2/ 2j3 .2j3 C 1/.2j3 C 2/ (5.11.52)

and

 2.J C 1/.J  2j1 /.J  2j2 /.J  2j3 C 1/ 1=2 j1 j2 j3 J D .1/ : 1 j3  1 j2 2j2 .2j2 C 1/.2j2 C 2/.2j3  1/2j3 .2j3 C 1/ (5.11.53)

5.11 Exercises

275

Finally, we introduce Wigner-9j symbols which occur in the coupling of four angular momenta (see, e.g., Edmonds 1964). The definition of the Wigner-9j symbol can be traced back to that of the Wigner-6j symbol. Definition 5.11.27 (Wigner-9j Symbols). Let j1 ; j2 ; j3 ; j4 ; j5 ; j6 ; j7 ; j8 ; j9 2 N0 and m1 ; m2 ; m3 ; m4 ; m5 ; m6 ; m7 ; m8 ; m9 2 Z. The expression 9 8   9 X  r > 0, we have Z lim

r!0C

q1 Sr .y/

G.x/

Z

lim

r!0C Sq1 .y/ r

@ Fq .jx  yj/ dS.x/ D  G.y/; @x

H.x/Fq .jx  yj/ dS.x/ D 0;

(6.2.20) (6.2.21)

q

where the (unit) normal field  is directed to the exterior of Br .y/. Proof. We restrict ourselves to dimensions q  3. The case q D 2 easily follows by analogous arguments. q Because of the continuity of the function H in each ball Br .y/, r < R, we find ˇZ ˇ ˇ ˇ

ˇ Z ˇ C jx  yj2q ˇ dS.x/ˇ  H.x/ jx  yj2q dS.x/ .q  2/kSq1 k kSq1 k jxyjDr jxyjDr D

C r 2q kSq1 kr q1 kSq1 k

DC r

(6.2.22)

for some positive constant C . This shows the second limit relation (6.2.21). For the first limit relation, we observe that the normal derivative can be understood as the radial derivative. Therefore, we obtain from the mean value theorem: Z Z r 1q jx  yj2q @ dS.x/ D  G.x/ G.x/ dS.x/ @x .q  2/kSq1 k kSq1 k jxyjDr jxyjDr D

r 1q q1 q1 r kS k G.xr / (6.2.23) kSq1 k

6 Spherical Harmonics in Rq

294 q1

for certain points xr 2 Sr continuity of G yields

.y/. The limit r ! 0 implies xr ! y, such that the lim G.xr / D G.y/:

(6.2.24)

r!0C

t u

This is the desired result.

Next, we want to apply the second Green theorem, i.e., Theorem 6.2.2, (for a regular region G with continuously differentiable boundary @G ) especially to the functions F W x 7! F .x/ D 1 ; x 2 G ;

(6.2.25)

G W x 7! G.x/ D Fq .jx  yj/ ; x 2 G n fyg;

(6.2.26)

where y 2 Rq is positioned in accordance with the following three cases: (i) Case y 2 G : For sufficiently small " > 0 we obtain by integration by parts, i.e., the second Green theorem (Theorem 6.2.2), that Z Z @ x Fq .jx  yj/ dV .x/ D Fq .jx  yj/ dS.x/ (6.2.27) x2G „ @ ƒ‚ … x x2@G jxyj" D0

Z C

x2G jxyjD"

@ Fq .jx  yj/ dS.x/: @x

In connection with Lemma 6.2.7 we obtain, by letting " ! 0; Z @G

@ Fq .jx  yj/ dS.x/ D 1: @x

(6.2.28)

(ii) Case y 2 @G : Again, by Theorem 6.2.2 we obtain, for " > 0, Z 

x2G jxyjD"

@ Fq .jx  yj/ dS.x/ D @x

Z x2@G

@ Fq .jx  yj/ dS.x/: @x (6.2.29)

By letting " ! 0 we now find in case of a continuously differentiable surface @G ; Z @ 1 (6.2.30) Fq .jx  yj/ dS.x/ D  : @ 2 x @G (iii) Case y … G : The second Green theorem (Theorem 6.2.2) now yields Z

Z G

x Fq .jx  yj/ dV .x/ D „ ƒ‚ … D0

@G

@ Fq .jx  yj/ dS.x/: @x

(6.2.31)

Summarizing all our results we obtain the following identity from (6.2.28), (6.2.30), and (6.2.31).

6.2 Integral Theorems for the Laplace Operator

295

Lemma 6.2.8. Let G  Rq be a regular region with continuously differentiable boundary @G . Then, 8 ˆ ˆ1
0, there exists an integer n D n."/ such that ˇ ˇ Z n N.qIk/ X X ˇ ˇ k ˇ sup ˇF ./  ˇn .q/ F . /Yk;j .qI / dS. / Yk;j .qI /ˇˇ  ": Sq1 2Sq1 kD0 j D1 „ ƒ‚ … Ddk;j

(6.5.56) This proves Corollary 6.5.12.

t u

Next, we are interested in the closure and the completeness in the Hilbert space L2 .Sq1 /. At first, we show the following lemma. Lemma 6.5.13. The system fYn;j .qI /gn2N0;j D1;:::;N.qIn/ is closed in C.Sq1 / with respect to k  kL2 .Sq1 / . Proof. Indeed, Lemma 6.5.13 immediately follows frompCorollary 6.5.12 by use of t the norm estimate for F 2 C.Sq1 /, i.e., kF kL2 .Sq1 /  kSq1 k kF kC.Sq1 / . u Finally, we arrive at the following result which is the q-dimensional analog of Corollary 4.4.12. Theorem 6.5.14. The system fYn;j .qI /gn2N0;j D1;:::;N.q;n/ of spherical harmonics is closed in the space L2 .Sq1 / with respect to k  kL2 .Sq1 / . Proof. C.Sq1 / is dense in L2 .Sq1 /, i.e., for every " > 0 and every F 2 L2 .Sq1 /, there exists a function G 2 C.Sq1 / with kF  GkL2 .Sq1 /  ". The continuous

6.5 Spherical Harmonics of Dimension q

321

function G 2 C.Sq1 / admits an arbitrarily close approximation by finite linear combinations of spherical harmonics. Therefore, the proof of the closure is clear. t u From constructive approximation (see Theorem 3.0.7 or, e.g., Davis 1963) we know the equivalence of closure and completeness within the Hilbert space L2 .Sq1 /. Therefore, we can finally state the following: Corollary 6.5.15. The orthogonal expansion of F 2 L2 .Sq1 / converges in the L2 .Sq1 /-norm to F , i.e., n N.qIk/ X X Z lim F 

n!1

F . /Yk;j .qI / dS. / Yk;j .qI / q1

S

kD0 j D1

D 0: (6.5.57)

L2 .Sq1 /

Associated Legendre Functions By virtue of the Funk–Hecke formula (i.e., Corollary 6.5.6) any spherical harmonic Yn .qI / 2 Harmn .Sq1 / given by Z Yn .qI / D Cn;l .q/

Sq2

 n   "q C i   .q1/ Yl .q  1I .q1/ / dS.q2/ . .q1/ / (6.5.58)

can be represented in the form Yn .qI / D Pn;l .qI t/ Yl .q  1I .q1/ /; where  D t "q C

(6.5.59)

p 1  t 2 .q1/ and Pn;l .qI / is given by Z

Pn;l .qI t/ D Cn;l .q/ kS

q3

k

1 1

n  p q4 t C i 1  t 2 s Pl .q  1I s/.1  s 2 / 2 ds: (6.5.60)

Note that l 2 f0; 1; : : : ; ng is arbitrary and the constant Cn;l is determined later (see (6.5.73)) such that the functions Pn;l .qI / are normalized. From Rodrigues’ formula (6.4.50) we are able to deduce that ! q2 n 2 2 (6.5.61) l .l C q2 2 / Z 1 nl p q4 l  .1  t 2 / 2 t C i 1  t 2s .1  s 2 /lC 2 ds:

il lŠ Pn;l .qI t/ DCn;l .q/ l 2

1

6 Spherical Harmonics in Rq

322

From the Laplace representation of Pnl .q C 2l; / given by

Pnl .q C 2l; t/ D

kSqC2l3 k kSqC2l2 k

Z

1 1

 nl p q4 t C i 1  t 2s .1  s 2 /lC 2 ds

(6.5.62)

we get ! q1 l il lŠ n 2 2  .1  t 2 / 2 Pnl .q C 2lI t/:  Pn;l .qI t/ D Cn;l .q/ l 2 l l C q1 2

(6.5.63)

By iterated use of (6.4.55), i.e., . q2 / l .l/ N.qI n/ 2 Pn .qI t/; Pnl .q C 2lI t/ D N.q C 2lI n  l/ .l C q2 /

(6.5.64)

we find Pn;l .qI t/ D Cn;l i

l

2

q1 2

l

. q1 / 2

.1  t 2 / 2 Pn.l/ .qI t/:

(6.5.65)

Until now, the coefficients Cn;l .q/ are arbitrary. Next, they will be determined in such a way that Z 1  2 q3 Pn;l .qI t/ .1  t 2 / 2 dt D 1: (6.5.66) 1

Using

! q1 2 2 il lŠ n ; Bn;l .q/ D Cn;l .q/ l 2 l .l C q1 / 2

(6.5.67)

we have l

Pn;l .qI t/ D Bn;l .q/.1  t 2 / 2 Pnl .q C 2lI t/

(6.5.68)

such that Z

1

1

 2 q3 2 Pn;l .qI t/ .1  t 2 / 2 dt DBn;l .q/

Z

1 1



2 qC2l3 Pn;l .q C 2lI t/ .1  t 2 / 2 dt

1 kSqC2l1 k qC2l2 kS k N.q C 2lI n  l/   1  2 l C q1 2 2  .q/  DBn;l : (6.5.69) l C q2 N.q C 2lI n  l/

2 DBn;l .q/

6.5 Spherical Harmonics of Dimension q

323

In other words, the coefficients Cn;l .q/ satisfying (6.5.66) are related to Bn;l .q/ given by s kSqC2l2 k Bn;l .q/ D N.q C 2lI n  l/ (6.5.70) kSqC2l1 k as follows: Cn;l .q/ D Bn;l .q/

l

il

2  lŠ nl

 lC 2

q1 2

q1 2

 :

(6.5.71)

Explicitly written out we have s Pn;l .qI t/ D

kSqC2l2 k l N.q C 2lI n  l/ .1  t 2 / 2 Pnl .q C 2lI t/: (6.5.72) kSqC2l1 k

Hence, under this normalization, we finally obtain Z

1 1

Pn;l .qI t/ Pm;l .qI t/ .1  t 2 /

q3 2

dt D ın;m :

(6.5.73)

Definition 6.5.16. The function Pn;l .qI / given by (6.5.72) is called (normalized) associated Legendre function of degree n and order l of dimension q. The system ˚  Pn;l .qI t/ Yl;j .q  1I .q1/ / lD0;:::nIj D1;:::;N.q1Il/

(6.5.74)

q1 q1 forms an L2 .S /, provided that the set n .S ˚  /-orthonormal system in Harm 2 q2 Yl;j .q  1; / lD0;:::;nIj D1;:::;N.q1;l/ forms an L .S /-orthonormal system.

Setting  D t "q C

p 1  t 2 .q1/ ;

we find   D ts C

D s"q C

p 1  s 2 .q1/ ;

p p 1  t 2 1  s 2 .q1/  .q1/ :

(6.5.75)

(6.5.76)

Therefore, the addition theorem of spherical harmonics (Theorem 6.5.2) gives us n X

Pn;l .qI t/Pn;l .qI s/

lD0

D

 N.q  1; l/  Pl q  1I .q1/  .q1/ q2 kS k

 p p N.qI n/  2 1  s2  qI ts C : (6.5.77) P 1  t 

n .q1/ .q1/ kSq1 k

6 Spherical Harmonics in Rq

324

By integration of (6.5.77) with respect to .q1/ 2 Sq1 we finally obtain Z

  p p q4 Pn qI ts C 1  t 2 1  s 2 u .1  u2 / 2 du D Pn .qI t/ Pn .qI s/:

1

1

(6.5.78) Further details can be found in M¨uller (1998).

Pointwise Expansion Theorem We start our considerations by remembering Corollary 3.4.10 which reads as follows when written in terms of Legendre polynomials of dimension q: 1 X

N.q; n/r n Pn .qI t/ D

nD0

1  r2 q

.1 C r 2  2rt/ 2

;

(6.5.79)

where q  3, jrj < 1, and t 2 Œ1; 1. From (6.5.79) it follows that Z

1

1

.1  r 2 /.1  t 2 /

q3 2

.1 C r 2  2rt/

q 2

Z

1

dt D

.1  t 2 /

q3 2

dt D

1

kSq1 k kSq2 k

(6.5.80)

for all r 2 R with 0  r < 1. This leads to Lemma 6.5.17. Let F be of class C.Œ1; 1/. Then, Z

.1  r 2 /F .t/.1  t 2 /

1

lim

r!1 1

.1 C r 2  2rt/

q3 2

q 2

kSq1 k : kSq2 k

(6.5.81)

dt D 0:

(6.5.82)

dt D F .1/

Proof. We have to verify that Z lim

1

r!1 1

.1  r 2 /.F .t/  F .1// .1 C

r2

 2rt/

q 2

.1  t 2 /

q3 2

In accordance with our assumption, there exists a positive function M with lims!0 M.s/ D 0, such that sup jF .t/  F .1/j D M.s/:

1st 1

(6.5.83)

Moreover, there exists a positive constant C such that sup jF .t/  F .1/j D C: t 2Œ1;1

(6.5.84)

6.5 Spherical Harmonics of Dimension q

325

Now, we have, for 1  t  1  s and r > 0, 1 C r 2  2rt D .1  r/2 C 2r.1  t/  2rs: In addition, we see that, for 1  t  1  s and 1  r2 .1 C

r2

 2rt/



q 2

1  r2 .2rs/

q 2

D

1 2

< r < 1,

.1 C r/.1  r/ q 2

.2r/ s

(6.5.85)

q 2

2

.1  r/ q

s2

:

(6.5.86)

We divide the interval Œ1; 1 into Œ1; 1  s and .1  s; 1. Then, we obtain Z

1s

.1  r 2 /.F .t/  F .1// q

.1 C r 2  2rt/ 2

1

.1  t 2 /

q3 2

 dt D O

kSq1 k s kSq1 k

 (6.5.87)

and Z

1

.1  r 2 /.F .t/  F .1// q

1s

.1 C r 2  2rt/ 2

2

.1t /

q3 2

 Z dt D O M.s/

1

1

.1  r 2 /.1  t 2 /

q3 2 q

 dt :

.1 C r 2  2rt/ 2 (6.5.88)

1

With s D .1  r/ q , Lemma 6.5.17 follows from the last estimate.

t u

Next, we are able to formulate the Poisson integral formula (like in Theorem 4.4.3). Theorem 6.5.18 (Poisson Integral Formula). Suppose that G is of class C.Sq1 /. Then, we have lim

r!1

1 kSq1 k

Z

.1  r 2 /G. / Sq1

q

.1 C r 2  2r   / 2

dS. / D G./

(6.5.89)

uniformly with respect to  2 Sq1 . Proof. Since Sq1 is compact, the continuity of G implies the existence of a positive function M , such that jG./  G. /j  M. / (6.5.90) holds for 1      1. For  D "q , we let Z F .t/ D Sq2

Then, we have

  p G t "q C 1  t 2 .q1/ dS.q2/ . .q1/ /: F .1/ D Sq2 G."q /:

(6.5.91)

(6.5.92)

From (6.5.90) it follows, for 1   t  1, that jF .1/  F .t/j  kSq2 k M. /:

(6.5.93)

6 Spherical Harmonics in Rq

326

The integral in (6.5.89) can be written in the form Z

1 kSq1 k

1

.1  r 2 /F .t/.1  t 2 / .1  r 2  2rt/

1

q3 2

dt:

q 2

(6.5.94)

In connection with Lemma 6.5.17 this yields, for  D "q , Z

1

lim

r!1 kSq1 k

.1  r 2 /G. / Sq1

Z

1

D lim

kSq1 k

r!1

q

.1 C r 2  2r  / 2 1

dS. /

.1  r 2 /F .t/.1  t 2 / q

.1 C r 2  2rt/ 2

1

q3 2

dt

kSq1 k F .1/ D G."q /: kSq1 k kSq2 k 1

D

(6.5.95)

Since every point on Sq1 can be transformed into "q , the argumentation is valid for all  2 Sq1 . It follows from (6.5.93) that the limit relation holds uniformly. t u Combining Theorem 6.5.18 and (6.5.79), we obtain the analog of Theorem 4.4.4. Theorem 6.5.19 (Expansion Theorem). Suppose that G is of class C.Sq1 /. Then,

G./ D lim

r!1

1 X

N.q;n/ Z

rn

X

j D1

nD0

Sq1

G. /Yn;j .qI / dS. / Ynj .qI /

(6.5.96)

holds uniformly with respect to  2 Sq1 . As a result of our considerations, we are able to solve the Dirichlet problem corresponding to continuous boundary values on the unit sphere. Theorem 6.5.20 (Dirichlet’s Problem). Let F be of class C.Sq1 /. Then, the series N.q;n/ Z 1 X X rn F . /Yn;j .qI / dS. / Ynj .qI / (6.5.97) nD0

j D1

Sq1

q

converges for all r  r0 < 1 absolutely and uniformly. The function U W B1 ! R given by U.x/ D U.r/ D

1 X nD0

N.q;n/ Z

rn

X

j D1

Sq1

F . /Yn;j .qI / dS. / Ynj .qI /

(6.5.98)

6.5 Spherical Harmonics of Dimension q

327

represents the uniquely determined solution of the Dirichlet problem  q  q U 2 C B1 \ C.2/ B1

(6.5.99)

q

with U D 0 in B1 satisfying the boundary condition lim U.r/ D F ./;  2 Sq1 ;

(6.5.100)

r!1

uniformly with respect to  2 Sq1 . Next, we are concerned with the pointwise representation of a function by its convergent orthogonal (Fourier) spherical harmonic expansion. Roughly speaking, the result is that the continuity of a function together with the convergence of its Fourier series expansion at a point on the unit sphere Sq1 assures the equality of the functional value and the value of its Fourier expansion at the point under consideration (cf. M¨uller 1998). Theorem 6.5.21 (Pointwise Expansion Theorem). Suppose F is continuous in the point 0 2 Sq1 and uniformly bounded on Sq1 . Assume that the sequence fSn ./gn2N0 with Sn ./ D

n N.qIk/ X X Z kD0 j D1

Sq1

F . /Yk;j .qI / dS. / Yk;j .qI /

(6.5.101)

converges for all  2 Sq1 . Then, F .0 / D lim Sn .0 /

(6.5.102)

n!1

D

1 N.qIk/ X X Z kD0 j D1

Sq1

F . / Yk;j .qI / dS. / Yk;j .qI 0 /:

Proof. From Theorem 6.5.19 it is known that

lim

r!1

1 N.q;k/ X X kD0 j D1

Z r

k Sq1

F . / Yk;j .qI / dS. / Yk;j .qI 0 / D F .0 /:

(6.5.103)

We rewrite the series expansion (6.5.103) as 1 X nD1

.r n .Sn .0 /  Sn1 .0 /// C S0 .0 / D .1  r/

1 X nD0

r n Sn .0 /:

(6.5.104)

6 Spherical Harmonics in Rq

328

Of course, we are able to use the knowledge of the limits lim .1  r/

r!1

1 X

r k Sk .0 / D F .0 /

(6.5.105)

kD0

and lim Sn .0 / D S1 .0 /:

n!1

(6.5.106)

For every " > 0, we have a number n0 ."/ such that S1 .0 /  "  Sn .0 /  S1 .0 / C "

(6.5.107)

holds true for all n > n0 ."/. This leads to the estimate 1 X r n0 r n0 .S1 .0 /  "/  .S1 .0 / C "/; r n Sn .0 /  1r 1r nDn

(6.5.108)

0

such that the limit r ! 1 gives S1 .0 /  "  F .0 /  S1 .0 / C ":

(6.5.109) t u

This proves the assertion of Theorem 6.5.21.

Asymptotic Relations for the Spherical Harmonic Coefficients Now, we discuss the pointwise representation theorem by means of spherical harmonics under the assumption of sufficient differentiability (“smoothness”) imposed on the function. If F is assumed to be continuously differentiable on Sq1 , then the first Green surface theorem (Theorem 6.2.1) shows, for all  2 Sq1 and Yn .qI / 2 Harmn .Sq1 / of the form Yn .qI / D

N.qI n/ kSq1 k

Z Sq1

F . /Pn .qI   / dS. /;

(6.5.110)

that ˇ ˇ Z ˇ N.qI n/ ˇ 1   ˇ r Pn .qI   /  r F . / dS. /ˇˇ : jYn .qI /j D ˇ q1 kS k n.n C q  2/ Sq1 (6.5.111)

6.5 Spherical Harmonics of Dimension q

329

The Cauchy–Schwarz inequality yields the estimate jYn .qI /j 

Z  12 ˇ ˇ2 N.qI n/ 1 ˇ  ˇ P .qI  

/ dS. / ˇr ˇ n kSq1 k n.n C q  2/ Sq1

Z  12 ˇ ˇ2 ˇ  ˇ  : (6.5.112) ˇr F . /ˇ dS. / Sq1

Now, the second Green surface theorem (Theorem 6.2.2) gives us Z Sq1

Z ˇ ˇ2 ˇ  ˇ ˇr Pn .qI   /ˇ dS. / D 

Sq1

Pn .qI   / Pn .qI   / dS. / Z

D n.n C q  2/ Sq1

D

.Pn .qI   //2 dS. /

kSq1 k n.n C q  2/: N.qI n/

(6.5.113)

From (6.5.112), in connection with (6.5.113), we obtain, for n > 0; kYn .qI /kC.Sq1 / D sup jYn .qI /j 2Sq1

 

N.qI n/ n.n C q  2/kSq1 k

 12 Z Sq1

 12 ˇ ˇ2 ˇ  ˇ : ˇr F . /ˇ dS. /

(6.5.114)

From the theory of spherical harmonics (see (6.4.26) and (6.4.27), and in particu lar (6.4.29)) we know that N.qI n/ D O nq2 for n ! 1 such that  q4  kYn .qI /kC.Sq1 / D O n 2 :

(6.5.115)

Assuming that F is twice continuously differentiable on Sq1 , we find, for n > 0, N.qI n/ kSq1 k D

Z Sq1

F . /Pn .qI   / dS. /

N.qI n/ 2 kSq1 k .n.n C q  2// 1

Z Sq1

(6.5.116)

    F . /  Pn .qI   / dS. /

such that Yn .qI / of the form (6.5.110) satisfies kYn .qI /kC.Sq1 / 

1 n.n C q  2/

(6.5.117) 

N.qI n/ kSq1 k

 12 Z Sq1

j F . /j2 dS. /

 12 :

6 Spherical Harmonics in Rq

330

This leads to the relation  q6  kYn .qI /kC.Sq1 / D O n 2

(6.5.118)

for n ! 1. Continuing in this way, we finally arrive at Lemma 6.5.22. If F is of class C.k/ .Sq1 /, then ˇ ˇ Z  q2k2  ˇ ˇ N.qI n/ ˇ ˇ F . / P .qI  

/ dS. / D O n 2 ; n ˇ ˇ kSq1 k q1 S

n ! 1: (6.5.119)

If k > q2 , then we are able to deduce from Lemma 6.5.22 that the series Z 1 X N.qI n/ nD0

kSq1 k

Sq1

F . / Pn .qI   / dS. /

(6.5.120)

is absolutely and uniformly convergent. Therefore, Theorem 6.5.21 in connection with the addition theorem (Theorem 6.5.2) shows that F ./ D

Z 1 X N.qI n/ nD0

D

kSq1 k

Sq1

1 N.qIn/ X X Z nD0 j D1

Sq1

F . / Pn .qI   / dS. /

(6.5.121)

F . / Yn;j .qI / dS. / Yn;j .qI /

holds true for all  2 Sq1 whenever fYn;j .qI /gn2N0;j D1;:::;N.qIn/ is an L2 .Sq1 /orthonormal system of spherical harmonics.

6.6 Integral Theorems for the Helmholtz–Beltrami Operator Next, we consider the eigenvalue problem . C /m Y D 0; Y 2 C.2m/ .Sq1 /; m 2 N:

(6.6.1)

Every Kn .qI / 2 Harmn .Rq / can be written in the form Kn .qI x/ D r n Yn .qI /, x D r,  2 Sq1 . In addition, an easy calculation shows   1q d q1 d n r r r D n.n C q  2/r n2 : (6.6.2) dr dr Thus, observing the representation (6.1.21) of the Laplace operator, we obtain   x Kn .qI x/ D 0 D r n2 n.n C q  2/ Yn .qI / C  Yn .qI / ;

(6.6.3)

6.6 Integral Theorems for the Helmholtz–Beltrami Operator

i.e.,

331

    C n.n C q  2/ Yn .qI / D 0

for all Yn 2 Harmn .S

q1

(6.6.4)

/, n 2 N0 .

Remark 6.6.1. It is noteworthy that the sequence fn.n C q  2/gn2N0 is monotonically increasing and its only “accumulation point” is at infinity. The differential equation (6.6.4) motivates the introduction of the Sq1 -sphere function, q  3, for the Helmholtz–Beltrami operator  C ;  2 R (cf. Freeden 2011). Note that the case q D 2 leads to the classical Fourier theory (see, e.g., Benedetto 1996; Butzer and Nessel 1971). Therefore, we always assume q  3, here. Moreover, for the case  D 0, the reader is referred to Definition 6.3.1. Definition 6.6.2. Let  be a real number. A function G. C I / W .; / 7! G. C I   /;

1    < 1;

(6.6.5)

is called the Green function for the Helmholtz–Beltrami operator  C  on the unit sphere Sq1 (briefly called, Sq1 -sphere function for  C ) if it satisfies the following properties: (i) For each fixed  2 Sq1 ; 7! G. C I   / is twice continuously differentiable with respect to 2 Sq1 , 1    6D 0, with . C /G. C I   / D 

X

X

N.qIn/

Yn;j .qI /Yn;j .qI /; (6.6.6)

n2K.;q/ j D1

where K.; q/ D fn 2 N0 W n.n C q  2/ D g. (ii) In the neighborhood of  2 Sq1 the following estimates are valid for q D 3: 1 ln.1    / D O.1/; kS2 k

(6.6.7)

1 r  ln.1    / D O.1/ kS2 k

(6.6.8)

G. C I   /  r  G. C I   /  and, for q  4,

3q

3q 2 2 1 .1    / 2 G. C I   / C q2 q  3 kS k 8 0; Z G. C I   / Yn .qI / dS. / (6.6.17) Sq1



Z



D Sq1

 N.qIn/ X 1 1  Y .qI /Y .qI

/ Yn .qI / dS. / n;j n;j kSq1 k n.n C q  2/ j D1

for all spherical harmonics Yn .qI / of degree n > 0. Therefore, the integral equation (6.6.16) has a solution which is uniquely determined by the conditions Z Sq1

G. C I   / Yn;j .qI / dS. / D 0

(6.6.18)

for j D 1; : : : ; N.qI n/. Finally, the analog to the Poisson differential equation in potential theory should be mentioned briefly. Lemma 6.6.4. Let F be a bounded function on the sphere Sq1 which satisfies a Lipschitz-condition in the neighborhood of the point  2 Sq1 . Then, Z

G. C I   / F . / dS. /

U./ D

(6.6.19)

Sq1

is twice continuously differentiable at  2 Sq1 with .

C / U./ D F ./ 

X

X

Z

N.qIn/

n2K.;q/ j D1

Yn;j .qI /

Sq1

F . / Yn;j .qI / dS. /: (6.6.20)

6 Spherical Harmonics in Rq

334

Indeed, the proof can be given by quite similar conclusions as known from potential theory (see, e.g., Kellogg 1929; Freeden and Gerhards 2012). The details are omitted here. Now, our purpose is to formulate a counterpart to the third Green theorem (Theorem 6.2.11) on the unit sphere Sq1 for the Beltrami operator  (see Freeden 1980a; Reuter 1982) and to derive some extensions to the operator  C ,  2 R. Suppose that F is a twice continuously differentiable function on Sq1 , i.e., F 2 C.2/ .Sq1 /. Then, for each sufficiently small " > 0 and for each number  2 R, the second Green surface theorem (Theorem 6.2.2) yields, for a spherical cap C.; "/ of radius " around  2 Sq1 , i.e., C.; "/ D f 2 Sq1 W 1  "     1g;

(6.6.21)

the identity Z C.;"/

Z D

 G. C I   /. C /F . /

(6.6.22)

  F . /. C /G. C I   / dS.q1/ . /

 @ F . / G. C I   / @

C.;"/  F . /

  @ G  C I  

@

 dS.q2/ . /;

where dS.q2/ denotes the surface element in Rq1 , while  is the (unit) vector normal to @C.; "/ D f 2 Sq1 W 1    D "g and tangential on Sq1 and directed into the exterior of the spherical cap C.; "/. Inserting the differential equations of the Sq1 -sphere function, we obtain Z C.;"/

  F . / . C /G. C I   / dS.q1/ . / X

D

(6.6.23)

N.qIn/ Z

X

n2K.;q/ j D1

F . / Yn;j .qI / dS.q1/ . / Yn;j .qI /: C.;"/

Observing the characteristic singularity of the Sq1 -sphere function, we are able to prove by analogous conclusions as known in potential theory, for " ! 0; Z

  @ G  C I  

F . / dS.q2/ . / D o.1/; @

@C.;"/

and, for " ! 0,

(6.6.24)

6.6 Integral Theorems for the Helmholtz–Beltrami Operator

Z F . / @C.;"/

 @   G  C I   dS.q2/ . / D F ./ C o.1/: @

335

(6.6.25)

Summarizing our results we obtain the following analog to the third Green theorem for the unit sphere Sq1 . Theorem 6.6.5 (Integral Formula for the Operator  C). If  2 R,  2 Sq1 , and F 2 C.2/ .Sq1 /, then F ./ D

X

N.qIn/ Z

X

n2K.;q/ j D1

Z

C Sq1

Sq1

F . / Yn;j .qI / dS. / Yn;j .qI /

   G  C I   . C /F . / dS. /;

(6.6.26)

where K.; q/ D fn 2 N0 W n.n C q  2/ D g (see also Remark 6.6.3). The integral formula (Theorem 6.6.5) may serve as a point of departure for purposes of numerical integration on the sphere (see, e.g., Freeden 1980a, 1981). It provides the explicit knowledge of a remainder term involving the Helmholtz–Beltrami operator  C . Finally, it should be mentioned that Theorem 6.6.5 can be formulated for iterated operators . C /m ,  2 R. For that purpose we introduce   Definition 6.6.6. Let G . C/m I   , m D 1; 2; : : :, be defined by the spherical convolution for m D 2; 3; : : :       G . C /m I   D G . C /m1 I   G. C I  / ./ (6.6.27) Z   D G . C /m1 I   G. C I  / dS. /; Sq1

and

    (6.6.28) G . C /1 I   D G  C I   :    m q1  m Then, G . C / I  is called the S -sphere function for . C / ;  2 R. In analogy to techniques known in potential theory (see, e.g., Helms 1969) it can be proved that   G . C /m ;  

8   q1 , G . C /m ;   is continuous on the whole sphere Sq1 . 2 Furthermore, for m > q1 , the bilinear expansion 2 X n2N0 nK.;q/

N.qIn/ X 1 Yn;j .qI / Yn;j .qI / .  n.n C q  2//m j D1

(6.6.30)

is absolutely and uniformly convergent both in  and as well as uniformly in  and together. Therefore, in connection with the addition theorem for spherical harmonics, we have the following result. Lemma 6.6.7. If m >

q1 2 ,

we have X

  G . C /m I   D

n2N0 nK.;q/

1 N.qI n/ Pn .qI   /: m .  n.n C q  2// kSq1 k (6.6.31)

Observing the differential equation     . C / G . C /m I   D G . C /m1 ;   ;

(6.6.32)

1    < 1, we obtain by successive integration by parts the following extension of Theorem 6.6.5. Theorem 6.6.8 (Integral Formula for the Operator . C /m ). Assume that F is of class C.2m/ .Sq1 / and  2 Sq1 . Then, N.qIn/ Z

X

F ./ D

X

n2K.;q/ j D1

Z

C Sq1

Sq1

F . / Yn;j .qI / dS. / Yn;j .qI /

    G . C /m I   . C /m F . / dS. /:

(6.6.33)

An immediate consequence of Theorem 6.6.8 is the following corollary. Corollary 6.6.9. Under the assumptions of Theorem 6.6.8 we have F ./ D

X n2K.;q/

N.qI n/ kSq1 k

Z

C Sq1

Z Sq1

F . / Pn .qI   / dS. /

(6.6.34)

   . C /m G . C /2m I   . C /m F . / dS. /:

The integral formula (Theorem 6.6.5) enables us to justify that the spherical harmonics are the only everywhere on the unit sphere Sq1 ; twice continuously differentiable, eigenfunctions of the Beltrami differential operator  .

6.7 Exercises

337

Lemma 6.6.10. Let F be of class C.2/ .Sq1 / satisfying . C /F ./ D 0;

 2 Sq1 :

(6.6.35)

(i) If  … Spect .Sq1 /, i.e.,  ¤ n.n C q  2/, for all n 2 N0 , then F D 0. (ii) If  2 Spect .Sq1 /, i.e.,  D n.n C q  2/, n 2 N0 , then F is a member of class Harmn .Sq1 /. Summarizing our results about harmonics, we are therefore led to the following conclusions: • The functions x 7! Hn .qI x/ D r n Yn .qI /, x 2 Rq , are polynomials in Cartesian coordinates which satisfy the Laplace equation Hn .qI / D 0 in Rq and are homogeneous of degree n. Conversely, every homogeneous harmonic polynomial of degree n is a spherical harmonic of degree n and dimension q when restricted to the unit sphere Sq1 . • The Legendre polynomial Pn .qI / is the only everywhere on the interval Œ1; 1; twice continuously differentiable, eigenfunction of the Legendre differential equation 

 .1  t/ „

2

d dt

2

 d  .q  1/t Cn.n C q  2/ Pn .qI t/ D 0; dt ƒ‚ …

t 2 Œ1; 1;

DLt

(6.6.36) n 2 N0 , which in t D 1 satisfy Pn .qI 1/ D 1. • The spherical harmonics Yn .qI / of degree n and dimension q are the everywhere on the unit sphere Sq1 ; twice continuously differentiable, eigenfunctions of the Beltrami (differential) equation     C n.n C q  2/ Yn .qI / D 0

(6.6.37)

corresponding to the eigenvalues n.n C q  2/, n 2 N0 .

6.7 Exercises (Cartesian Generation of Spherical Harmonics, Best Approximations) Next, we come to some exercises involving spherical harmonics of dimension q. At first, we develop an algorithm for the exact Cartesian generation of spherical harmonics. Then, best approximate integration on the sphere Sq1 is discussed.

6 Spherical Harmonics in Rq

338

Exact Cartesian Generation of Spherical Harmonic Bases The generation of spherical harmonics has a long history in geosciences. Traditionally, the basis involving associated Legendre functions (see Examples 4.3.32 and 4.3.33) are chosen for numerical implementation. In what follows we present a different approach (cf. Freeden and Reuter 1984, 1990). It gives an exact generation of spherical harmonics in Cartesian coordinates that can be implemented in symbolic computation. Exercise 6.7.1. Prove the following statement: Let An and An1 , respectively, be members of class Homn .Rq1 / and Homn1 .Rq1 /. For j D 0; : : : ; n  2 set recursively Anj 2 .x.q1/ / D 

1 x Anj .x.q1/ /: .j C 1/.j C 2/ .q1/

Then, Hn .qI x.q/ / D

n X

xqj Anj .x.q1/ /

(6.7.1)

(6.7.2)

j D0

constitutes a homogeneous harmonic polynomial of degree n in dimension q, i.e., Hn 2 Harmn .Rq /. Exercise 6.7.2. Let An;1 .x.q1/ /; : : : ; An;M.q1In/ .x.q1/ / be an ordered set of the M.q  1I n/ monomials of degree n, i.e., ˛

.q1/ ; An;l .x.q1/ / D x.q1/

q1

where Œ˛.q1/  D n, ˛.q1/ 2 N0

l D 1; : : : ; M.q  1I n/;

(6.7.3)

and

M.q  1I n/ D

  nCq2 : n

(6.7.4)

Verify that the union n

.n/

[n

o

Hn;l

lD1;:::;M.q1In/

.n1/

Hn;l

o lD1;:::;M.q1In1/

(6.7.5)

is a basis of Harmn .Rq /, where bn=2c .n1/

Hn;l

.x.q/ / D

X

xq2k1 An.2k1/;l .x.q1/ /;

l D 1; : : : ; M.q  1I n  1/;

kD1

(6.7.6)

6.7 Exercises

339 bn=2c

.n/

Hn;l .x.q/ / D

X

xq2k An2k;l .x.q1/ /;

l D 1; : : : ; M.q  1I n/:

(6.7.7)

kD0

For l D 1; : : : ; M.q  1I n  1/ in the case of (6.7.6) or for l D 1; : : : ; M.q  1I n/ in the case of (6.7.7) Anj 2;l .x.q1/ / D 

1 Anj;l .x.q1/ /: x .j C 1/.j C 2/ .q1/

(6.7.8)

Note that the polynomials Anj;l which result from (6.7.8) and are used in the sums in (6.7.6) and (6.7.7) are not necessarily monomials for j > 1. .n/

.n1/

Exercise 6.7.3. Let fHn;k gkD1;:::;M.q1In/ [ fHn;l glD1;:::;M.q1In1/ be the members of Harmn .Rq / as known from Exercise 6.7.2. Show that D E .n1/ .n/ Hn;l ; Hn;k

Homn .Rq /

D0

(6.7.9)

for all l D 1; : : : ; M.q  1I n  1/ and all k D 1; : : : ; M.q  1I n/. Exercise 6.7.4. Let the binary digits bin.k/ given by ( bin.k/ D

0 ;

k even

1 ;

k odd:

(6.7.10)

Write an algorithm by use of bin.k/ for the following splitting of the set of all multiindices of degree n and dimension q: q

M .qI n/ D f˛ 2 N0 W Œ˛ D ng

(6.7.11)

M0 .qI n/ D f˛ 2 M .qI n/ W ˛q eveng

(6.7.12)

M1 .qI n/ D f˛ 2 M .qI n/ W ˛q oddg

(6.7.13)

M0;0 .qI n/ D f˛ 2 M0 .qI n/ W ˛q1 eveng

(6.7.14)

M0;1 .qI n/ D f˛ 2 M0 .qI n/ W ˛q1 oddg

(6.7.15)

M1;0 .qI n/ D f˛ 2 M1 .qI n/ W ˛q1 eveng

(6.7.16)

M1;1 .qI n/ D f˛ 2 M1 .qI n/ W ˛q1 oddg:

(6.7.17)

Note that Mbin.j / .qI n/ contains the same patterns in the components ˛2 ; : : : ; ˛q as in Mbin.j /.qI n  1/, but the multi-indices differ in ˛1 . As usual for sets, #Mbin.j / .qI n/ indicates the number of multi-indices contained in the set Mbin.j / .qI n/.

6 Spherical Harmonics in Rq

340

Exercise 6.7.5. Assume that a basis B.qI n/ of Harmn .Rq / is split into 2q1 subsets Bj .qI n/, j D 0; : : : ; 2q1  1 as follows: ( Bj .qI n/ D Hn D

X

) C˛ x W ˛ 2 Mbin.j /.qI n/ ˛

 B.qI n/:

(6.7.18)

˛

Prove that Bj .qI n/ ? Bk .qI n/;

j 6D k;

(6.7.19)

q

in the Homn .R /-sense. In other words, B.qI n/ is split into 2q1 subsets Bk .qI n/, k D 0; : : : ; 2q1  1, such that 2q1 [1

Bk .qI n/ D B.qI n/;

(6.7.20)

kD0

Bk .qI n/ \ Bj .qI n/ D ;

and Bk .qI n/ ? Bj .qI n/;

k 6D j:

(6.7.21)

Exercise 6.7.6. Show that Algorithm 6.7.7 provides a basis of the space Harmn .Rq /. Algorithm 6.7.7. Do for l D n  1 to n generate M .q  1I l/ do for k D 0 to 2q2  1 generate Mbin.k/ .q  1I l/ enddo do for k D 0 to 2q2  1 mk D #Mbin.k/ .q  2I l/ .l/ Bk .qI n/ D ; do for j D 1 to mk Hn .x.q/ / D Al;j .x.q1/ /  xqnl do for i D n  l to n  2 step 2 1 Ani 2;j .x.q1/ / D  .i C1/.i C2/ x.q1/ Ani;j .x.q1/ / Hn .x.q/ / D Hn .x.q/ / C Ani 2;j .x.q1/ /  xqi C2 enddo .l/ .l/ Bk .qI n/ D Bk .qI n/ [ fHn g enddo enddo enddo Exercise 6.7.8. Implement in symbolic computation the Gram-Schmidt orthonormalizing process to get an Homn .Rq /-orthonormal basis corresponding to the basis of Exercise 6.7.6.

6.7 Exercises

341

Remark 6.7.9. A discussion of the amount of the computational work that is needed for our method of Homn .Rq /-orthonormal bases can be found in Freeden and Reuter (1984, 1990). Remark 6.7.10. Finally, we add a test example (q D 3, n D 5) for a better understanding. We start with the sets of multi-indices M .2I 4/ D f.4; 0/; .3; 1/; .2; 2/; .1; 3/; .0; 4/g;

(6.7.22)

M .2I 5/ D f.5; 0/; .4; 1/; .3; 2/; .2; 3/; .1; 4/; .0; 5/g:

(6.7.23)

They are split into M0 .2I 4/ D f.4; 0/; .2; 2/; .0; 4/g;

(6.7.24)

M1 .2I 4/ D f.3; 1/; .1; 3/g;

(6.7.25)

M0 .2I 5/ D f.5; 0/; .3; 2/; .1; 4/g;

(6.7.26)

M1 .2I 5/ D f.4; 1/; .2; 3/; .0; 5/g:

(6.7.27)

From each of these sets the corresponding homogeneous harmonic polynomials are derived. They are shown in Table 6.1, written down in a schematic manner. The first polynomial reads as follows: H1 .x/ D 1  x14 x20 x31  2  x12 x20 x33 C

1 5

 x10 x20 x35 :

(6.7.28)

Additionally, the orthonormalized set of polynomials is tabulated in similar fashion. The first of these reads:

 q 384  K1 .x/ D 1  x14 x20 x31  2  x12 x20 x33 C 15  x10 x20 x35 (6.7.29) 5 : Note that all calculations (except the normalization) can be performed exactly using integers in symbolic computation. The normalization factor (denoted by nf in Table 6.1) in the last step demands a calculation of the square root of a quotient of two integers.

Best Approximate Integration on the Unit Sphere Best approximations are important geoscientific applications if an integral over the sphere has to be determined from only a limited number of functional values at scattered positions. The concept dates back to Sard (1949). The essential tool for our numerical realization in the spherical case is the Green function with respect to the iterated Beltrami operator (see Definition 6.3.5). The optimality is established via the method of Lagrange multipliers (see Freeden 1980a).

6 Spherical Harmonics in Rq

342

Table 6.1 Table of the coefficients corresponding to the example in Remark 6.7.10 Generating set M0 .2I 4/

M1 .2I 4/

M0 .2I 5/

Not orthonormalized C ˛1 ˛2 1 4 0 2 2 0 1=5 0 0 1 2 2 1=3 0 2 1=3 2 0 1=15 0 0

˛3 1 3 5 1 3 3 5

1 2 1=5

0 0 0

4 2 0

1 3 5

1 1 1 1

3 1 1 1

1 1 3 1

1 3 1 3

1 10 5 1 3 1 1

5 3 1 3 1 3 1

0 0 0 2 2 0 0

0 2 4 0 2 2 4

1 6 1

1 1 1

4 2 0

0 2 4

Orthonormalized C ˛1 ˛2 1 4 0 2 2 0 1=5 0 0 1 2 2 1=3 0 2 1=12 2 0 1=24 0 0 1=8 4 0 1 0 4 3=2 0 2 1=8 0 0 1=8 4 0 1=4 2 0 3=2 2 2 1 3 1 1 1 1 1 1 3 1=2 1 1 1=2 3 1 1 5 0 10 3 0 5 1 0 1 3 2 3 1 2 1=4 3 0 3=8 1 0 1=8 5 0 1 1 4 3=2 1 2 1=8 1 0 1=8 5 0 1=4 3 0 3=2 3 2

˛3 1 3 5 1 3 3 5 1 1 3 5 1 3 1 1 3 1 3 1 0 2 4 0 2 2 4 0 0 2 4 0 2 0

nf p 384=5

p 6

p 63 p 12 p 9 p 1920

p 54

p 63

(continued)

6.7 Exercises

343

Table 6.1 (continued) Generating set M1 .2I 5/

Not orthonormalized C ˛1 ˛2 1 4 1 6 2 1 1 0 1 1 2 3 1 0 3 3 2 1 1 0 1 1 0 5 10 0 3 5 0 1

Orthonormalized C ˛1 ˛2 1 4 1 6 2 1 1 0 1 1 2 3 1 0 3 1=2 0 1 1=2 4 1 1 0 5 5 0 3 15=8 0 1 15=8 0 1 15=8 4 1 15=4 2 1 5 2 3

˛3 0 2 4 0 2 2 4 0 2 4

˛3 0 2 4 0 2 4 0 0 2 4 4 0 2 0

nf p 192 p 36

p 945

Exercise 6.7.11. Let 1 ; : : : ; N 2 Sq1 be given nodes. Deduce from the integral formula for the iterated Beltrami operator . /m (Theorem 6.3.6) that the approximate integration formula N X kD1

1 ak F . k / D q kS k

Z F ./ dS./

(6.7.30)

Sq1

Z

N X

C Sq1

ak G





m

 I k ; . /m F . / dS. /

kD1

holds true for all F 2 C.2m/ .Sq1 /, m 2 N, and all coefficients a1 ; : : : ; aN 2 R with N X

ak D 1:

(6.7.31)

kD1

Prove the following statement: For m >

q1 , 4

m 2 N,

ˇ ˇ Z N ˇ ˇ X 1 ˇ ˇ F ./ dS./  ak F . /ˇ ˇ .q1/ ˇ kS ˇ k Sq1

(6.7.32)

kD1



N X N X kD1 lD1



 2m

ak al G . / I k ; l

!1=2 Z 

  m 2 . / F . / dS. / Sq1

holds for all coefficients a1 ; : : : ; aN satisfying (6.7.31).

1=2

6 Spherical Harmonics in Rq

344

Table 6.2 Table of the vertices of the regular polyhedra T1 to T5 in R3 Tj

Coordinates on S2 ˛.1; 1; 1/, ˛.1; 1; 1/, ˛.1; 1; 1/, ˛.1; 1; 1/, ˛ D

T1

Polyhedron Tetrahedron

#Tj 4

T2

Octahedron

6

.˙1; 0; 0/, .0; ˙1; 0/, .0; 0; ˙1/

T3

Cube

8

˛.˙1; ˙1; ˙1/, ˛ D

T4

Icosahedron

12

Dodecahedron

20

3

3

˛.0; ˙ ; ˙1/, ˛.˙1; 0; ˙ /, ˛.˙ ; ˙1; 0/, D

T5

1 p

1 p

p 1C 5 , 2

˛D

p 1 1C 2

˛.˙ 1 ; ˙ ; 0/, ˛.0; ˙ 1 ; ˙ /, ˛.˙ ; 0; ˙ 1 /, q p ˛.˙1; ˙1; ˙1/, D 1C2 5 , ˛ D 13

Exercise 6.7.12. The best approximate integration formula corresponding to the given nodes 1 ; : : : ; N 2 Sq1 , N X

ı

ak F . k /;

(6.7.33)

kD1

to the integral

Z

1 kSq1 k ı

F . / dS./

(6.7.34)

Sq1

ı

is the solution a 1 ; : : : ; aN of the quadratic optimization problem N X N X

  ak al G . /2m I k ; l ! min

(6.7.35)

kD1 lD1

under the constraints (6.7.31). Write down the linear system obtained from the Lagrange method of multipliers. Exercise 6.7.13. Show that the Lagrange multiplier  admits the representation

D

N N X X

  ı ı ak al G . /2m I k ; l ;

(6.7.36)

kD1 lD1

such that ˇ ˇ  12 Z N ˇ 1 Z ˇ X 1 ı ˇ ˇ  m 2 F ./ dS./  ak F . k /ˇ   2 .. / F . // dS. / : ˇ q1 ˇ kS k Sq1 ˇ Sq1 kD1 (6.7.37)

6.7 Exercises

345

Exercise 6.7.14. The regular polyhedra Tj  R3 , j D 1; : : : ; 5, can be situated such that their vertices are contained by the unit sphere S2 (see Table 6.2). Verify that the best approximate coefficients corresponding to the regular polyhedra satisfy ı

ı

a 1 ; : : : ; a#Tj D ı

1 ; #Tj

j 2 f1; 2; 3; 4; 5g:

(6.7.38)

Note that the weights ak , k D 1; : : : ; #Tj , are independent of the special choice of the iteration order m, whereas the Lagrange multiplier  is not (for more details see Freeden and Reuter 1982).

Chapter 7

Classical Bessel Functions

The classical theory of Bessel functions is closely connected with the investigation of the integral (see (7.3.16)) 1 2

Z

2

cos.k sin.u/  ku/ du

(7.0.1)

0

by Bessel (1824). He took k as an integer and obtained many results. After the time of Bessel, investigations of these integrals, which by then bore his name, became numerous. G.N. Watson consolidated these results in a monograph entitled “A Treatise on the Theory of Bessel Functions” (Watson (1944) is the second edition). In what follows, we give a brief insight into the classical theory, thereby concentrating on specific features such as integral and series expressions, recurrence relations, and orthogonality relations. The exercises of Sect. 7.4 present applications of Bessel functions to discontinuous integrals and the modeling of electrons in a magnetic field. The theory is extended in Chap. 8 to the Helmholtz equation in q dimensions. For further properties and details in classical Bessel functions we refer to Abramowitz and Stegun (1972), Lebedev (1973), Watson (1944), and Whittaker and Watson (1948).

7.1 Derivation and Definition of Bessel Functions We consider vibrations of a membrane fixed in a circular frame, i.e., a disk B21 of radius 1. Let Z be the amplitude of the membrane which follows the wave equation x Z D

1 @2 Z c 2 @t 2

in B21 ;

Z D 0 on @B21 D S1 ;

W. Freeden and M. Gutting, Special Functions of Mathematical (Geo-)Physics, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-0348-0563-6 7, © Springer Basel 2013

(7.1.1)

347

348

7 Classical Bessel Functions

p where c D = is the phase-velocity. Here,  is the surface tension of the membrane and  is its mass-density. We are interested in time-harmonic vibrations, i.e., Z.x; t/ D Z.x/ exp.i!t/

(7.1.2)

which gives us .x Z.x// exp.i!t/ D

1 .! 2 /Z.x/ exp.i!t/; c2

(7.1.3)

or equivalently, x Z.x/ C

!2 Z.x/ D 0 c2

in B21

(7.1.4)

and Z D 0 on S1 . Thus, Z has to fulfill the Helmholtz equation in B21 , i.e., x Z C k 2 Z D 0 in B21 ;

ZD0

on S1

(7.1.5)

with k D !=c being the wave number. The spherical geometry leads us to polar coordinates and we use separation of variables, i.e., Z.x/ D U.r/˚.'/;

where x D Œr cos.'/; r sin.'/T :

(7.1.6)

We get 0 D x Z C k 2 Z D

1 @2 1 @ @2 Z C Z C Z C k2Z @r 2 r @r r 2 @' 2

1 1 D U 00 .r/˚.'/ C U 0 .r/˚.'/ C 2 U.r/˚ 00 .'/ C k 2 U.r/˚.'/ r r D

rU 0 .r/ ˚ 00 .'/ r 2 U 00 .r/ C C C r 2k2: U.r/ U.r/ ˚.'/

(7.1.7)

This can only be true for all r 2 Œ0; R and for all ' 2 Œ0; 2/ if ˚ 00 .'/ D  ˚.'/

(7.1.8)

r 2 U 00 .r/ rU 0 .r/ C C r 2 k 2 D : U.r/ U.r/

(7.1.9)

and

7.1 Derivation and Definition of Bessel Functions

349

The first ordinary differential equation, ˚ 00 .'/ C ˚.'/ D 0;

(7.1.10)

possesses a solution if and only if  D n2 , n 2 N0 . The solution is given by ˚.'/ D A cos.n'/ C B sin.n'/:

(7.1.11)

Inserting  D n2 into the second ordinary differential equation gives us r 2 U 00 .r/ C rU 0 .r/ C .r 2 k 2  n2 /U.r/ D 0:

(7.1.12)

Substitute now x D kr and we find x 2 U 00 .x/ C xU 0 .x/ C .x 2  n2 /U.x/ D 0:

(7.1.13)

Definition 7.1.1. The equation   d2 d z2 2 C z C .z2   2 / Y .z/ D 0 dz dz

(7.1.14)

with z;  2 C and without loss of generality Re./  0, is called Bessel’s differential equation. For the solution, we choose the following approach with a0 ¤ 0, ˛ 2 C, Y .z/ D z˛

1 X

ak zk D

kD0

1 X

ak zkC˛ ;

(7.1.15)

kD0

which gives us   d2 d 0 D z2 2 C z C .z2   2 / Y .z/ dz dz D

1 X

  ak .k C ˛/.k C ˛  1/zkC˛ C .k C ˛/zkC˛ C .z2   2 /zkC˛

kD0

D

1 X kD0

1 X   ak .k C ˛/2   2 zkC˛ C ak2 zkC˛ :

(7.1.16)

kD2

Thus, we get by comparing the coefficients for k D 0: a0 .˛ 2   2 / D 0;

(7.1.17)

350

7 Classical Bessel Functions

1

J0 J1 J2

0.75

J1/2 J1 J−1

0.6 0.4

0.5 0.2 0.25

0

0

−0.2

−0.25 −0.5

−0.4

0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

Fig. 7.1 Bessel functions of first kind

i.e., ˛ 2 D  2 or ˛ D ˙. For k D 1 we have  a1 .˛ C 1/2  a1  2 D 0;

(7.1.18)

such that a1 .1 C 2˛/ D 0. Now, we investigate two cases: Case 1: If ˛ D , then a1 .1 C 2/ D 0 leads to a1 D 0 since Re./  0. For k > 1,   ak .k C ˛/2   2 C ak2 D 0; (7.1.19) which gives us ak D a2l D .1/l

ak2 . k.kC2/

22l lŠ.

Therefore, a2lC1 D 0 for l 2 N and

a0 C 1/. C 2/ : : : . C k/

The freedom of choice for a0 remains. We set a0 D of the Gamma function.

for l 2 N :

1 2 .C1/

(7.1.20)

and use the properties

Theorem 7.1.2. For z 2 C n R 0, ;  2 Sq1 , er D

1 X

N.qI n/Jn .qI r/Pn .q;   /:

(8.1.40)

nD0

(ii) For x D jxj, y D jyj, J0 .qI jx  yj/ D

1 X

N.qI n/Jn .qI jxj/Jn .qI jyj/Pn .qI   /:

(8.1.41)

nD0

(iii) Jn .qI / is a solution of the Bessel differential equation Jn00 .qI r/ C

  q1 0 n.n C q  2/ Jn .qI r/ C 1  Jn .qI r/ D 0: r r2

(8.1.42)

(iv) Jn .qI / satisfies the recursion relation .n C q  2/JnC1 .qI r/ C .2n C q  2/Jn0 .qI r/  nJn1 .qI r/ D 0

(8.1.43)

and n n

Jn .qI r/ D .1/ r Jn1 .qI r/ D r 2qn



1 d r dr

n

J0 .qI r/;

 d  nCq2 r Jn .qI r/ : dr

(v) For n ! 1 the Bessel function Jn .qI / behaves such that   Jn .qI r/ n C q2   lim D 1: n!1  q2 . r2 /n

(8.1.44) (8.1.45)

(8.1.46)

(vi) For homogeneous harmonic polynomials Hn .qI / of degree n, Hn .qI rx /J0 .qI jxj/ D .1/n Jn .qI jxj/Hn .qI /:

(8.1.47)

8 Bessel Functions in Rq

370

The traditional notation due to Watson (1944) discussed in Chap. 7, which is mostly used in the literature, leads to the observation that the Bessel functions of integral order n and dimension q are expressible as Bessel functions of order n C q2 2 and dimension 2. Now, this aspect is investigated in more detail: letting, as usual (see Theorem 7.1.2), J .r/ D J .2I r/ D

1  r  X .1/k

2



kD0

 r 2k 2

;  0;

(8.1.48)

; n 2 N0 ;

(8.1.49)

 . C k C 1/

and observing that Jn .qI r/ D

 r n 2



1 q  X .1/k

2



kD0

 r 2k 2

 .n C k C q2 /

we are led by comparison to the relation Jn .qI r/ D 

 q   r  2q 2 2

2

JnC q2 .2I r/; n 2 N0 : 2

(8.1.50)

In particular, we have r J0 .3I r/ D

 J 12 .2I r/ J1 .1I r/ : p D 2 r r

(8.1.51)

Furthermore, we see that J0 .3I / is the sinc-function 1

J0 .3I r/ D

.1/k r 2k 1p X    2 22k  .k C 1/ k C 32 kD0

D

1 X kD0

sin.r/ .1/k r 2k D D sinc.r/: .2k C 1/Š r

(8.1.52)

8.2 Modified Bessel Functions Solutions of the equation U  U D 0 can be obtained in an analogous way. Indeed, standard polar coordinates x D r, r D jxj,  2 Sq1 , together with the Funk–Hecke formula (see Theorem 6.5.5), imply the following radial and angular separation of the integral 1 kSq1 k

Z Sq1

ex Yn .qI / dS./ D In .qI r/Yn .qI /;

(8.2.1)

8.2 Modified Bessel Functions

371

where In .qI / is given by 

q  2

Z

. r2 /n

In .qI r/ D p   n C

q1  2

1

1

ert .1  t 2 /nC

q3 2

dt:

(8.2.2)

Definition 8.2.1. In .qI / as given by (8.2.2) is called a modified Bessel function of order n and dimension q. By comparison we find the relation In .qI r/ D in Jn .qI ir/. Hence, the properties of the modified Bessel function of order n and dimension q can be collected in the following lemma. Lemma 8.2.2. The modified Bessel function In .qI / satisfies the following relations: (i) For r > 0; ;  2 Sq1 ; er D

1 X

N.qI n/In .qI r/Pn .q;   /:

(8.2.3)

nD0

(ii) For x D jxj; y D jyj, I0 .qI jx  yj/ D

1 X

N.qI n/In .qI jxj/In .qI jyj/Pn .qI   /:

(8.2.4)

nD0

(iii) In .qI / is a solution of the Bessel differential equation In00 .qI r/ C

  q1 0 n.n C q  2/ In .qI r/  1 C In .qI r/ D 0: r r2

(8.2.5)

(iv) In .qI / satisfies the recursion relation .n C q  2/InC1 .qI r/ C .2n C q  2/In0 .qI r/  nIn1 .qI r/ D 0 and In .qI r/ D r

n



1 d r dr

n

I0 .qI r/:

(8.2.6)

(8.2.7)

(v) In .qI / behaves for n ! 1 such that   In .qI r/ n C q2   lim D 1: n!1  q2 . r2 /n

(8.2.8)

(vi) For homogeneous harmonic polynomials Hn .qI / of degree n, Hn .qI rx /I0 .qI jxj/ D In .qI jxj/Hn .qI /:

(8.2.9)

8 Bessel Functions in Rq

372

In particular, we have for q D 3; I0 .3I r/ D and In .3I r/ D r

n



sinh.r/ r

1 d r dr

n

(8.2.10)

sinh.r/ : r

(8.2.11)

8.3 Hankel Functions The Hankel functions can be deduced by two different complexifications of the Funk–Hecke formula (Theorem 6.5.5) due to M¨uller (1998). The complexification requires two subsets of the complex unit sphere as defined in the following way for  2 Sq1 : n o p q1 S1 ./ D  2 Cq W  D 1  y 2   y; y 2 Eq1 ./ ( p j1  y 2 j1=2 2 1y D ijy 2  1j1=2

with

and

;

y 2 < 1;

;

y 2 > 1;

n o p q1 S2 ./ D  2 Cq W  D 1  y 2   y; y 2 Eq1 ./ ( p j1  y 2 j1=2 1  y2 D ijy 2  1j1=2

with

;

y 2 < 1;

;

y 2 > 1;

(8.3.1)

(8.3.2)

(8.3.3)

(8.3.4)

where Eq1 ./ are hyperplanes given by  ˚ Eq1 ./ D y 2 Rq W y   D 0;  2 Sq1 :

(8.3.5)

p q1 q1 The sets S1 ./, S2 ./ only differ by the understanding of 1  y 2 , y 2 < 1. q1 q1 The set S1 ./ contains the hemisphere y   > 0, whereas the set S2 ./ contains q1 y   < 0. Keeping in mind the definition of Sj ./, j D 1; 2, we are able to discuss the following integrals (see also (8.1.1)) with x D r,  2 Sq1 : Un.j / .x/

in D 2.1/ kSq1 k j

Z q1

Sj

./

eix Yn .qI / dS.q1/ ./;

(8.3.6)

where Yn .qI / 2 Harmn .Sq1 / and dS.q1/ is understood to be the canonical complexification of the surface element (see M¨uller (1998) for further details). The

8.3 Hankel Functions

373

Fig. 8.1 The two integration paths of the Hankel functions .1/ .2/ Hn .qI / (left) and Hn .qI / (right), respectively

0

.j /

integrals Un

1

-1

0

exist in the half plane x   > 0,  2 Sq1 . Obviously, we have . C 1/Un.j / D 0

(8.3.7)

in this half plane. From the complex variant of the Funk–Hecke formula (cf. M¨uller 1998) we are able to deduce that Un.j /.x/ D Hn.j / .qI r/Yn .qI /; .1/

j D 1; 2;

(8.3.8)

.2/

where x D r. For r > 0, the functions Hn .qI /, Hn .qI / are given by Z q3 kSq2 k 1C1i irt e Pn .qI t/.1  t 2 / 2 dt; (8.3.9) q1 kS k 1C0i Z q3 kSq2 k 1C1i irt Hn.2/ .qI r/ D 2in q1 e Pn .qI t/.1  t 2 / 2 dt: (8.3.10) kS k 1C0i Hn.1/ .qI r/ D 2in

.1/

.2/

Definition 8.3.1. The functions Hn .qI /, Hn .qI / are called Hankel functions of the first and second kind of order n and dimension q, respectively. It is well-known that the two paths of integration in this definition may be deformed .2/ to the curves depicted in Fig. 8.1. For example, we get for the path of Hn .qI /; 1 n kSq1 k .2/ i H .qI r/ D 2 kSq2 k n

Z

0 1

eirt Pn .qI t/.1  t 2 /

Z

1

Ci 0

D .1/

n

C inC1

q3 2

dt

ers Pn .qI is/.1 C s 2 /

Z

1 0

Z

q3 2

eirt Pn .qI t/.1  t 2 /

1 0

(8.3.11) ds

q3 2

dt

ers Pn .qI is/in .1 C s 2 /

q3 2

ds:

8 Bessel Functions in Rq

374

Remark 8.3.2. The representation of the Legendre polynomial as a power series shows us that in Pn .qI is/ is a polynomial of degree n in the variable s possessing exclusively real coefficients (see (8.4.15))    .q  1/ 2n1  n C q2   2 q  s n C O .1 C s/n2 ; in Pn .qI is/ D  2  .n C q  2/

(8.3.12)

where s 2 Œ0; 1/, n C q  3. From Definition 8.3.1 it follows that kSq1 k .2/ H .qI r/ D 2in kSq2 k n

Z 0

Z

C 2i D

1

eirt Pn .qI t/.1  t 2 / 1

0

q3 2

dt

(8.3.13)

ers Pn .qI is/in .1 C s 2 /

q3 2

ds

kSq1 k .1/ Hn .qI r/: kSq2 k

The Hankel functions have certain characteristic properties for r ! 1 and fixed n. With the substitution t D 1 C is, s 2 Œ0; 1/, we get Hn.1/ .qI r/

Z D En .qI r/

1

0

ers Pn .qI 1 C is/.s.s C 2i//

where En .qI r/ D 2

q3 2

ds;

kSq2 k i.rn    .q1// 2 4 e : kSq1 k

(8.3.14)

(8.3.15)

Note that t D 1 C is yields .1  t 2 /

q3 2

D .is.2 C is//

q3 2



D e 4 i.q3/ .2s/

q3 2

C ::: ;

(8.3.16)

such that by standard arguments of complex analysis (according to our definition of the root function) .1  t 2 /

q3 2



D e 4 i.q3/ .2s/

q3 2

C ::: :

(8.3.17)

With the abbreviation F .s/ D .2 C is/

Z

we have Hn.1/ .qI r/

q3 2

D En .qI r/

0

Pn .qI 1 C is/

1

ers F .s/s

q3 2

(8.3.18)

ds:

(8.3.19)

8.3 Hankel Functions

375

It can be deduced from well-known results for the Legendre polynomial (see M¨uller 1998) that F .s/ D 2

q3 2

  .2n C q  1/.2n C q  3/ is C ::: : 1C q1 4

(8.3.20)

Moreover, we borrow from M¨uller (1998) the following result. Lemma 8.3.3. For n fixed and r ! 1, we have  q1 qC1 2 2 i.rn  .q1/  / 2 4 C O.r  2 / e q1 kS k r   q1  qC1   . q2 / 2 2 n q1 ir 2 e C O r 2 ; i D p  r   q1  qC1   . q2 / 2 2 nC q1 ir .2/ 2 e Hn .qI r/ D p i C O r 2 :  r

Hn.1/ .qI r/ D

2



(8.3.21)

(8.3.22)

Observing the identity Jn .qI r/ D

 1  .1/ Hn .qI r/ C Hn.2/ .qI r/ ; 2

(8.3.23)

we obtain the following lemma from Lemma 8.3.3. Lemma 8.3.4. For n fixed and r ! 1,  .q/ Jn .qI r/ D p 2 

  q1   qC1    2 2  cos n C .q  1/  r C O r  2 : (8.3.24) r 2 4

Based on techniques due to Watson (1944) the O-term in Lemma 8.3.4 can be written out in more detail. Lemma 8.3.5. For n 2 N0 ; m 2 N fixed and r ! 1,  . q2 / 1q Jn .qI r/ D 3q p r 2 2 2 

m1 X

.n C q2 2 ; l/ (8.3.25) e .2ir/l lD0 ! m1   X .n C q2 ; l/ q2 q1 i.r 2 .nC 2 / 4 / 2 .mC 2 / ; Ce C O r .2ir/l i.r 2 .nC

q2  2 / 4

/

lD0

where in terms of the Pochhammer factorial (see Definition 2.4.1) as well as in terms of the Gamma function

8 Bessel Functions in Rq

376

 nC

q2 2 ;l



D D



.1/l lŠ

1 2

n

 

q2 2 l

1 2

CnC



q2 2 l

q1 2 C l/ : q1 C 2  l/

 .n C lŠ .n

(8.3.26)

Next, we are interested in an asymptotic relation for n ! 1 and fixed r. Lemma 8.3.6. For n ! 1 and fixed r > 0, .1/

q 

lim

n!1 i  

2

Hn .qI r/   D 1:  n C q2  1 . 2r /nCq2

(8.3.27)

Proof. Rodrigues’ rule yields Hn.1/ .qI r/

 . q2 /  D 2 p  n C

q1 2



 r n Z 2

1C1i 1C0i

eirt .1  t 2 /nC

q3 2

dt: (8.3.28)

Obviously, Z

1C1i

1C0i

Z

0

D 1

eirt .1  t 2 /nC 2 nC

e .1  t / irt

q3 2

q3 2

dt

(8.3.29) Z

dt C

1i 0i

eirt .1  t 2 /nC

q3 2

dt:

The first integral is uniformly bounded and does not give any contribution to the assertion. The second integral can be written as Z

1i 0i

eirt .1  t 2 /nC

Z

q3 2

1

dt D i 0

ers .1 C s 2 /nC

q3 2

ds:

(8.3.30)

In order to prove Lemma 8.3.6, we observe that R1 lim

n!1

0

q3

ers .1 C s 2 /nC 2 ds R1 D 1: rs s 2nCq3 ds 0 e

(8.3.31)

Moreover, we have Z

1 0

ers s 2nCq3 ds D

 .2n C q  2/ : r 2nCq2

(8.3.32)

Now, two estimates come into play .1 C s/2k D .1 C 2s C s 2 /k > .1 C s 2 /k ;

s > 0; k > 0;

(8.3.33)

8.3 Hankel Functions

377

and .1 C s 2 /k  s 2k  k.1 C s 2 /k1  k.1 C s/2k2 :

(8.3.34)

Note that the estimate (8.3.34) follows from the mean value theorem of onedimensional analysis. This leads to the estimate Z 0

1

 Z 1   q3 q3 ers .1 C s 2 /nC 2  s 2nCq3 ds  n C ers .1 C s/2nCq5 ds 2 0 Z 1  q3 eru u2nCq5 du:  er n C 2 0

(8.3.35)

The last integral can be determined explicitly. Z 1    q3 q  3  .2n C q  4/ ru 2nCq5 r e u du D e n C e nC 2 2 r 2nCq4 0 r

D

er 2r 2nCq4

 .2n C q  2/ : 2n C q  4

(8.3.36)

Consequently, we are able to see that ˇZ ˇ ˇ ˇ

1

e

rs

0

2 nC

.1 C s /

q3 2



ˇ  .2n C q  2/ ˇˇ ds  ˇ r 2nCq2 er r 2nCq4

(8.3.37)

 .2n C q  2/ : 2n C q  4

In connection with (8.3.29) and (8.3.30) we obtain for n ! 1; R 1i

R 1i

q3

irt 2 nC 2 dt 0i e .1  t / R q3 1 rs n!1 nC 2 2 i 0 e .1 C s / ds

lim

D lim

n!1

0i

eirt .1  t 2 /nC

q3 2

i  r.2nCq2/ 2nCq2

dt

D 1: (8.3.38)

Lemma 8.3.6 follows with the help of the duplication formula p  .2n C q  2/ D 22nCq3 

    q1 q2  nC nC 2 2

known from the theory of the Gamma function (see Lemma 2.3.3).

(8.3.39) t u

Next, we come to the Neumann function which together with the Bessel function implies the Hankel functions.

8 Bessel Functions in Rq

378

Definition 8.3.7. For r > 0, the Neumann function Nn .qI / of order n and dimension q is defined by Z

 q3  Pn .qI t/.1  t 2 / 2 dt sin rt  n 2 0 q2 Z 1 q3 kS k  2in q1 ert Pn .qI it/.1  t 2 / 2 dt: kS k 0

kSq2 k N.qI r/ D 2 q1 kS k

1

(8.3.40)

As already announced, Hankel functions can be obtained by combination of Bessel and Neumann functions. More explicitly, from (8.3.11) and (8.3.13) we get the following identities. Lemma 8.3.8. For r > 0, Hn.1/ .qI r/ D Jn .qI r/ C iN.qI r/;

(8.3.41)

Hn.2/ .qI r/ D Jn .qI r/  iN.qI r/:

(8.3.42)

Let Cn .qI / stand for any of the so-called cylinder functions Jn .qI /, Nn .qI /, .1/ .2/ Hn .qI / and Hn .qI /. Then, for r > 0, the following recursion relation holds true: 2n C q  2 Cn1 .qI r/ C CnC1 .qI r/ D Cn .qI r/: (8.3.43) r Furthermore, we have .2n C q  2/Cn0 .qI r/ D nCn1 .qI r/  .n C q  2/CnC1 .qI r/:

(8.3.44)

These identities immediately follow from the recursion relation for the Legendre polynomials.

8.4 Kelvin Functions The Kelvin functions can be obtained by another complexification of the Funk– Hecke formula (cf. Richter 1971). This time we consider the following subset of the complex unit sphere n o p q1 (8.4.1) S3 ./ D  2 Cq W  D 1 C y 2   iy; y 2 Eq1 ./ p where 1 C y 2 D j1 C y 2 j1=2 and Eq1 ./ is defined by (8.3.5). Now, the point of departure is the function (see also (8.1.1) and (8.3.6)) i1q Wn .x/ D q1 kS k

Z q1

S3

./

ex Yn .qI / dS.q1/ ./;

(8.4.2)

8.4 Kelvin Functions

379

where dS.q1/ again is understood to be the canonical complexification of the surface element (as described by M¨uller 1998). For x   > 0, we immediately obtain .  1/Wn D 0: (8.4.3) From the complex variant of the Funk–Hecke formula (due to M¨uller 1998) we get, by using polar coordinates x D r,  2 Sq1 ; Wn .x/ D Kn .qI r/Yn .qI / Z

where Kn .qI r/ D

1 1

ert Pn .qI t/.t 2  1/

(8.4.4) q3 2

dt:

(8.4.5)

Definition 8.4.1. For r > 0, the function Kn .qI / given by (8.4.5) is called the Kelvin function (or modified Hankel function) of order n and dimension q. It can be shown that Hn.1/ .qI ir/ D 2

kSq2 k 1qn i Kn .qI r/; kSq1 k

(8.4.6)

Hn.2/ .qI ir/ D 2

kSq2 k nCq1 Kn .qI r/: i kSq1 k

(8.4.7)

Moreover, we find, for r > 0, (see (8.3.43) and (8.3.44), respectively) Kn1 .qI r/  KnC1 .qI r/ D 

2n C q  2 Kn .qI r/ r

(8.4.8)

and nKn1 .qI r/ C .n C q  2/KnC1.qI r/ D .2n C q  2/Kn0 .qI r/:

(8.4.9)

This again follows from the recursion relation (6.4.56) for the Legendre polynomial. Keeping r fixed, we obtain the following lemma by similar techniques as used for the proof of Lemma 8.3.6. Lemma 8.4.2. For n ! 1 and fixed r > 0, we have lim

q1 n!1 1 2  . 2 /

Kn .qI r/    nCq2 D 1: n C q2  1 2r

(8.4.10)

Proof. Rodrigues’ rule gives the integral representation

Kn .qI r/ D p





q1 2



  n C

rn q1 2

Z 

1

1

ert .t 2  1/nC

q3 2

dt:

(8.4.11)

8 Bessel Functions in Rq

380

Now, we have

R1 lim

1

n!1

q3

ert .t 2  1/nC 2 dt R1 D 1; rt t 2nCq3 dt 0 e

(8.4.12)

where the integral in the denominator can be calculated explicitly: Z

1

0

ert t 2nCq3 dt D

 .2n C q  2/ : r 2nCq2

(8.4.13)

From the estimate 0  t 2k  .t 2  1/k  kt 2k2 ;

t  1;

(8.4.14)

we get Z 0

1 1

ert .t 2  1/nC

q3 2

Z

1

dt  1

ert t 2nCq3 dt

 Z 1 q3   nC ert t 2nq5 dt 2 0

(8.4.15)

such that Z 0

1 1

ert .t 2  1/nC

q3 2

Z

1

dt  0

ert t 2nCq3 dt

(8.4.16)

    q  3  .2n C q  4/ 1  .2n C q  4/ 1 2nCq4  nC D  : 2 r 2nCq4 2 2n C q  4 r This is the desired result.

t u

The Hankel functions as well as the Kelvin functions are not defined at the origin, but they have characteristic singularities there. Lemma 8.4.3. For r ! 0 and n C q  3  0, we have      nCq2   q2 q 2 nC  C O r nqC4 ; (8.4.17) 2 2 r   nCq2    2  .q  1/ q2 Kn .qI r/ D q1  q   n C C O r nqC4 : (8.4.18) 2 r 2  2

Hn.1/ .qI r/

i D  

Proof. First, we deal with the asymptotic relation (8.4.17). For that purpose, we observe that from (6.4.36)

8.4 Kelvin Functions

Pn .qI t/ D D

381

n1

 .q  1/ 2 nŠ    q2 .n C q  3/Š an0 .q/t n

bn=2c 

X



lD0

 l  n  l C 1 4

q2 2



lŠ.n  2l/Š

 C :::;

t n2l (8.4.19)

where (see (8.3.12)) an0 .q/ D Therefore,

 2n1  n C

q2 2



 .q  1/   :  .n C q  2/ q2

(8.4.20)

  in Pn .qI it/ D an0 .q/t n C O .1 C t/n2 ;

(8.4.21)

such that 2in

kSq2 k kSq1 k

D 2nCq2

Z

1 0

ert Pn .qI it/.1 C t 2 /

  nC

q2 2





q  2

 r nCq2

q3 2

dt

(8.4.22)

  C O r nqC4 :

We now discuss the asymptotic relation (8.4.18). For n C q  3  0, we obtain after some manipulations Kn .qI r/ D er

Z

1 0

D an0 .q/ Replacing the integral n C q  3  0,

Z

ers Pn .qI 1 C s/.s.1 C s//nCq5 ds 1

0

R1 0

ers s nCq3 ds C O

1 0

: : : in the second term by

 .q  1/ Kn .qI r/ D q1  q   2  2 This is the desired result.

Z

(8.4.23)

 ers .s C 1/nCq5 ds :

R1 1

: : :, we find, for all n with

  nCq2    2 q2 C O r nqC4 : nC 2 r

(8.4.24) t u

8 Bessel Functions in Rq

382

The identity Z K0 .2I r/ D

1 1

1

ert .t 2  1/ 2 dt D

D er ln.r/ C

Z

1 r

Z

1

r

1

eu .u2  r 2 / 2 du

  p eu ln u C u2 C r 2 du

(8.4.25)

shows us that K0 .2; r/ has a logarithmic singularity. In fact, for n D 0, q D 2, we have K0 .2I r/ D  ln.r/ C O.1/: (8.4.26) In the same way we obtain .1/

H0 .2I r/ D

2i ln.r/ C O.1/: 

(8.4.27)

Remark 8.4.4. The function x 7! K0 .qI jxj/, x 2 Rq n f0g, shows the same singularity behavior at the origin as the fundamental solution x 7! Fq .jxj/, x 2 Rq n f0g, for the Laplace operator in Rq . For r ! 1 and n  0 fixed, we obtain Z Kn .qI r/ D t D1Cs

1 1

D e

ert Pn .qI t/.t 2  1/

Z

r

1 0



D er 2

q3 2

dt

(8.4.28)

ers Pn .qI 1 C s/..2 C s/s/

q3 2



 r

q1 2



q1 2

C O.r 

qC1 2

q2 2

ds

 / :

This leads to the following asymptotic relations. Lemma 8.4.5. For r ! 1 and n  0 fixed, er  Kn .qI r/ D 2



q1 2

    q1  qC1   2 2 : C O r 2 r

(8.4.29)

Moreover, for r ! 1, we have   qC1 Kn0 .qI r/  Kn .qI r/ D O er r  2 :

(8.4.30)

The properties of the Kelvin function relevant for our purposes in the analytic theory of numbers can be summarized as follows. Lemma 8.4.6. The Kelvin function Kn .qI / satisfies the following relations:

8.4 Kelvin Functions

383

(i) For x D jxj; y D jyj, K0 .qI jx  yj/ D

1 X

N.qI n/In .qI jxj/Kn .qI jyj/Pn .qI   /:

(8.4.31)

nD0

(ii) Kn .qI / is a solution of the differential equation Kn00 .qI r/

  q1 0 n.n C q  2/ Kn .qI r/ 1 C Kn .qI r/ D 0: r r2

(8.4.32)

(iii) Kn .qI / satisfies the recursion relations .n C q  2/KnC1 .qI r/ C .2n C q  2/Kn0 .qI r/ C nKn1 .qI r/ D 0;

(8.4.33)

2n C q  2 Kn .qI r/ D 0: r

(8.4.34)

Kn1 .qI r/  KnC1 .qI r/ C

(iv) The following relations exist between KnCm.qI / and Kn .qI / with m 2 N: .1/m r nm KnCm .qI r/ D .1/m r nmCq2 Knm .qI r/ D

 

1 d r dr 1 d r dr

m m

.r n Kn .qI r// ; 

 r nCq2 Kn .qI r/ :

(8.4.35) (8.4.36)

(v) For the limit n ! 1, it holds that lim

n!1





q1 2



2 Kn .qI r/  . 2r /nCq2  n C

q 2

1

 D 1:

(8.4.37)

(vi) For n C q  3 > 0 fixed and r ! 0,  .q  1/ Kn .qI r/ D q1  q   2  2

  nCq2  2 q2 nC CO.r nqC4 / 2 r

(8.4.38)

and, for n D 0, q D 2, and r ! 0, K0 .qI r/ D  ln.r/ C O.1/:

(8.4.39)

(vii) For r ! 1 and n  0 fixed, er  Kn .qI r/ D 2



q1 2

    q1  qC1   2 2 : C O r 2 r

(8.4.40)

8 Bessel Functions in Rq

384

In particular, K0 .3I r/ D and Kn .3I r/ D r n



er r

1 d r dr

(8.4.41)

n

er : r

(8.4.42)

It is interesting to relate the Kelvin and Hankel functions of dimension q to their counterparts of dimension 2. For 2 R, we have (see Magnus et al. 1966; Watson 1944) p  r  Z 1  1   K .r/ D K .2I r/ D ert .t 2  1/  2 dt:  C 12 2 1

(8.4.43)

By comparison we find

Kn .qI r/ D





q1 2

p 

Similarly, Hn.1/ .qI r/ D 



 r  2q 2 2

 q   r  2q 2 2

2

KnC q2 .r/: 2

H

.1/

nC

q2 2

.r/;

(8.4.44)

(8.4.45)

where  r  Z 1C1i 1 2   eirt .1t 2 /  2 dt: H .1/ .r/ D H .1/ .2I r/ D p  C 12 2 1C0i

(8.4.46)

Finally, we come to some interrelations between the cylinder functions, which are originated by their differential equation (of Sturm–Liouville type). For that purpose we introduce the notion of the so-called Wronskian determinant W .F; G/ D F G 0  F 0 G

(8.4.47)

for a pair of (scalar) continuously differentiable functions F; G. Theorem 8.4.7. For r > 0, the Bessel functions satisfy the equations  q1 2 q   .1/  .2/ 2 .2/ ; W Hn .qI r/; Hn .qI r/ D 2i r   q1 2 q    .2/ 2 .1/ W Jn .qI r/; Hn .qI r/ D i ; r   q1 2 q    .2/ 2 .1/ : W N.qI r/; Hn .qI r/ D  r 

(8.4.48) (8.4.49) (8.4.50)

8.5 Expansion Theorems

385

For r > 0, the modified functions satisfy W .In .qI r/; Kn .qI r// D 2 r 1q  .q  1/:

(8.4.51)

.i /

Proof. The Bessel functions Jn .qI /, N.qI /, Hn .qI /, i D 1; 2, abbreviated by Cn .qI /, satisfy the differential equation .r q1 Cn0 .qI r//0  n.n C q  2/r q3 Cn .qI r/ C r q1 Cn .qI r/ D 0;

(8.4.52)

whereas the modified functions In , Kn satisfy .r q1 Cn0 .qI r//0  n.n C q  2/r q3 Cn .qI r/  r q1 Cn .qI r/ D 0:

(8.4.53)

For any two solutions Cn .qI /; CQ n .qI / of either equation, the identity

is valid such that

 d  q1  r W Cn .qI r/; CQ n .qI r/ D 0 dr

(8.4.54)

  r q1 W Cn .qI r/; CQ n .qI r/ D const:

(8.4.55)

By virtue of the asymptotic expansions for r ! 1 this implies the assertion of Theorem 8.4.7. For example, for r ! 1,     .1/ W Hn.1/ .qI r/; Hn.2/ .qI r/ D W Hn.1/ .qI r/; Hn .qI r/ D Hn.1/ .qI r/

(8.4.56)

d .1/ d .1/ Hn .qI r/  Hn .qI r/ Hn.1/ .qI r/ dr dr

 q1 2 q  .2/ 2 C O.r q /: D 2i r 

Because of (8.4.55) the O-term in (8.4.56) is zero. This proves Theorem 8.4.7.

t u

8.5 Expansion Theorems The solutions of the Helmholtz equation can be subdivided into three classes depending on their domain of definition: q

• Solutions for the inner space of a fixed ball BR ; R > 0; q • Solutions for the outer space of a fixed ball BR ; R > 0; • Entire solutions valid for the whole Euclidean space Rq .

8 Bessel Functions in Rq

386

From survey articles about the Helmholtz equation (cf. M¨uller 1969; Niemeyer 1962) we borrow the following expansion theorems for inner and/or outer space solutions involving Bessel and Hankel functions. Theorem 8.5.1 (Expansion for the Inner Space of a Ball). Let U be a function of q q class C.2/ BR such that U C U D 0 in BR . There exists a sequence of spherical harmonics fYn .qI /gn2N0 such that U.x/ D

1 X

Jn .qI r/Yn .qI /;

(8.5.1)

nD0

x D r;  2 Sq1 , where the series is absolutely and uniformly convergent for all q x 2 Br with r D jxj < R. q Conversely, if (8.5.1) holds uniformly for all x 2 Br with r D jxj < R, then U q q .2/ is of class C .BR / satisfying U C U D 0 in BR . Proof. The functions Un;j ; j D 1; : : : ; N.qI n/, given by Z Un;j .r/ D

Sq1

U.r/Yn;j .qI / dS./

(8.5.2)

are twice continuously differentiable for all r 2 Œ0; R . Applying Green’s formula we obtain Z   @ q1 0 x dS.x/ (8.5.3) U.x/Yn;j qI jxj r Un;j .r/ D q1 @ Sr Z D

     x x qI .U.x//  U.x/Y qI dV .x/ Y n;j n;j jxj jxj q

Br

such that 0 r q1 Un;j .r/ D 

Z 0

r



 s q1 Un;j .s/  n.n C q  2/s q3 Un;j .s/ ds:

(8.5.4)

By differentiation with respect to r we are able to show that 00 Un;j .r/

  q1 0 n.n C q  2/ Un;j .r/ C 1  Un;j .r/ D 0: C r r2

(8.5.5)

Apart from a multiplicative constant, the only solution of (8.5.5) that is bounded for r ! 0 is the Bessel function Jn .qI /. Hence, there exists a constant An;j such that Un;j .r/ D An;j Jn .qI r/

(8.5.6)

8.5 Expansion Theorems

387

for all r 2 Œ0; R/. We let N.qIn/

Yn .qI / D

X

Cn;j Yn;j .qI /;

 2 Sq1 ;

(8.5.7)

j D1

and Cn2 D

Z Sq1

jYn .qI /j2 dS./:

(8.5.8)

The L2 .Sq1 /-orthonormality of the spherical harmonics then yields Cn2 D

N.qIn/

X ˇ ˇ ˇCn;j ˇ2 :

(8.5.9)

j D1

For r 2 Œ0; R/, it follows from Lemma 8.5.2 that the series ˚.rI / D

1 X

Jn .qI r/Yn .qI /

(8.5.10)

nD0

is convergent for all r 2 Œ0; R/. More precisely, the series (8.5.10) is absolutely and q uniformly convergent on every ball Br , r 2 Œ0; R/. In addition, Z Sq1

.U.r/  ˚.rI // Yn;j .qI / dS./ D 0

(8.5.11)

for all n; j . The completeness of the system of spherical harmonics, therefore, implies U.r/ D ˚.rI /;

r 2 Œ0; R/;

(8.5.12)

which shows the first part of Theorem 8.5.1. The second part follows from the fact that all partial sums of U satisfy the q q Helmholtz equation for every Br  BR , r < R, and converge uniformly to U .  q Under these assumptions (see, e.g., Niemeyer 1962) U is of class C.2/ Br for every q r 2 Œ0; R/ and, in addition, U satisfies the differential equation U C U D 0 in BR . t u Lemma 8.1.7 allows us to formulate the following expansion theorem. Lemma 8.5.2. Let fYn .qI /gn2N0 be a sequence of spherical harmonics of dimension q such that 1 X Jn .qI R/Yn .qI /;  2 Sq1 ; (8.5.13) nD0

8 Bessel Functions in Rq

388

is convergent for R > 0. Then, 1 X

n˛ Jn .qI r/Yn .qI /;

 2 Sq1 ;

(8.5.14)

nD0

is convergent for all r 2 Œ0; R/ and ˛  0, where the series (8.5.14) is absolutely q q and uniformly convergent for all Br  BR . Proof. According to (8.5.13), we have lim jJn .qI R/Yn .qI /j D 0;

n!1

(8.5.15)

i.e., for sufficiently large n, jYn .qI /j 

1 : jJn .qI R/j

(8.5.16)

From the limit relation (Lemma 8.1.7) we obtain, for n ! 1,   q2 q2 q  n  : R jYn .qI /j D O 2nC 2 n 2  n C 2

(8.5.17)

Hence, there exist constants A and N D N.R/ such that jn˛ Jn .qI r/Yn .qI /j  A n˛

 r n R

for all n  N and r < R. This proves Lemma 8.5.2.

(8.5.18) t u

Next, we come to series expansions by products of Hankel functions and spherical harmonics in the exterior of a ball. Theorem 8.5.3 (Expansion for the Outer Space of a Ball). Let U be of class  q q C.2/ Rq nBR , such that U CU D 0 in Rq nBR . There exist sequences of spherical .i / harmonics fYn .qI /gn2N0 , i D 1; 2, such that U.x/ D

1 X  .1/  Hn .qI r/Yn.1/ .qI / C Hn.2/ .qI r/Yn.2/ .qI / ;

(8.5.19)

nD0

x D r;  2 Sq1 , where the series (8.5.19) is absolutely and uniformly convergent q for all x 2 Br with r D jxj > R. q Conversely, if (8.5.19) holds uniformly for all x 2 Rq n Br with r D jxj > R,   q q then U is of class C.2/ Rq n BR satisfying U C U D 0 in Rq n BR .

8.5 Expansion Theorems

389

Proof. The functions Un;j , j D 1; : : : ; N.qI n/, given by Z Un;j .r/ D

Sq1

U.r/ Yn;j .qI / dS./

(8.5.20)

q

are defined for all Rq n Br , r > R, and satisfy the Bessel differential equations 00 .r/ Un;j

  q1 0 n.n C q  2/ Un;j .r/ C 1  Un;j .r/ D 0: C r r2

(8.5.21)

.i /

Hence, there are coefficients An;j , i D 1; 2, such that, for all r > R, .1/

.2/

Un;j .r/ D An;j Hn.1/ .qI r/ C An;j Hn.2/ .qI r/:

(8.5.22)

It follows that Z

1 N.qIn/ ˇ2 X X ˇˇ .1/ ˇ .2/ jU.r/j dS./ D ˇAn;j Hn.1/ .qI r/ C An;j Hn.2/ .qI r/ˇ : 2

Sq1

(8.5.23)

nD0 j D1

Therefore, we are able to verify that the series 1 N.qIn/  X X  .1/ .2/ An;j Hn.1/ .qI r/ C An;j Hn.2/ .qI r/ Yn;j .qI /

(8.5.24)

nD0 j D1 q

is absolutely and uniformly convergent on R3 n Br for all r > R. Letting Yn.i / .qI / D

N.qIn/

X

.i /

Cn;j Yn;j .qI /;

i D 1; 2;

(8.5.25)

j D1

we finally get 1 X  .1/  Hn .qI r/Yn.1/ .qI / C Hn.2/ .qI r/Yn .qI / ; U.x/ D

(8.5.26)

nD0

as required for the first part of Theorem 8.5.3. The second part follows by analogous arguments as described in the proof of Theorem 8.5.1. t u Evidently, for entire solutions, both theorems (i.e., Theorem 8.5.1 as well as Theorem 8.5.3) hold true.

8 Bessel Functions in Rq

390

8.6 Exercises (Helmholtz Equation, Entire Solutions, Bessel Function Like Asymptotics) In the following, we deal with a truncated series expansion of type (8.5.1) as preparation for integral representations that serve as entire solutions of the Helmholtz equation.

Entire Solutions of the Helmholtz Equation and Asymptotic Expansions Exercise 8.6.1. Let U .N / be given by U .N / .x/ D .2/

1q 2

kSq1 k

N X

in Jn .qI r/Yn .qI /

(8.6.1)

nD0

with x D r, r D jxj,  2 Sq1 , Yn .qI / 2 Harmn .Sq1 / and N 2 N0 . Show that U .N / is an entire solution of the Helmholtz equation U .N / C U N D 0

(8.6.2)

in Rq . Exercise 8.6.2. Let U .N / be given by (8.6.1). Use the asymptotic expansion of the regular Bessel functions to prove that r

q1 2





U .N / .r/ D ei.r.q1/ 4 / F .N / .qI / C ei.r.q1/ 4 / F .N / .qI / C o.1/; (8.6.3)

where F .N / .qI / D

N X

Yn .qI /:

(8.6.4)

nD0

Exercise 8.6.3. Show by use of the asymptotic expansions for the regular Bessel functions that  r  q1 Z 1q q1 2 eir dS./ D i 2 eir C i 2 eir C o.1/ 2 Sq1 holds for r ! 1 and uniformly with respect to all  2 Sq1 . Exercise 8.6.4. Show the following generalization of Lemma 8.3.4:

(8.6.5)

8.6 Exercises

391

Let U be an entire solution of the Helmholtz equation U C U D 0 in Rq . Then, Z  r  q1 1q q1 2 U.R/eir dS./ D i 2 eir U.R/ C i 2 eir U.R/ C o.1/ 2 Sq1 (8.6.6) holds true for arbitrary R > 0 and r ! 1 uniformly with respect to all  2 Sq1 . Exercise 8.6.5. Let F be of class C.k/ .Sq1 /, k 

q 2

C 1. Prove that

 r  q1 Z 1q q1 2 F ./eir dS./ D i 2 eir F ./Ci 2 eir F ./Co.1/ 2 Sq1

(8.6.7)

holds for r ! 1 and uniformly with respect to all  2 Sq1 . Exercise 8.6.6. Verify the following statement: q Let U be a solution of the Helmholtz equation U C U D 0 in Rq n BC for some C > 0. Suppose that (8.6.6) of Exercise 8.6.4 is valid for arbitrary R > 0 and r ! 1 uniformly with respect to all  2 Sq1 . Then, U is an entire solution of U C U D 0 in Rq . Remark 8.6.7. More details can be found in M¨uller (1952, 1969) and Hartman and Wilcox (1961) and the references therein.

Chapter 9

Lattice Functions in R

The Euler summation formula expresses a finite sum of integral points in terms of the integral and derivatives of the function with explicit knowledge of the error in integral form involving the Bernoulli polynomial. The classical summation formula due to Euler (1736a,b) and MacLaurin (1742) is of great importance in many branches of periodic modeling and simulation, in lattice point summation, in constructive approximation, and in approximate integration, which motivates us to include it in this textbook. In this chapter we lay the one-dimensional foundation which is generalized to higher dimensions in Chap. 10. Our considerations show that the Euler summation formula (Sect. 9.4) and the Poisson summation formula (Sect. 9.6) on finite intervals Œa; b of R are equivalent. In consequence, periodization can be understood equivalently from both approaches. This part of the chapter is a condensed version of the material presented by Freeden (2011). The asymptotic criteria for the validity of the Poisson summation formula in all of R are strongly influenced by the notes of Mordell (1928, 1929) and their extensions to the multi-dimensional case derived in Freeden (2011). Our investigations are followed by exercises concerned with aspects in numerical integration by virtue of trapezoidal rules and constructive approximation by means of projections to polynomial spaces within one-dimensional periodic Sobolev spaces in Sect. 9.8.

9.1 Bernoulli Polynomials We start with the classical introduction of Bernoulli polynomials by the recursion relation d  B .x/ D .k C 1/Bk .x/; dx kC1

x 2 Œ0; 1;

W. Freeden and M. Gutting, Special Functions of Mathematical (Geo-)Physics, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-0348-0563-6 9, © Springer Basel 2013

(9.1.1)

395

9 Lattice Functions in R

396

for k 2 N, corresponding to the “initial function” B0 W R ! R given by B0 .x/ D 1 ;

x 2 Œ0; 1:

(9.1.2)

It follows that Bk W Œ0; 1 ! R, k 2 N, is a polynomial of degree k that can be written in the form Bk .x/ D x k C ck1 x k1 C : : : C c0 :

(9.1.3)

 We notice that, for known Bk .x/, (9.1.1) determines BkC1 .x/ up to an additive integration constant. By convention, we choose the integration constants in such a way that   B2kC1 .0/ D B2kC1 .1/ D 0 (9.1.4)

for all k 2 N. In doing so we get uniqueness in the determination of all Bk (see, e.g., Magnus et al. 1966; Rademacher 1973). This fact becomes clear from the observation that  B2k1 .x/ D x 2k1 C c2k2 x 2k2 C : : : C c1 x C c0

(9.1.5)

implies  B2kC1 .x/ D x 2kC1 C

.2k C 1/.2k/ c2k2 x 2k C : : : C .2k C 1/cx C d; (9.1.6) .2k/.2k  1/

  where the conditions B2kC1 .0/ D B2kC1 .1/ D 0 uniquely determine the constants c and d . The resulting polynomials (of degrees k D 1; 2; 3; 4) defined on the interval Œ0; 1 read as follows:

B1 .x/ D x  12 ;

(9.1.7)

B2 .x/ D x 2  x C 16 ;

(9.1.8)

B3 .x/ D x 3  32 x 2 C 12 x; B4 .x/

D x  2x C x  4

3

2

(9.1.9) 1 : 30

(9.1.10)

Definition 9.1.1. The polynomials Bk W Œ0; 1 ! R, k 2 N0 , as described by the conditions (9.1.1), (9.1.2), and (9.1.4), are called Bernoulli polynomials. The reals Bk .0/ are called the Bernoulli numbers. The Bernoulli polynomials satisfy the symmetry relation .1/k Bk .1  x/ D Bk .x/ ;

x 2 Œ0; 1:

(9.1.11)

In other words, the polynomials x 7! .1/k Bk .1  x/, x 2 Œ0; 1, obey the same recursion formula (9.1.1). Moreover, (9.1.4) remains valid.

9.1 Bernoulli Polynomials

397

Furthermore, (9.1.11) tells us that the relation Bk .0/ D Bk .1/, k 2 N, that holds true for odd k, is also valid for even k. Hence, it follows that for all k 2 N: Z

1 0

Bk .t/ dt D

 1    BkC1 .1/  BkC1 .0/ D 0: kC1

(9.1.12)

Lemma 9.1.2. For n 2 N, we have that 1  .x/ > 0, where (i) .1/n B  2n1  0 < x < 2, n   (ii) .1/ B2n .x/  B2n .0/ > 0, where 0 < x < 1,  (iii) .1/nC1 B2n .0/ > 0.

Proof. We verify .i / by induction. Clearly, .i / is true for n D 1. Now, let it be true for n 2 N. We have to show .i / for n C 1. To this end we notice that the Bernoulli   .0/ D B2nC1 . 12 / D 0 which follows from (9.1.4) and (9.1.11). polynomials B2nC1 Because of (9.1.1) we obtain .1/n

d2   B .x/ D .2n C 1/.2n/.1/n B2n1 .x/ > 0; dx 2 2nC1

(9.1.13)

 is strictly convex where we use the induction assumption. Therefore, .1/n B2nC1 1  .x/ < 0 in .0; 2 /. Together with the values at the boundary this implies .1/n B2nC1 1 for x 2 .0; 2 /, i.e., we obtain .i /. Now, we know that .i / holds true for all n 2 N, 0 < x < 12 . Then, because of (9.1.1), Z  .1/n   .1/n x d   B2n .x/  B2n B .t/ dt .0/ D 2n 2n 0 dx 2n Z x  B2n1 .t/ dt > 0: (9.1.14) D .1/n 0

In connection with (9.1.11) it then follows that (9.1.14) is also valid for x 2 . 12 ; 1/ such that (ii) becomes valid. Even more, by aid of (9.1.12) Z 1      B2n .t/  B2n .1/nC1 B2n .0/ D .1/n .0/ dt > 0 (9.1.15) 0

t u

which shows relation (iii).

By use of the floor function x 7! bxc we are able to extend the Bernoulli polynomial Bn of degree n defined on Œ0; 1 to the so-called Bernoulli function Bn of degree n. Note that bxc denotes that integer for which bxc  x < bxc C 1. Definition 9.1.3. The function Bn W R ! R given by Bn .x/ D Bn .x  bxc/ ; is called the Bernoulli function of degree n.

x 2 R;

(9.1.16)

9 Lattice Functions in R

398

Clearly, Bk .0/ D Bk .0/ D Bk .1/ D Bk .1/, i.e., Bk is a piecewise polynomial of degree k and it is Z-periodic. As usual, we set Bk D Bk .0/. More explicitly, B1 D  12 ;

(9.1.17)

B2 D

1 6;

(9.1.18)

B3 D 0;

(9.1.19)

B4 D 

(9.1.20)

1 : 30

9.2 Periodic Polynomials Let Z denote the one-dimensional lattice of integral points, i.e., the additive group of points in R having integral coordinates (the addition being, of course, the one derived from the vector structure of R). The fundamental cell F of the integer lattice Z is given by     1 1 : F D  12 ; 12 D x 2 R W   x < 2 2

(9.2.1)

Definition 9.2.1. A function F W R ! C is called Z-periodic if F .x C g/ D F .x/

(9.2.2)

holds for all x 2 F and g 2 Z. Definition 9.2.2. The function ˚h W R ! C; h 2 Z; given by x 7! ˚h .x/ D e2ihx

(9.2.3)

is Z-periodic: ˚h .x C g/ D e2ih.xCg/ D e2ihx e2ihg D e2ihx D ˚h .x/

(9.2.4)

for all x 2 F and all g 2 Z. These functions are called Z-periodic polynomials. .m/

The space of all F 2 C.m/ .R/ that are Z-periodic is denoted by CZ .R/, .0/ 0  m  1, we write CZ .R/ for CZ .R/. L2Z .R/ is the space of all F W R ! C that are Z-periodic and are Lebesgue-measurable on F with Z kF kL2 .R/ D Z

jF .x/j dx 2

F

 12

< 1:

(9.2.5)

9.2 Periodic Polynomials

399

Clearly, the space L2Z .R/ is the completion of CZ .R/ with respect to the norm k  kL2 .R/ , i.e., Z

L2Z .R/ D CZ .R/

kkL2 .R/ Z

:

(9.2.6)

An easy calculation shows that the system f˚h gh2Z is orthonormal with respect to the L2Z .R/-inner product ( Z 1 ; h D h0 ; (9.2.7) ˚h .x/˚h0 .x/ dx D ıh;h0 D h˚h ; ˚h0 iL2 .R/ D Z 0 ; h ¤ h0 : F In more detail, Z

Z F

˚h .x/˚h0 .x/ dx D

1 2

0

e2ihx e2ih x dx D

 12

Z

1 2

 12

0

e2i.hh /x dx D ıh;h0 :

(9.2.8) d An elementary calculation yields (with rx D dx as the one-dimensional gradient d2 and x D dx 2 as the one-dimensional Laplacian) rx ˚h .x/ D

d d 2ihx ˚h .x/ D e D 2ih ˚h .x/ dx dx

(9.2.9)

such that x ˚h .x/ D

d2 ˚h .x/ D .2ih/2 ˚h .x/ D 4 2 h2 ˚h .x/; dx 2

h 2 Z; x 2 R: (9.2.10) By convention we say that  is an eigenvalue of the lattice Z with respect to the operator  of the second order derivative (i.e., the one-dimensional Laplace operator), if there is a non-trivial solution U of the differential equation . C /U D 0

(9.2.11)

satisfying the “boundary condition” of periodicity U.x C g/ D U.x/

(9.2.12)

for all x 2 F and g 2 Z. From classical Fourier analysis (see, e.g., Hilbert 1912; Courant and Hilbert 1924) we know that the eigenspectrum of the operator  (with respect to Z) is given by ˚ Spect .Z/ D 4 2 h2 W h 2 Z :

(9.2.13)

The orthonormal system f˚h gh2Z of (eigen)functions ˚h W x 7! ˚h .x/ D e2ihx , where x 2 R, is closed in the space CZ .R/, i.e., for every " > 0 and every F 2 CZ .R/, there exist an integer N D N."/ > 0 and a linear combination

9 Lattice Functions in R

400

X

ah ˚h

(9.2.14)

h2Z\B1N

ˇ ˇ X ˇ ˇ sup ˇF .x/  ah ˚h .x/ˇ  ":

such that

x2F

(9.2.15)

h2Z\B1N

By virtue of the norm estimate Z kF kL2 .R/ D Z

jF .x/j dx 2

F

 12

 sup jF .x/j D kF kCZ .R/ ;

F 2 CZ .R/;

x2F

(9.2.16)   in the  closure of the system f˚h gh2Z in CZ .R/; k  kCZ .R/ implies the closure  CZ .R/; k  kL2 .R/ . By standard arguments we obtain the completeness in L2Z .R/; Z  k  kL2 .R/ . Z After these preliminaries on Z-periodic functions (i.e., functions with period 1) we now come to the definition of the Z-lattice function with respect to the oned dimensional Laplacian  D r 2 ; r D dx , i.e., Green’s function with respect to  corresponding to Z-periodic “boundary conditions”. Based on the constituting properties of this function the Euler summation formula can be developed by integration by parts. The Poisson summation formula is seen to be an equivalent tool for lattice point summation over finite intervals.

9.3 Lattice Functions We start with the definition of the Z-lattice function for the Laplacian. Definition 9.3.1. A function G.I / W R ! R is called the Green function for the operator  with respect to the Z-periodicity (in brief, Z-lattice function for ) if it satisfies the following properties: (i) (Periodicity) G.I / is continuous in R, and G.I x C g/ D G.I x/

(9.3.1)

for all x 2 R and g 2 Z. (ii) (Differential equation) G is twice continuously differentiable with G.I x/ D 1 for all x … Z.

(9.3.2)

9.3 Lattice Functions

401

(iii) (Characteristic singularity) 1 x 7! G.I x/  x sign.x/ 2

(9.3.3)

is continuously differentiable for all x 2 F . (iv) (Normalization) Z G.I x/ dx D 0:

(9.3.4)

F

First, we prove that the Z-lattice function for the operator of the second order derivative, i.e., for the one-dimensional Laplace operator, is uniquely determined by its constituting properties. Lemma 9.3.2. G.I / is uniquely determined by the properties (i)–(iv). Proof. Denote by D.I / the difference between two Z-lattice functions for . Then, we have the following properties: (i) D is continuous in R and, for all x 2 R and g 2 Z, D.I x C g/ D D.I x/:

(9.3.5)

(ii) D.I / is twice continuously differentiable for all x … Z with D.I / D 0:

(9.3.6)

(iii) D.I / is continuously differentiable in R. (iv) Z D.I x/ dx D 0:

(9.3.7)

F

The properties (i)–(iii) show that D.I / is a constant function. The last condition .i v/ shows us that the constant must be zero. Thus, G is uniquely determined. u t It is not hard to see (see, e.g., Magnus et al. 1966) that G.I / is explicitly available in elementary form. In fact, the function x 7! G.I x/ D 

x  bxc 1 .x  bxc/2 C  ; 2 2 12

x2R

(9.3.8)

satisfies all defining properties .i /–(iv) of the Z-lattice function for the one-dimensional Laplace operator . Apart from a multiplicative constant, G.I / coincides with the Bernoulli function B2 of degree 2 given by x 7! B2 .x/ D B2 .x  bxc/ D .x  bxc/2  .x  bxc/ C

1 : 6

(9.3.9)

9 Lattice Functions in R

402 Fig. 9.1 The illustration of the Z-lattice function G.I / for 

y

-1/2

1/2

x

Theorem 9.3.3. The Z-lattice function G.I / for  possesses the explicit representation G.I x/ D 

x  bxc 1 .x  bxc/2 C  ; 2 2 12

x 2 R:

(9.3.10)

Observing the Z-periodicity (i) and the characteristic singularity (iii) of the Z-lattice function for , we obtain by applying integration by parts Z hG.I /; ˚h iL2 .R/ D Z

F

G.I x/˚h .x/ dx D

1 ; 4 2 h2

(9.3.11)

provided that h ¤ 0. Thus, the classical representation theorem of one-dimensional Fourier theory gives us the pointwise Fourier series representation. Lemma 9.3.4. For all x 2 R, G.I x/ D

X h2Znf0g

X 1 1 ˚ .x/ D ˚h .x/: h 4 2 h2 4 2 h2

(9.3.12)

h2Znf0g

The series on the right-hand side of (9.3.12) is absolutely and uniformly convergent on each compact interval I  R. Remark 9.3.5. Note that we are formally able to write (see (9.3.2)of Definition 9.3.1) G.I x/ D ı.x/  1; where ı.x/ D

X

˚h .x/

(9.3.13)

(9.3.14)

h2Z

has to be understood in distributional sense. Figure 9.1 gives an illustration of the Z-lattice function for . Obviously, for all x 2 R, we have

9.3 Lattice Functions

G.I x/ D

403

1 X

1  2inx  1 X 1 1 2inx e D  C e cos.2nx/: (9.3.15) 4 2 n2 2 2 nD1 n2 nD1

Moreover, the following property of G.I / should be noted: G.I / is a piecewise polynomial of degree 2 and we have G.I g/ D 

1 ; 12

g 2 Z:

(9.3.16)

Since the Fourier series of G.I / converges absolutely and uniformly for all x 2 R, elementary differentiation yields 1 rx G.I x/ D G.rI x/ D  .x  bxc/ C ; 2

x 2 R n Z:

(9.3.17)

Remark 9.3.6. In the literature the function G.rI / is known as the Bernoulli function B1 of degree 1 given by x 7! B1 .x/ D x  bxc  12 , x 2 R (see Definition 9.1.3). The Fourier series of G.rI / reads X

G.rI x/ D

h2Znf0g

1 ˚h .x/; 2ih

(9.3.18)

where the equality is understood in the L2Z .R/-sense. Using the well-known representation of the sin-function, we find X h2Znf0g

  X 1 1 X 1 1 e2inx e2inx sin.2nx/ D ˚h .x/ D  : 2ih 2i n n n nD1 nD1

(9.3.19)

We mention the following result concerning the pointwise convergence of the series (9.3.19) and its explicit representation (see Fig. 9.2). Lemma 9.3.7. The series (9.3.19) converges uniformly in each compact interval I  .g; g C 1/; g 2 Z. Moreover, for x 2 I with I a compact subset of .g; g C 1/; g 2 Z, we have 1 d  .x  bxc  / D 2 dx „ ƒ‚ …



1 X cos.2nx/ nD1

2 2 n2

! D

1 X sin.2nx/ nD1

n

:

(9.3.20)

DB1 .x/

The proof can be given by partial summation (see, e.g., Rademacher 1973).

9 Lattice Functions in R

404 Fig. 9.2 The derivative G.rI x/ D rG.I x/ of the Z-lattice function G.I x/

y

1/2 -1/2 x

1/2

-1/2

9.4 Euler Summation Formula First, we want to verify the Euler summation formula in its classical form related to the one-dimensional operator of the first order derivative, i.e., the gradient r, and the one-dimensional operator of the second order derivative, i.e., the Laplace operator , respectively. A particular role is played by the Z-lattice function for  as introduced by Definition 9.3.1. We begin with the formulation of the classical one-dimensional Euler summation formula. Theorem 9.4.1. If F 2 C.1/ .Œ0; n/; n 2 N, then Z

X

n

F .g/ D 0

g2Z\Œ0;n

 1 F .x/ dx C F .n/ C F .0/  2

Z

n

G.rI x/ rF .x/ dx : 0

(9.4.1) If F 2 C.2/ .Œ0; n/; n 2 N, then Z

X

F .g/ D

n

F .x/ dx C 0

g2Z\Œ0;n

 1 F .n/ C F .0/ 2

 1 .rF /.n/  .rF /.0/ C C 12

(9.4.2) Z

n

G.I x/ F .x/ dx: 0

Let F W Œa; b ! C; a < b; be a twice continuously differentiable function, i.e., we require F 2 C.2/ .Œa; b/. Then, X0

Z F .g/ D

g2Z\Œa;b

Z

b

b

F .x/ dx C a

G.I x/ F .x/ dx a

C .F .x/.rG.I x//  G.I x/ .rF .x///jba ;

(9.4.3)

9.4 Euler Summation Formula

where

X0

405

F .g/ D

g2Z\Œa;b

X

F .g/ C

g2Z\.a;b/

1 2

X

F .g/;

(9.4.4)

g2Z\fa;bg

and the last sum in (9.4.4) occurs only if a and/or b are lattice points. Otherwise, it is assumed to be zero. Proof. We restrict ourselves to the proof of the formula (9.4.3). First, we are concerned with the case that both endpoints a; b are non-integers. Applying integration by parts we get, for every (sufficiently small) " > 0, Z .G.I x/F .x/  F .x/G.I x// dx (9.4.5) x2Œa;bnA.1;Z;"/

ˇb D .G.I x/ .rF .x//  F .x/ .rG.I x/// ˇa X ˇg" C .G.I x/ .rF .x//  F .x/ .rG.I x/// ˇgC" ; g2Z\.a;b/

S where we use the notation A.1; Z; "/ D g2Z B1" .g/. By virtue of the differential equation G.I x/ D 1; x 2 R n Z, it follows that Z

Z F .x/ G.I x/ dx D  x2Œa;bnA.1;Z;"/

F .x/ dx :

(9.4.6)

x2Œa;bnA.1;Z;"/

By letting " ! 0 and observing the (limit) values of the Z-lattice function for  and its derivatives in the lattice points, we obtain X

Z F .g/ D a

g2Z\.a;b/

b

Z

b

F .x/ dx C

G.I x/ F .x/ dx

(9.4.7)

a

ˇb C .F .x/ .rG.I x//  G.I x/ .rF .x/// ˇa : Second, the cases where a and/or b are integers, i.e., members of the lattice Z, follow by obvious modifications, where .G.rI x/F .x//jba is understood in the usual sense of one-sided limits: G.rI b/F .b/  G.rI aC/F .a/: This is the desired result.

(9.4.8) t u

Extended Stirling’s Formula Involving the Lattice Function Next, we come to a generalization of Stirling’s formula by use of Euler summation: Let F be of class C.1/ .Œ0; n/. Then, we know from Theorem 9.4.1 that

9 Lattice Functions in R

406 n X

F .k/ 

kD0

1 .F .0/ C F .n// D 2

Z

Z

n

n

F .x/ dx  0

.rG.I x// .rF .x// dx; 0

(9.4.9) i.e., the Euler Œ0; n with respect to the operator  summation formula on the interval d rx D dx : In particular, with F .x/ D .1 C x/1 ; x 2 Œ0; n, we get n X kD0

1 1 1 D C C ln.n C 1/ C kC1 2.n C 1/ 2

Z

n

G.rI x/ .1 C x/2 dx: (9.4.10)

0

An immediate consequence is the following representation of Euler’s constant in terms of the function G.rI /:  D lim

n!1

! Z 1 n X 1 1  ln.n/ D C G.rI x/ .1 C x/2 dx: k 2 0

(9.4.11)

kD1

For x > 0, we know from Lemma 2.4.3 that 

1 ln  .x/

 D lim

x

n!1

ln n

x

!! n  Y kCx k

kD1

:

(9.4.12)

This is equivalent to ln . .x// D lim

x ln.n/ C

n!1

n1 X

ln.1 C k/ 

kD0

n X

! ln.x C k/ :

(9.4.13)

kD0

Now, again with the above variant (9.4.9) of the Euler summation formula, we obtain n X

Z

n

ln.xCk/ D 0

kD0

1 ln.xCt/ dt C .ln.x C n/ C ln.x// 2

Z

n

G.rI t/.xCt/1 dt:

0

(9.4.14) Thus, elementary calculations yield, for x > 0, Z

1

ln. .x// D 1  0

Z

1

C

  1 ln.x/ G.rI t/.1 C t/1 dt  x C x  2

G.rI t/.x C t/1 dt:

(9.4.15)

0

The integral

R1 0

G.rI t/.x C t/1 dt exists for x > 0. Even more, for x > 0, we get

9.4 Euler Summation Formula

407

 0 .x/ 1 d ln . .x// D D ln.x/   dx  .x/ 2x

Z

1

G.rI t/ .x C t/2 dt: (9.4.16)

0

In connection with (9.4.11) and Lemma 2.1.4 we obtain Lemma 9.4.2 (Euler’s Constant). Z

0

 D  .1/ D 

1

et ln.t/ dt:

(9.4.17)

0

Furthermore, we have Z

1

G .rI t/ .x C t/1 dt D O

0

  1 x

(9.4.18)

for x ! 1. This leads us to the formula     Z 1 1 1 ln.x/ C O : ln. .x// D 1  G.rI t/.1 C t/1 dt  x C x  2 x 0 (9.4.19) From Stirling’s formula (Theorem 2.3.1) we get     p 1 1 ln. .x// D ln. 2/ C x  ln.x/  x C O : 2 x Thus, by combination we find Z 1 p G.rI t/.1 C t/1 dt D ln. 2/: 1

(9.4.20)

(9.4.21)

0

All in all, we have eln. .x// D eln.

p R1 1 2/ x ln.x x 2 / 0 G.rIt /.xCt /1 dt

e

e

e

:

(9.4.22)

Consequently, we get the following extension of Stirling’s formula involving the derivative G.rI / of the lattice function G.I /. Theorem 9.4.3. For x > 0, R1 p 1 1  .x/ D 2 x x 2 ex e 0 G.rIt /.xCt / dt :

(9.4.23)

Euler Summation to Periodic Boundary Conditions In the following we want to present an extension of the Euler summation formula. This can be achieved by replacing the integer lattice Z by a “translated lattice” Z C fxg D fg C x W g 2 Zg based at x 2 R.

9 Lattice Functions in R

408

Corollary 9.4.4. Let x be a point of R. Suppose that F is of class C.2/ .Œa; b/. Then, X0

Z

Z

b

F .g C x/ D

F .y/ dy C a

gCx2Œa;b g2Z

b

G.I x  y/ y F .y/ dy

(9.4.24)

a

  ˇb C F .y/.ry G.I x  y//  .ry F .y//G.I x  y/ ˇa :

An easy consequence is the following corollary (note that the operators ry D and y D ry2 D

d2 dy 2

d dy

apply to the y-variable).

Corollary 9.4.5. Let x be a point of the fundamental cell F of the lattice Z. Suppose that F is of class C.2/ .F /. Then, Z

Z F .y/ dy C

F .x/ D F

F

G.I x  y/ y F .y/ dy

(9.4.25)

  ˇ1   C F .y/ ry G.I x  y/  .ry F .y//G.I x  y/ ˇ2 1 : 2

Next, we mention the following result involving Z-periodic functions. .2/

Corollary 9.4.6. Assume that x 2 F and F 2 CZ .R/. Then, Z Z F .x/ D F .y/ dy C G.I x  y/ y F .y/ dy; F

(9.4.26)

F

Now, we introduce the lattice function to the iterated Laplace operator. Definition 9.4.7. The lattice function to the iterated Laplace operator 2 (in brief, Z-lattice function for 2 ) G.2 I / W x 7! G.2 I x/, x 2 R, is defined by periodic convolution   G 2 I x  y D G.I x  /  G.I   y/ (9.4.27) Z D G.I x  z/G.I z  y/ dz: F

Obviously, the bilinear series of G.2 I x  y/ reads as follows: X X ˚h .x/ ˚k .y/ Z G.2 I x  y/ D ˚h .z/˚k .z/ dz 4 2 h2 4 2 k 2 F k2Znf0g h2Znf0g

D

X

X

k2Znf0g h2Znf0g

D

X h2Znf0g

˚h .x/ ˚k .y/ ıh;k 4 2 h2 4 2 k 2

1 ˚h .x  y/: .4 2 h2 /2

(9.4.28)

9.4 Euler Summation Formula

Moreover, we have

409

  y G 2 I x  y D G.I x  y/:

(9.4.29)

It follows from Corollary 9.4.6 that Z

Z



F .y/ dy C

F .x/ D F

F

  y G.2 I x  y/ y F .y/ dy:

(9.4.30)

By virtue of the Cauchy–Schwarz inequality we get from (9.4.30) ˇ Z ˇ ˇF .x/  ˇ

F

ˇ Z ˇ    1 F .y/ dy ˇˇ  G 2 I 0 2

jF .y/j2 dy

 12 :

(9.4.31)

F

Corollary 9.4.6 leads back to the well-known solutions of the following differential equations corresponding to periodic boundary conditions. Later on, the formula (9.4.30) enables us to introduce periodic splines comparable to the cubic splines of ordinary algebraic polynomial theory (see Sect. 10.9). .2/

Corollary 9.4.8. Assume that F is of class CZ .R/ satisfying F .y/ D 0, y 2 F . Then, F is a constant function. More explicitly, Z F .x/ D

F .y/ dy; F

x 2 F:

(9.4.32)

Corollary 9.4.9. Assume that H is of class CZ .R/ such that Z H.y/ dy D 0:

(9.4.33)

F .2/

Let F 2 CZ .R/ satisfy F .y/ D H.y/, y 2 F , such that Z F .y/ dy D 0:

(9.4.34)

F

Then,

Z G.I x  y/H.y/ dy;

F .x/ D F

x 2 F:

(9.4.35)

Next, we mention the explicit representation of the lattice function G.2 I / for 2 : x 7! G.2 I x/, x 2 R. To this end, we observe that   1 X 1 G 2 I x D cos.2hx/; 8 4 h4 h2N

(9.4.36)

9 Lattice Functions in R

410

such that (see (9.4.31)) G.2 I 0/ D

1 : 720

(9.4.37)

  Lemma 9.4.10. The function G 2 I  with respect to the operator 2 D r 4 is Z-periodic, twice continuously differentiable in R such that   G 2 I x D G.I x/;

x 2 R;

(9.4.38)

and we have   1 1 1 1 : (9.4.39) G 2 I x D  .x  bxc/4 C .x  bxc/3  .x  bxc/2 C 24 12 24 720 In analogy to (9.4.27) we define the l-iterated Z-lattice functions G.l I /, l 2 N, by   G lC1 I x  y D

Z

  G l I x  z G.I z  y/ dz;

x; y 2 R:

F

Clearly, we have

  G lC1 I x D G.l I x/;

x 2 R;

(9.4.40)

(9.4.41)

so that the l-iterated Z-lattice function is nothing more than the Z-lattice function to the iterated operator l ; X   G l I x D h2Znf0g

1 ˚h .x/; .4 2 h2 /l

x 2 R:

(9.4.42)

Furthermore, it is clear that     rG l I x D G r 2l1 I x ;

x 2 R;

(9.4.43)

for every l 2 N. An easy calculation shows that (cf. Magnus et al. 1966) X cos.2hx/   G l I x D 2.1/l .2h/2l

(9.4.44)

X sin.2hx/   rG l I x D 2.1/l1 ; .2h/2l1

(9.4.45)

h2N

and

h2N

such that

  G r k I x D 2

1 X 1  : cos 2hx  k .2/k hk 2 h2N

(9.4.46)

9.5 Riemann Zeta Function

411

  Remark 9.4.11. The functions Bk defined by Bk .x/ D  kŠ G r k I x ; x 2 R; are known as Bernoulli functions of degree k. In more detail, we have, by Definition 9.1.3, 1 B1 .x/ D B1 .x  bxc/ D x  bxc  ; 2

(9.4.47)

1 B2 .x/ D B2 .x  bxc/ D .x  bxc/2  .x  bxc/ C ; 6 3 1 B3 .x/ D B3 .x  bxc/ D .x  bxc/3  .x  bxc/2 C .x  bxc/; 2 2 B4 .x/ D B4 .x  bxc/ D .x  bxc/4  2.x  bxc/3 C .x  bxc/2 

(9.4.48) (9.4.49) 1 : 30 (9.4.50)

Bk is a piecewise polynomial of degree k and it is Z-periodic. Obviously, we have that     1 Bk D G r k I 1 ; (9.4.51) G rkI 0 D  kŠ where B2lC1 D 0 for l 2 N and B2l D 2.1/lC1 .2l/Š

 1  X 1 2l ; 2k

l 2 N:

(9.4.52)

kD1

Note that the integer values have to be exempted for the Bernoulli function of degree 1 because of the discontinuity, i.e., the characteristic singularity of the lattice   function G.rI / D rG r 2 I  for x 2 Z. For more details the reader is referred to, e.g., Magnus et al. (1966), Rademacher (1973) and the literature therein.

9.5 Riemann Zeta Function In number theory, the Riemann Zeta function has become the basis of the whole theory of the distribution of primes. In addition, it has significant relations to the lattice function which are of particular interest for us (for more details see, e.g., Mordell 1928; Titchmarsh 1951). In the following, the one-dimensional Riemann Zeta function is introduced in such a way that its extension to the multi-variate case (see Sect. 10.5) becomes transparent. For s 2 C with Re.s/ > 1, we consider the Riemann Zeta function  given by the series 1 X 1 s 7! .s/ D : (9.5.1) ns nD1

9 Lattice Functions in R

412

P Re.s/ The series is absolutely convergent since it can be dominated by 1 , nD1 n which shows us that the convergence of (9.5.1) is uniform in every half-plane fs 2 C W Re.s/  1 C ı; ı > 0g:

(9.5.2)

Thus,  is holomorphic in these half-planes (note that we define x s , where s 2 C with Re.s/ > 0, as es ln.x/ with  2 < Im.ln.s// < 2 ). For the function x 7! x1s ; x > 0; we have x

1 d2 1 D D s.s C 1/ x s2 ; xs dx 2 x s

x > 0:

(9.5.3)

The Euler summation formula (Theorem 9.4.1) yields X

0

n2Z\Œ%;N 

1 D ns



ˇ Z N 1 1 ˇN x 1s ˇ C s.s C 1/ G.I x/ sC2 dx (9.5.4) % 1s x % 

C

ˇ 1 1 ˇN .rG.I x// C s G.I x/ ˇ s sC1 % x x

for all % 2 R with 0 < %  1. The explicit representation of the lattice function gives   ˇN 1 1 ˇˇN 1 ˇ rG.I x/ˇ D s .x  bxc/ C (9.5.5) ˇ % xs x 2 % and s

  ˇN 1 x  bxc 1 ˇˇN .x  bxc/2 ˇ G.I x/ D s C   ˇ : ˇ % x sC1 x sC1 2 2 12 % 1

(9.5.6)

jG.rI /j and jG.I /j are bounded such that the terms (9.5.5) and (9.5.6) do not contribute as N ! 1. For s 2 C with Re.s/ > 1, we obtain .s/ D

Z 1 1 1 %1s C s.s C 1/ G.I x/ sC2 dx s1 x %     2 1 % 1 1 1 % C s sC1  C : C s % % 2 % 2 2 12

(9.5.7)

With the help of the Fourier expansion of G.I / (see Lemma 9.3.4) the integral Z

N

G.I x/ %

1 x sC2

dx

(9.5.8)

9.5 Riemann Zeta Function

413

permits the representation 

X h2Znf0g

Therefore, the integral

Z

1 4 2 h2

Z

N %

1

G.I x/ %

1 2ihx e dx: x sC2

(9.5.9)

1 dx x sC2

(9.5.10)

converges not only in the half-plane of all s 2 C with Re.s/ > 1, but also for all s 2 C with Re.s/ > 1. Thus, (9.5.7) furnishes an analytic continuation of  into the half-plane fs 2 C W Re.s/ > 1g showing the pole at s D 1 as the only singularity. In addition, the expression (9.5.8) is convergent as % ! 0, provided that Re.s/ < 0. Hence,  can be continued by (9.5.7) to any point in the s-plane and  emerges a meromorphic function with the simple pole s D 1. Even more, we are able to formulate the following lemma. Lemma 9.5.1. For s 2 C with 1 < Re.s/ < 0, we have Z

1

.s/ D s.s C 1/

G.I x/ 0

1 dx: x sC2

(9.5.11)

Now, if s 2 C with 1 < Re.s/ < 0, we get by integration by parts .s/ D

X Z h2Znf0g

1

e2ihx

0

1 dx: xs

(9.5.12)

For h 6D 0 we have (see, e.g., Freeden 2011) Z

1 0

8  < 1 1s  .1  s/e i2 .1s/ 1 2h 2ihx e dx D 1 1s i s : x  .1  s/e 2 .1s/ 2.h/

; h > 0; ; h < 0:

(9.5.13)

For all s 2 C with 1 < Re.s/ < 0, it follows that .s/ D 2s  s1  .1  s/ sin

1

s X 1 : 2 h1s

(9.5.14)

hD1

Now, the left-hand side of (9.5.14) is a meromorphic function with the only pole at s D 1, i.e., this equation provides an analytic continuation of the right-hand member as a meromorphic function over the whole s-plane and  appears as a meromorphic function with only the simple pole at s D 1 (cf. Titchmarsh 1951).

9 Lattice Functions in R

414

Theorem 9.5.2 (Functional Equation of the Riemann Zeta Function). The Zeta function  given by .s/ D

1 X 1 ; ns nD1

s 2 C; Re.s/ > 1;

(9.5.15)

can be extended analytically to a meromorphic function with the pole 1 s1

(9.5.16)

to the whole complex plane C. Moreover,  satisfies the functional equation .s/ D 2s  s1  .1  s/ sin

s 2

.1  s/:

(9.5.17)

The functional equation can be put into a more illuminating form if we make use of the duplication formula (Lemma 2.3.3) of the Gamma function   1 1 D 2 2 22s  .2s/:  .s/ s C 2

(9.5.18)

Replacing here 2s by 1  s, we obtain by using (2.4.32) for the second step 1 2

 .1  s/ 2 sin s

s 2

 D



s  . 1s / 1 s s 2  1  sin D : 2 2 2 2  . 2s / (9.5.19)

Thus, Theorem 9.5.2 can be reformulated into the following form. Corollary 9.5.3. Under the assumptions of Theorem 9.5.2, 

s 2

.s/ D 

s 12

 

1s 2

 .1  s/:

(9.5.20)

Remark 9.5.4. Note that the function given by s

.s/ D   2 

s 2

.s/

(9.5.21)

fulfills the functional equation .s/ D .1  s/:

(9.5.22)

9.6 Poisson Summation Formula for the Laplace Operator

415

The identity (9.5.7) is also valid for s 2 C with Re.s/ > 1 and we get from (9.5.7) 1 .0/ D  : 2

(9.5.23)

Even more, our considerations enable us to deduce that, for n 2 N; .n/ D  .2n/ D

BnC1 ; nC1 0;

(9.5.24) (9.5.25)

B2n : 2n

(9.5.26)

1  0 .0/ D  ln.2/: 2

(9.5.27)

.2n C 1/ D  In addition, an easy calculation yields

Clearly, .2n/ is proportional to the Bernoulli numbers B2n D .2n/Š G.n I 0/ (see (9.4.52) and (9.5.24)) .2n/ D .1/nC1

.2/2n 1 B2n D .4 2 /n G.n I 0/: 2.2n/Š 2

(9.5.28)

Furthermore, we have .2n C 1/ D 

1 B2n D .2n  1/Š G.n I 0/: 2n

(9.5.29)

The zeros of  at 2l, l 2 N, are often called the “trivial zeros”, as they are easily found. The role of the “trivial zeros” of  is evident in (9.5.20). For Re.s/ < 0, we have Re.1  s/ > 1, such that the right-hand member of (9.5.20) is regular. However,  . 2s / has poles for s D 2n, n 2 N, which are just neutralized by the zeros of . Only for s D 0, we have a pole of first order on both sides of (9.5.20) since .0/ D  12 .

9.6 Poisson Summation Formula for the Laplace Operator Next, we discuss some variants of the one-dimensional Poisson summation formula. First, we prove that the Euler summation formula and the Poisson summation formula is equivalent on finite intervals Œa; b  R.

9 Lattice Functions in R

416

Theorem 9.6.1. Let F be twice continuously differentiable, i.e., F 2 C.2/ .Œa; b/. Then, X XZ b 0 F .g/ D F .x/˚h .x/ dx (9.6.1) where

P0

g2Z\Œa;b

a

h2Z

is defined in (9.4.4).

Proof. Using the Fourier expansion of the Z-lattice function G.I / we get Z

b

X Z

G.I x/ F .x/ dx D

a

b

.F .x// a

h2Znf0g

˚h .x/ dx: 4 2 h2

(9.6.2)

Note that the Fourier expansion is absolutely and uniformly convergent in R, hence, summation and integration can be interchanged. As already pointed out in  d 2 d Sect. 1.1, partial integration gives us (with r D dx ,  D r 2 D dx , and .x C 4 2 h2 /˚h .x/ D 0, x 2 R): Z

b a

ˇb .F .x// ˚h .x/ dx D .rF .x//˚h .x/ˇa  .2ih/

Z

b

.rF .x// ˚h .x/ dx a

ˇb ˇ D .rF .x//˚h .x/  .2ih/F .x/˚h .x/ ˇ

a

Z

b

C.2ih/2

F .x/˚h .x/ dx:

(9.6.3)

a

Thus, we obtain from the context of the Euler summation formula (Theorem 9.4.1) Z a

b

ˇb G.I x/F .x/ dx D ..rF .x//G.I x/  F .x/rG.; x// ˇa C

X Z h2Znf0g

b

F .x/˚h .x/ dx:

(9.6.4)

a

Combining all results we get the Poisson summation formula X

0

Z

b

F .g/ D

F .x/ dx C a

g2Z\Œa;b

D as announced.

h2Znf0g

XZ h2Z

X Z

b

F .x/˚h .x/ dx

(9.6.5)

a

b

F .x/˚h .x/ dx; a

t u

9.6 Poisson Summation Formula for the Laplace Operator

417

Clearly, the arguments of the aforementioned proof in the backward direction immediately lead to the Euler summation formula, that shows the equivalence. Example 9.6.2. We know from the Poisson summation formula (Theorem 9.6.1) that X XZ b 0 1D ˚h .x/ dx: (9.6.6) g2Z\Œa;b

This gives us

X

Z

0

h2Z

b

1D

dx C a

g2Z\Œa;b

a

X Z h2Znf0g

b

˚h .x/ dx:

(9.6.7)

a

In particular, for Œa; b D ŒR; R;

R > 0;

(9.6.8)

we obtain as a one-dimensional counterpart of the so-called Hardy–Landau identity (see Freeden 2011) X

0

1 D 2R C 2R

g2Z\ŒR;R

X J1 .1I 2hR/ : 2hR

(9.6.9)

h2Znf0g

In other words, we have X

0

1 D 2R

g2Z\ŒR;R

X J1 .1I 2hR/ h2Z

2hR

;

(9.6.10)

where J1 .1I 2hR/ D 2hR

r

 J 12 .2I 2hR/ sin.2hR/ : p D J0 .3I 2hR/ D sinc.2hR/ D 2 2hR 2hR (9.6.11)

It should be noted that the Hardy–Landau series on the right-hand side of (9.6.9) is alternating. Nevertheless, the (pointwise) convergence of the alternating series (9.6.9) can be readily seen from the derived Poisson summation formula for the interval ŒR; R. Moreover, it is uniformly convergent on each compact interval I  .g; g C 1/; g 2 Z. Since the formulation of the Poisson summation formula (Theorem 9.6.1) on finite domains is not straightforward in higher dimensions (because of the behavior of the Fourier series of the lattice function), the convergence will bother us much more for the two- and certainly for higher-dimensional counterparts of the Hardy–Landau series. A way out will be found by formulating a Poisson summation formula corresponding to an adaptive “wave number”  of a suitably chosen Helmholtz operator  C . For more details concerning the twodimensional case the reader is referred to Sect. 10.9, multi-dimensional variants are discussed in Freeden (2011). Next, we discuss the Poisson summation formula for R (due to Mordell 1929).

9 Lattice Functions in R

418

Theorem 9.6.3. Let F be twice continuously differentiable in R. Moreover, suppose that F .x/ ! 0; x ! ˙ 1;

(9.6.12)

rF .x/ ! 0; x ! ˙ 1:

(9.6.13)

Furthermore, assume that the limits Z

Z F .x/ dx D lim

N !1 N

R

and

Z

Z jF .x/j dx D lim

N

N !1 N

R

exist. Then,

N

X

F .g/ D

g2Z

XZ h2Z

R

F .x/ dx

(9.6.14)

jF .x/j dx

(9.6.15)

F .x/˚h .x/ dx:

(9.6.16)

Theorem 9.6.3 admits a canonical generalization to the multi-dimensional case (see Chap. 10). A modification of Theorem 9.6.3 is formulated in the next corollary. Corollary 9.6.4. Let y ! 7 F .x C y/; x 2 R; satisfy the assumptions of Theorem 9.6.3. Then, XZ XZ X F .g C x/ D F .y/˚h .y/ dy ˚h .x/ D F .y/˚h .y/ dy ˚h .x/: g2Z

h2Z

R

h2Z

R

(9.6.17) Proof. From Theorem 9.6.3 it follows that X

F .g C x/ D

g2Z

XZ h2Z

R

F .x C y/˚h .y/ dy:

(9.6.18)

Clearly, under the assumptions of Theorem 9.6.3, Z

Z R

F .x C y/˚h .y/ dy D

This is the desired result.

Z R

F .z/˚h .z  x/ dz D ˚h .x/

R

F .y/˚h .y/ dy: (9.6.19) t u

9.7 Theta Function

419

9.7 Theta Function It is not difficult to see that, for x 2 R; 2 C; Re. / > 0; Z

e

jyxj2

e2ihy dy D e2ihx

Z

R

u2

e e2ihu du:

(9.7.1)

R

Thus, it follows that Z

e

jyxj2

e2ihy dy D

p 2 2  e2ihx e2  h :

(9.7.2)

R

In particular, for h D 0,

p1 

R R

e

jyxj2

dy D 1.

Lemma 9.7.1. For all x 2 R and all 2 C with Re. / > 0, X 1 X  jgxj2 2 2 p e D e  h e2ihx :  g2Z h2Z

(9.7.3)

Lemma 9.7.1 motivates us to introduce the Theta function (see also Mordell 1929) Definition 9.7.2. For all 2 C with Re. / > 0 and for all x; y 2 R, the function #.I x; yI Z/ given by #. I x; yI Z/ D

X

e jgxj e2igy 2

(9.7.4)

g2Z

is called the Theta function (of degree 0 and dimension 1). Obviously, for all 2 C with Re. / > 0, we set #. / D #. I 0; 0I Z/. Moreover, from Lemma 9.7.1 we are able to deduce the functional equation #





1  I x; 0I Z

1

D . / 2 # . I 0; xI Z/ :

(9.7.5)

More generally, in accordance with our previous considerations, the one-dimensional Poisson summation formula yields X

X  1 2 2 e jgxj e2igy D p e2ixy e jyChj e2ihy :

g2Z h2Z

(9.7.6)

This leads us to the following functional equation of the one-dimensional Theta function (see Theorem 10.8.4 for the generalization to dimension q).

9 Lattice Functions in R

420

Lemma 9.7.3 (Functional Equation of the Theta Function). The Theta function #.I x; yI Z/ is holomorphic for all 2 C with Re. / > 0. Furthermore, #.I x; yI Z/ satisfies the functional equation e2ixy #. I x; yI Z/ D p #



 1 I y; xI Z :

(9.7.7)

9.8 Exercises (Trapezoidal Rule, Periodic Sobolev Spaces, Projection Method) The application of the Euler summation formula has a long history starting from Euler (1736a,b) and MacLaurin (1742). Two perspectives are offered, namely (i) To compute (slowly) converging infinite series as well as to specify convergence criteria, (ii) To evaluate integrals numerically as well as to estimate the error.

Trapezoidal Rule and Lattice Function We are interested in the lattice functions to the operators k , k 2 N. Definition 9.8.1. The iterated Z-lattice function G.k I x/ W R ! R is defined by periodic convolution Z G.k I x/ D

G.I x  y/G.k1 I y/ dy;

k  2;

(9.8.1)

F

G.k I x/ D G.I x/;

k D 1;

(9.8.2)

as in (9.4.40). Exercise 9.8.2. (a) Prove that the explicit representation of G.2 I / in elementary functions is given by (9.4.39) in Lemma 9.4.10. (b) Formulate the constituting properties of G.2 I /, i.e., periodicity, differential equation, characteristic singularity, and normalization. (c) Show that the Fourier expansion of G.k I /, k  1, is given by (9.4.44). (d) Verify that (9.4.41) holds true for k  2, i.e., x G.k I x/ D G.k1 I x/;

x 2 R:

(9.8.3)

9.8 Exercises

421

Exercise 9.8.3. Prove that the extended Euler summation formula X

0

Z

b

F .g/ D

Z

a

g2Œa;b\Z

C

b

F .x/ dx C

G.m I x/m x F .x/ dx

a m1 X kD0

(9.8.4)

  ˇb ˇ d d ˇ G.kC1 I x/.kx F .x//  G.kC1 I x/ kx F .x/ ˇ dx dx a

holds true for all F 2 C.2m/ .Œa; b/. Exercise 9.8.4. For a positive real , set Z D fg D h W h 2 Zg. Formulate the defining properties of the Z-lattice function for the operator , i.e., the Green function G Z .I  / with respect to the operator  corresponding to Z-periodicity. Exercise 9.8.5. Show that G Z .I x  y/ D

X 1 h2 Znf0g

holds for all x; y 2 R, where ˚h .x/ D

1 ˚h .x/˚h .y/ 4 2 h2

p1 e2ihx ,

x 2 R.

Definition 9.8.6. Let G Z .k I  / be defined by Z

2 G Z .k I x/ D G Z .I x  y/G Z .k1 I y/ dy; 

(9.8.5)

k  2;

(9.8.6)

2

G Z .k I x/ D G Z .I x/;

k D 1:

(9.8.7)

Note that the fundamental cell of (9.2.1) is scaled by , i.e., for the lattice Z we have F Z D Œ 2 ; 2 /. Exercise 9.8.7. Verify that G Z .k I x  y/ D

X 1 h2 Znf0g

1 ˚h .x/˚h .y/ .4 2 h2 /k

is valid for all x; y 2 R and all k 2 N, where ˚h .x/ D

p1 e2ihx ,

(9.8.8)

x 2 R.

Exercise 9.8.8. Prove the extended Euler summation formula for F 2 C.2m/ .Œa; b/ and the lattice D Z, > 0, i.e., X

0

Z F .g/ D

b

Z

a

g2Œa;b\ Z

C

m1 X kD0

b

F .x/ dx C a



G Z .m I x/m x F .x/ dx

(9.8.9)

 ˇb ˇ d d G Z .kC1 I x/.kx F .x//  G Z .kC1 I x/ kx F .x/ ˇˇ : dx dx a

9 Lattice Functions in R

422

Exercise 9.8.9. Let F be of class C.2m/ .Œa; b/ such that at the end points of the interval F .l/ .a/ D F .l/ .b/ for l D 1; : : : ; 2m  1 and supx2Œa;b jF .2m/ .x/j  M . Set D ba n . Prove that the trapezoidal rule  Tn .F / D

1 1 F .a/ C F .a C / C : : : C F .a C .n  1/ / C F .b/ 2 2

 (9.8.10)

satisfies the following estimate for the remainder ˇZ ˇ   ˇ b ˇ b  a 2m ˇ ˇ jRn .F /j D ˇ F .x/ dx  Tn .F /ˇ  4M  .2m/; (9.8.11) ˇ a ˇ 2 n where .2m/ D

1 X 1 2m n nD1

(9.8.12)

is the functional value of the Riemann Zeta function at 2m. Exercise 9.8.10. Apply the trapezoidal rule to the integral Z

Z

1

1

F .x/ dx D 0

0

1 dx D ln.2/ 1Cx

(9.8.13)

(a) By calculating Tn .F / for n D 10, (b) By calculating Sn .F / D Tn .F / 

h2 0 h4 .F .1/  F 0 .0// C .F .3/ .1/  F .3/ .0//; (9.8.14) 12 720

(c) By calculating the error Rn .F / D Tn .F /  Sn .F /. Remark 9.8.11. For more details see, e.g., Davis and Rabinowitz (1967), H¨ammerlin and Hoffmann (1992), Freund and Hoppe (2007).

.s/

Approximation in Sobolev Spaces HZ .R/ In the following, we are interested in one-dimensional Z-periodic Sobolev spaces and their induced Hilbert space framework. We start with classical Fourier expan.1/ sions of functions F belonging to the space CZ .R/: F D

X h2Z

F ^ .h/˚h

(9.8.15)

9.8 Exercises

423

with

Z

F ^ .h/ D

F

F .x/˚h .x/ dx:

(9.8.16)

.s/

Definition 9.8.12. The Z-periodic Sobolev spaces HZ .R/, s 2 R, are understood .1/ as the closure of CZ .R/ with respect to the norm 0

X

kF kH .s/ .R/ D @jF ^ .0/j2 C Z

1 12 jF ^ .h/j2 .4 2 h2 / 2 A ; s

(9.8.17)

h2Znf0g

i.e., .s/

.1/

HZ .R/ D CZ .R/

kk

.s/

HZ .R/

:

(9.8.18)

.s/

HZ .R/ equipped with the inner product hF; GiH .s/ .R/ D F ^ .0/G ^ .0/ C Z

X

s

F ^ .h/G ^ .h/ .4 2 h2 / 2

(9.8.19)

h2Znf0g

is a Hilbert space. .sC˛/

Note that the inner product (9.8.19) extends to a duality between HZ .s˛/ .0/ HZ .R/ for arbitrary ˛ 2 R. Moreover, HZ .R/ D L2Z .R/.

.R/ and

Exercise 9.8.13. The projection method uses an increasing sequence of finite truncations of the Fourier series which can be formulated by means of finite dimensional spaces PN D spanf˚h gh2Z\ŒN;N 

(9.8.20)

and the corresponding projections given by SN .F I  / D

X

F ^ .h/˚h :

(9.8.21)

h2Z\ŒN;N  .s/

Prove that fSN gN 2N0 is a convergent family of operators in every HZ .R/, s 2 R, i.e., lim kF  SN .F I  /kH .s/ .R/ D 0

N !1 .s/

for every F 2 HZ .R/.

Z

(9.8.22)

9 Lattice Functions in R

424

.s/

Exercise 9.8.14. Show that for r 2 .1; s/ and F 2 HZ .R/ we have the approximation property kF  SN .F I  /kH .r/ .R/  CN rs kF kH .s/ .R/ ; Z

Z

(9.8.23)

where the constant C is independent of N and F . Exercise 9.8.15. Show that for every s  r the inverse property kP kH .s/ .R/  CN rs kP kH .s/ .R/ Z

(9.8.24)

Z

holds for all P 2 PN , where C is independent of P and N .

Numerical Integration of Fourier Coefficients Exercise 9.8.16. Let xi , i D 1; : : : ; n and wi , i D 1; : : : ; n, be the nodal points and weights, respectively, of the Gauß–Legendre quadrature formula on [1,1] (see Exercise 3.8.3). Show that Gaussian quadrature applied to M equidistant subintervals of Œ0; 1 leads to the approximate integration formula  M 1    n m xl C 1 X 1 X xl C 1 m ^ Q C F .h/ D wl ˚h F ˚h M 2M 2M M M mD0

(9.8.25)

lD1

for the Fourier coefficient F ^ .h/, h 2 Z. Exercise 9.8.17. Verify that the quadrature error can be estimated by X

FQ ^ .h/  F ^ .h/ D

n^ .g/F ^ .h C gM /;

(9.8.26)

g2Znf0g

where

1X D wl ˚h 2 n

n^ .g/



lD1

xl C 1 gM 2M

 (9.8.27)

for g 2 Z n f0g, provided that the Fourier expansion of R F is absolutely convergent. Note that j n^ .g/j  1 is small since it approximates F ˚g .x/ dx D 0. Remark 9.8.18. The sum M 1 X mD0

 F

m xl C 1 C 2M M

 ˚h

m M

(9.8.28)

can be computed by the numerically stable FFT-algorithm (see Cooley and Tukey 1965).

9.8 Exercises

425

Exercise 9.8.19. Prove that X ˇ2 ˇ jı0;h C 2ihj2r ˇFQ ^ .h/  F ^ .h/ˇ  CN 2.rs/ kF k2

.s/

HZ .R/

h2Z\ŒN;N  .s/

holds true for all F 2 HZ .R/, s > F and N .

1 2,

(9.8.29)

where the constant C is independent of

Remark 9.8.20. For higher dimensional generalizations of this method and applications to partial differential equations the reader is referred to, e.g., Lamp et al. (1985).

Chapter 10

Lattice Functions in Rq

If an attempt is made to generalize the one-dimensional theory to a higher dimensional case, we are confronted with pointwise convergence problems for the bilinear series of the multi-variate counterpart of G.I  /. Nonetheless, as we have already seen in the one-dimensional case in Chap. 9, we are able to circumvent any possible calamities by paying close attention to the defining constituents. However, the q-dimensional theory remains more complicated, since the characteristic singularity of the lattice function in lattice points becomes much harder to handle with increasing dimension. In conclusion, the proof of the Euler summation formula associated to the Laplace operator as well as the specification of sufficient criteria for validity of the Poisson summation formula is a matter of multi-dimensional potential theory. The results obtained in such a way are applicable in many branches, e.g., the calculation of certain lattice point sums involving charged particles, functional equations of Zeta and Theta functions, etc. Some of the applications are worked into exercises in Sect. 10.9, where we also explain the interrelations between Green and spline functions. Our multi-periodic approach is based on the concepts of metaharmonic lattice point theory as presented in Freeden (2011).

10.1 Lattices in Euclidean Spaces Let g1 ; : : : ; gq be linearly independent vectors in the q-dimensional Euclidean space Rq . Definition 10.1.1. The set  (more precisely, .q/ ) of all points g D n1 g1 C : : : C nq gq 2 Rq ;

(10.1.1)

ni 2 Z; i D 1; : : : ; q, is called a lattice in Rq with basis g1 ; : : : ; gq 2 Rq .

W. Freeden and M. Gutting, Special Functions of Mathematical (Geo-)Physics, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-0348-0563-6 10, © Springer Basel 2013

427

10 Lattice Functions in Rq

428

Clearly, the vectors "1 ; : : : ; "q form a lattice basis of Zq : Trivially, a lattice basis fg1 ; : : : ; gq g is related to the canonical basis f"1 ; : : : ; "q g in Rq via the formula gi D

q X

.gi  "r / "r :

(10.1.2)

rD1

Definition 10.1.2. The half-open parallelotope F (more precisely, F ) consisting of the points x 2 Rq with x D t1 g1 C : : : C tq gq ;



1 1  ti < ; 2 2

(10.1.3)

i D 1; : : : ; q; is called the fundamental cell of the lattice  (see Fig. 10.1 for the two-dimensional case). Remark 10.1.3. Obviously, there are infinitely many cells of  reflecting the -periodicity. F , as specified by (10.1.3), is both simple and appropriate for our purposes. From linear algebra (see, e.g., Davis 1963) it is well-known that the volume of F is equal to the quantity Z kF k D F

dV.q/ D

q

  det .gi  gj /i;j D1;:::;q :

(10.1.4)

For each g 2 , we have F C fgg D fy C g W y 2 F g, such that kF k D kF C fggk:

(10.1.5)

Clearly, because of .F C fgg/ \ .F C fg 0 g/ D ; for g 6D g 0 , g; g 0 2 ; we have Rq D

[

.F C fgg/ D

g2

[

.F  fgg/ :

(10.1.6)

g2

Since the vectors g1 ; : : : ; gq 2 Rq are assumed to be linearly independent, there exists a system of vectors h1 ; : : : ; hq 2 Rq such that ( hj  gi D ıi;j D

0

;

i ¤ j;

1

;

i D j:

(10.1.7)

In more detail, for i; j D 1; : : : ; q; we let gi  gj D ij :

(10.1.8)

10.1 Lattices in Euclidean Spaces

429

Fig. 10.1 Two-dimensional lattice and its fundamental cell F g2

0

g1

The scalars  ij ; i; j D 1; : : : ; q; are defined by q X

 ij j k D ıi;k :

(10.1.9)

j D1

The vectors hj ; j D 1; : : : ; q; given by hj D

q X

 j k gk ;

j D 1; : : : ; q;

(10.1.10)

kD1

satisfy the equations hj  gi D

q X

jk

 gk  gi D

kD1

q X

 j k ki D ıj;i ;

(10.1.11)

kD1

i; j D 1; : : : ; q. Moreover, we find for i; j D 1; : : : ; q; hi  hj D

q X kD1

 i k gk 

q X lD1

 j l gl D

q X

 jl

lD1

q X

 i k kl D  j i :

(10.1.12)

kD1

Definition 10.1.4. The lattice with basis h1 ; : : : ; hq 2 Rq given by (10.1.10) is called the inverse (or dual) lattice 1 to . The inverse lattice 1 consists of all vectors h 2 Rq such that the inner product h  g is an integer for all g 2 . Obviously,  D .1 /1 :

(10.1.13)

10 Lattice Functions in Rq

430

Moreover, for the fundamental cell F1 of the inverse lattice 1 (throughout this work, denoted by F 1 ), we have  1  F  D kF k1 :

(10.1.14)

For more details see, e.g., Cassels (1968) and Lekkerkerker (1969). Example 10.1.5. Let  D Zq be the lattice which is generated by the dilated basis "1 ; : : : ; "q , where  is a positive number and "1 ; : : : ; "q forms the canonical orthonormal basis in Rq . Then, the volume of the fundamental cell of Zq is kF k D  q . Generating vectors of the inverse lattice 1 are  1 "1 ; : : : ;  1 "q . The volume of the fundamental cell of the inverse lattice is given by  1  F  D  q D kF k1 :

(10.1.15)

In particular, for  D 1, i.e., the lattice  D Zq , we have 1 D Zq D  such that  1  F  D 1 D kF k:

(10.1.16)

This fact has been used permanently in our one-dimensional theory (see Chap. 9), and it is always obvious throughout this work whenever  D Zq .

10.2 Periodic Polynomials First, the standard -periodic polynomials (orthonormal in L2 .Rq /-sense) are listed. Equivalent conditions for the closure and completeness are formulated within the space L2 .Rq / of square-integrable -periodic functions in Rq (see, e.g., Stein and Weiss 1971). Definition 10.2.1. Let  be a lattice in Rq . The functions ˚h ; h 2 1 , defined by 1 ˚h .x/ D p e2ihx ; kF k

x 2 Rq ;

(10.2.1)

are -periodic, i.e., ˚h .x C g/ D ˚h .x/

(10.2.2)

for all x 2 Rq and all g 2 . They are called periodic polynomials. .m/

The space of all F 2 C.m/ .Rq / that are -periodic is denoted by C .Rq / with .0/ p 0  m  1. As usual we write C .Rq / for C .Rq /. L .Rq /, 1  p < 1, is the

10.2 Periodic Polynomials

431

space of all F W Rq ! C that are -periodic and are Lebesgue-measurable on F with Z  p1 p p kF kL .Rq / D jF .x/j dV.q/ .x/ < 1: (10.2.3) 

F

p L .Rq /. As is well-known, L2 .Rq / is the completion of C .Rq /

q

Clearly, C .R /  with respect to the norm k  kL2 .Rq / : 

L2 .Rq / D C .Rq /

kkL2 .Rq / 

:

(10.2.4)

An easy calculation shows that Z F

˚h .x/˚h0 .x/ dV.q/ .x/ D ıh;h0 :

(10.2.5)

In other words, the system f˚h gh21 of multi-dimensional “periodic polynomials” is orthonormal with respect to the L2 .Rq /-inner product: Z h˚h ; ˚h0 iL2 .Rq / D 

F

˚h .x/˚h0 .x/ dV.q/ .x/ D ıh;h0 :

(10.2.6)

An elementary calculation yields   x C 4 2 h2 ˚h .x/ D 0;

h 2 1 :

(10.2.7)

We shall say that  is an eigenvalue of the lattice  with respect to the operator  if there is a non-trivial solution U of the differential equation . C / U D 0

(10.2.8)

satisfying the “boundary condition of periodicity” U.x C g/ D U.x/

(10.2.9)

for all g 2 . The function U is then called an eigenfunction of the lattice  with respect to the eigenvalue  and the operator . In analogy to the one-dimensional case, we are able to see that the set of all eigenvalues with respect to , i.e., the spectrum Spect ./, is given by  ˚ Spect ./ D 4 2 h2 W h 2 1 :

(10.2.10)

Clearly, an analog of Theorem 3.0.7 also holds true for the multi-dimensional case, i.e., the system f˚h gh21 is closed and complete in the pre-Hilbert space .C .Rq /I k  kL2 .Rq / / as well as in the Hilbert space .L2 .Rq /I k  kL2 .Rq / /. In this 



10 Lattice Functions in Rq

432

respect, the fundamental result in Fourier analysis is that each F 2 L2 .Rq / can be represented by its (orthogonal) Fourier series in the sense   lim  F 

N !1

  F^ .h/˚h 

X q

h21 \BN

L2 .Rq /

D 0;

(10.2.11)

where the Fourier coefficients F^ .h/ of F are given by F^ .h/

Z D F

F .x/˚h .x/ dV.q/ .x/;

h 2 1 :

(10.2.12)

The Parseval identity then tells us that, for each F 2 L2 .Rq /; Z F

jF .x/j2 dV.q/ .x/ D

X ˇ ˇ ˇF ^ .h/ˇ2 : 

(10.2.13)

h21

10.3 Lattice Function for the Laplace Operator Our considerations start with the definition of the -lattice function by its constituting ingredients. In accordance with the one-dimensional case (see Sect. 9.2) and seen from the point of mathematical physics, the -lattice function as introduced in Definition 10.3.1 is nothing more than the Green function for the Laplace operator  in Euclidean space Rq corresponding to the “boundary condition of periodicity” with regard to the lattice . Definition 10.3.1. G.I / W Rq n  ! R is called the -lattice function for the operator  if it satisfies the following properties: (i) (Periodicity) For all x 2 Rq n  and g 2 , G.I x C g/ D G.I x/:

(10.3.1)

(ii) (Differential equation) G.I / is twice continuously differentiable for all x …  and we have 1 : (10.3.2) G.I x/ D  kF k (iii) (Characteristic singularity) In the neighborhood of the origin 8 ˆ ; ˆ 2m:

(10.3.17)

The differential equation   G .m I x/ D G m1 I x ;

x 62 ;

(10.3.18)

m D 2; 3; : : :, represents a recursion relation relating the -lattice function for the operator m to the -lattice function for the operator m1 . The series expansion of G .m I / in terms of eigenfunctions, which is equivalent to the (formal) Fourier expansion, reads for iteration orders m D 2; 3; : : : X 1 ˚h .x/ p : kF k h21 nf0g .4 2 h2 /m

(10.3.19)

10 Lattice Functions in Rq

436

Therefore, for m > q2 , it follows that there is a constant C > 0 such that ˇ ˇ ˇ ˇ

X h21 nf0g

ˇ X ˚h .x/ ˇˇ 1  C < 1: ˇ 2 2 m .4 h / .1 C h2 /m 1

(10.3.20)

h2

Thus, it is clear that the Fourier series converges absolutely and uniformly in Rq and G .m I / is continuous in Rq , provided that m > q2 . Altogether, we are able to formulate Lemma 10.3.7. For m > q2 , the -lattice function G .m I / is continuous in Rq and its bilinear series reads G .m I x  y/ D

X h21 nf0g

˚h .x/˚h .y/ ; .4 2 h2 /m

x; y 2 Rq :

(10.3.21)

10.4 Euler Summation Formula for the Laplace Operator Let  be an arbitrary lattice in Rq . Suppose that G  Rq is a regular region as introduced in Definition 6.1.1. Let F be a function of class C.2/ .G /; G D G [ @G . Then, for every (sufficiently small) " > 0, the second Green theorem (Theorem 6.2.2) gives (see Fig. 10.2) Z G nA.q;;"/

.F .x/ .G .I x//  G .I x/ .F .x/// dV.q/ .x/

(10.4.1)

     @F @ G .I x/  G .I x/ .x/ dS.q1/ .x/ D F .x/ @ @ @G nA.q;;"/      X Z @F @ G .I x/  G .I x/ .x/ dS.q1/ .x/; F .x/ C q1 @ @ G \S" .g/ Z

g2G \

S q where  is the outer (unit) normal field and A.q; ; "/ D x2 B" .x/. Observing the differential equation (condition .ii/ of Definition 10.3.1), we get Z G nA.q;;"/

F .x/G .I x/ dV.q/ .x/ D 

1 kF k

Z G nA.q;;"/

F .x/ dV.q/ .x/: (10.4.2)

Hence, on passing to the limit " ! 0 and observing the characteristic singularity of the -lattice function (i.e., Condition .iii/ of Definition 10.3.1), we obtain in connection with Lemma 6.2.7

10.4 Euler Summation Formula for the Laplace Operator

437

Fig. 10.2 The geometric situation of Euler summation in Eq. (10.4.1)

Theorem 10.4.1 (Euler Summation Formula for the Laplace Operator ). Let  be an arbitrary lattice in Rq . Suppose that G  Rq is a regular region. Let F be twice continuously differentiable on G , G D G [ @G . Then, X0

F .g/ D

g2G \

1 kF k Z C @G

where

Z

Z G

F .x/ dV.q/ .x/ C

G

G .I x/ F .x/ dV.q/ .x/

(10.4.3)

     @F @ G .I x/  G .I x/ .x/ dS.q1/ .x/; F .x/ @ @ X0

X

F .g/ D

g2G \

˛.g/ F .g/

(10.4.4)

g2G \

and ˛.g/ is the solid angle subtended at g 2 G by the surface @G . This formula provides a comparison between the integral over a regular region G and the sum over all functional values of the twice continuously differentiable function F in the lattice points g 2 G under explicit knowledge of the remainder term in integral form. In fact, the formula for the Laplace operator  (Theorem 10.4.1) is an immediate generalization to the multi-dimensional case of the one-dimensional Euler summation formula (Theorem 9.4.1), where G.I / takes the role of the Bernoulli polynomial of degree 2.  Provided that F is of class C.2kC2/ G ; k D 1; : : : ; m  1, on G D G [ @G , G a regular region, we get from the second Green theorem (Theorem 6.2.2) by aid of the differential equation (10.3.18) Z

  G kC1 I x kC1 F .x/ dV.q/ .x/ 

G nA.q;;"/

D

   G kC1 I x k F .x/ dV.q/ .x/

G nA.q;;"/

 @ k  F .x/ dS.q1/ .x/ @ @G nA.q;;"/   Z @  kC1  I x k F .x/ dS.q1/ .x/  G  @G nA.q;;"/ @

Z

  G kC1 I x



Z

10 Lattice Functions in Rq

438

C

X Z q1

g2G \



G \S"

.g/



X Z q1

g2G \

  G kC1 I x

G \S"

.g/



@ k  F .x/ @

 dS.q1/ .x/

 @  kC1  I x k F .x/ dS.q1/ .x/ G  @

(10.4.5)

for every (sufficiently small) " > 0. From classical potential theory (see, e.g., Kellogg 1929) we know that the integrals over all hyperspheres around the lattice points tend to 0 as " ! 0. This leads to the recursion formula Z G

Z

  G kC1 I x kC1 F .x/ dV.q/ .x/ D

G

  G k I x k F .x/ dV.q/ .x/

(10.4.6)

 @ k  F .x/ dS.q1/ .x/ @ @G  Z  @  kC1  I x k F .x/ dS.q1/ .x/: G   @G @ Z

C

  G kC1 I x



 From (10.4.6) we easily obtain, for F 2 C.2m/ G ; m 2 N; Z

Z G

G .I x/ F .x/ dV.q/ .x/ D

G

C

  G m I x m F .x/ dV.q/ .x/

@G

kD1





m1 XZ

m1 XZ kD1

@G

(10.4.7)

 @  kC1  I x k F .x/ dS.q1/ .x/ G  @

  G kC1 I x



 @ k  F .x/ dS.q1/ .x/: @

In connection with Theorem 10.4.1 we obtain the Euler summation formula with respect to the operator m . m Theorem 10.4.2 (Euler Summation Formula for the Operator , m 2 N). Let   G  Rq be a regular region. Suppose that F is of class C.2m/ G ; G D G [ @G . Then, Z Z X0   1 F .g/ D F .x/ dV.q/ .x/ C G m I x m F .x/ dV.q/ .x/ kF k G G g2G \

C

m1 XZ kD0

@G



 @  kC1  G  I x k F .x/ dS.q1/ .x/ @

10.5 Zeta Functions

439



m1 XZ kD0

where

P0



G 

kC1

Ix



@G



@ k  F .x/ @

 dS.q1/ .x/;

(10.4.8)

is defined by (10.4.4).

Applications of Theorem 10.4.2 in geothermal research (heat transport) can be found in Ostermann (2011). Generalizations to more general (elliptic) partial differential operators are due to Freeden (1982), Freeden and Fleck (1987), and Ivanow (1963).

10.5 Zeta Functions We start with some preparatory material (see, e.g., Epstein 1903, 1907; M¨uller 1954; q Wienkamp 1958 and the literature therein) concerning the Zeta function n .I I / of dimension q  3 and degree n. q

Definition 10.5.1. The Zeta function n .I I / of dimension q  3 and degree n is defined by s 7!

nq .sI I /

X

D

g2nf0g

  g 1  ; Pn qI jgjs jgj

(10.5.1)

where s 2 C satisfies Re.s/ > q, is an arbitrary, but fixed, element of the unit sphere Sq1 , and Pn .qI / is the Legendre polynomial of degree n and dimension q. For each positive value % < infx2F jxj, sufficiently large positive N , and for all s 2 C with Re.s/ > q; the Euler summation formula (Theorem 10.4.2) gives us in terms of the auxiliary function F defined by x 7! F .x/ D

  x 1 

; qI P n jxjs jxj

2 Sq1 ; n 2 N0 ; x ¤ 0;

(10.5.2)

an identity for directionally dependent poles in lattice points X0 q g2\B%;N

F .g/ D

1 kF k

Z

Z C

q

F .x/ dV .x/

B%;N

q

G .m I x/ .m F .x// dV .x/

B%;N

C



m1 XZ kD0

q1

SN

  @  kC1   k G  Ix  F .x/ dS.x/ @

(10.5.3)

10 Lattice Functions in Rq

440 m1 XZ



kD0

C

q1

 q1

S%

m1 XZ



kD0



SN

m1 XZ kD0

  G kC1 I x

q1

@ k  F .x/ @

 dS.x/

  @  kC1   k G  Ix  F .x/ dS.x/ @

  G kC1 I x

S%



@ k  F .x/ @

 dS.x/; q

where m 2 N is chosen such that m > q=2 and  is the outward unit normal to B%;N . First, we want to calculate the second integral on the right-hand side of (10.5.3). It is not difficult to see that x F .x/ D .s C n/.s  n  q C 2/ such that m x F .x/ D As;n;m where As;n;m D

F .x/ ; jxj2

F .x/ ; jxj2m

m1 Y

m1 Y

j D0

j D0

.s C n C 2j /

x ¤ 0;

(10.5.4)

x ¤ 0;

(10.5.5)

.s  n C 2j  q/:

(10.5.6)

For m > q=2, we find in connection with the absolutely and uniformly convergent bilinear expansion of the -lattice function in Rq ; Z G .m I x/ m F .x/ dV .x/ (10.5.7) q B%;N

As;n;m D kF k

X h21 nf0g

1 .4 2 h2 /m

 e2ixh Pn qI 

Z q

B%;N

jxjsC2m

x jxj

 dV .x/:

From the theory of Bessel functions (see Chap. 8) we obtain after some elementary calculations Z q

B%;N

G .m I x/ m F .x/ dV .x/

D As;n;m

in kSq1 k . q2 / 2

2q 2

kF k

(10.5.8)

 Z 2jhjN J q2 .r/  1 h nC 2 P 

qI n q dr: qs .2jhj/ jhj 2jhj% r sC2m 2 h21 nf0g X

Remembering the asymptotic behavior of the Bessel function, we see that the last series converges uniformly with respect to % and N for all parameters s 2 C with 2mC q2 C 12 < Re.s/ < min.0I 2mCq Cn/. Consequently, with s 2 C indicated as before, it follows that

10.5 Zeta Functions

Z Rq

441

G .m I x/ m F .x/ dV .x/

D As;n;m

in kSq1 k . q2 / 2

2q 2

kF k

(10.5.9)

 Z 1 J q2 .r/  h 1 nC 2 

Pn qI dr: q qs .2jhj/ jhj r sC2m 2 0 h21 nf0g X

In connection with the following formula (see, e.g., Magnus et al. 1966) Z

1 0

. 2 /2 1 J .r/ dr D ; r C1 .  12 C 1/

0<
q. Even more, for Re.s/ > q, the integrals (10.5.12) tend to zero as N ! 1. Thus, it follows for Re.s/ > q that lim

N !1

X0 q

g2\B%;N

  g 1 

Pn qI jgjs jgj (

D

) Z F .x/ ; nD0 G .m I x/ dV .x/ C As;n;m 2m q jxj ; n>0 B%

kSq1 k %qs kF k sq

0

C

m1 X



kD0



Z As;n;k

kD0 m1 X

q1

S%

Z As;n;k

(10.5.13)

q1

S%

 @  kC1  F .x/ dS.x/ G  Ix @ jxj2k

  G kC1 I x



@ F .x/ @ jxj2k

 dS.x/:

10 Lattice Functions in Rq

442

q

Consequently, the right-hand side of (10.5.13) shows that n .I I / can be continued by the left-hand side of (10.5.13) to the half plane Re.s/ > 2m C q2 C 12 . It remains to investigate the sums m1 X

As;n;k

q1

S%

kD0





Z

m1 X

 @ F .x/ kC1 G. I x/ dS.x/ @ jxj2m

Z As;n;k

kD0

q1

  G kC1 I x

S%



@ F .x/ @ jxj2m

(10.5.14)

 dS.x/:

Because of the singularity behavior of the -lattice function at 0, the sums (10.5.14) tend to 0 as % ! 0, provided that s 2 C with Re.s/ < 2m C q. Thus, for all values q s 2 C with 2m C q2 C 12 < Re.s/ < minf0; 2m C q C ng, n .I I / admits the representation q

nq .sI I /

D

/ in  s 2 . nCqs 2 kF k . sCn / 2

X h21 nf0g

  h 1  : Pn qI jhjqs jhj

(10.5.15)

Summarizing our results we obtain Theorem 10.5.3 (Functional Equation of the q-Dimensional Zeta Function). q For n > 0, the Zeta function n .I I / of dimension q and degree n defined by s 7! nq .sI I / D

X g2nf0g

  g 1 

; qI P n jgjs jgj

Re.s/ > q;

(10.5.16)

admits a holomorphic continuation that represents an entire function in C. q q For n D 0, the continuation of n .I I /, i.e., 0 .I I /, is a meromorphic function showing the single pole kSq1 k 1 : (10.5.17) kF k s  q q

For n 2 N0 , n .I I / satisfies the functional equation nq .sI I / D q

nCqs q . / q in 2  s 2 n .q  sI I 1 /: sCn kF k . 2 /

(10.5.18)

Proof. n .I I / is holomorphic for Re.s/ > 2m C q2 C 12 (except for s D q in case of n D 0). According to (10.5.15), the functional equation holds true for all q s 2 C with 2m C q2 C 12 < Re.s/ < minf0; 2m C q C ng. n .q  I I 1 / is holomorphic (except for s D q in the case of n D 0) for all s 2 C with Re.s/ < 2m C q2  12 . As the functional equation is valid in the aforementioned strip, it is valid everywhere. t u

10.6 Integral Asymptotics for Lattice Functions

443

q

Remark 10.5.4. 0 .I I / is independent of 2 Sq1 . Hence, we simply write q q 0 .I / instead of 0 .I I /. Furthermore, 02 .I I / coincides with .I /.

10.6 Integral Asymptotics for Lattice Functions Next, we come to integral estimates involving fundamental solutions of the iterated Laplacian which play an essential role in asymptotic relations for Euler and Poisson summation in Euclidean spaces Rq . We begin with some potential theoretic preliminaries. For x; y 2 Rq , we introduce the polar coordinates x D r ; y D s ;

2 D 1;

r D jxj;

2

D 1;

(10.6.1)

s D jyj;

characterizing the geometric situation as illustrated by Fig. 10.3. We are interested in an estimate for the scalar product  if x and y satisfy jx yj  ı with (sufficiently small) fixed positive ı. Clearly, jx yj2  ı 2 is equivalent to r 2 Cs 2 2rs   ı 2 , such that   2 s ı2   C .  /2  .  /2 : 1 2  (10.6.2) r r Thus, we are led to the following estimate. Lemma 10.6.1. For x 2 Rq fixed with jxj  ı > 0, the inequality jx  yj  ı implies in the polar coordinates (10.6.1) r

  .r; ı/ D

1

ı2 : r2

(10.6.3)

Based on Lemma 10.6.1 we are able to prove the following lemma. Lemma 10.6.2. Suppose that ı is a fixed real number with ı 2 .0; 1/. Then, Z q1

SN

q

\Bı .y/

1 kSq2 k ı q1k dS .x/  .q1/ jx  yjk q  1  k .N; ı/

(10.6.4)

holds true for all N  1 and all integers k with 0  k < q  1. Note that .N; ı/ D q 2 1  Nı 2 as defined in Lemma 10.6.1. q1

Proof. We already know that the surface element of the sphere SN with radius N around 0 is equal to dS.q1/ .x/ D N q1 dS.q1/ . /, x D N , where dS.q1/ . / is the surface element of the unit sphere Sq1 . By virtue of Lemma 10.6.1 we get

10 Lattice Functions in Rq

444 Fig. 10.3 The geometric situation as discussed by Lemma 10.6.1

Z q1

SN

q

\Bı .y/

1 dS.q1/ .x/ D jx  yjk

where we use the notation % D

jyj N

Z

Z

N .q1/k

  .N;ı/

2Sq1

. Now, we have

N .q1/k

  .N;ı/

2Sq1

k

.1 C %2  2%.  // 2

D kSq2 kN .q1/k

dS.q1/ . /; (10.6.5)

s N

D

k

.1 C %2  2%.  // 2

Z

1

dS.q1/ . /

.1  t 2 /

 .N;ı/

(10.6.6)

q3 2 k

.1 C %2  2%t/ 2

dt:

We observe the inequality 1 C %2  2%t D .%  t/2 C 1  t 2  1  t 2 :

(10.6.7)

Because of .N; ı/  t  1 we obtain N .q1/k

Z

1

.1  t 2 /

 .N;ı/

q3 2

.1 C %2  2%t/

k 2

dt  N .q1/k

Z

1

.1  t 2 /

q3k 2

.1  t 2 /

q3k 2

dt (10.6.8)

 .N;ı/

N

.q1/k

Z

1  .N;ı/

t dt: .N; ı/

Now, an elementary calculation gives Z

1

.1  t 2 /

q3k 2

t dt D 

 .N;ı/

D

1 2

1

.1  t 2 / q1k

q1k 2

2

ı q1k 1 : q  1  k N q1k

Combining (10.6.6) and (10.6.9), we obtain the desired result.

ˇ1 ˇ ˇ

 .N;ı/

(10.6.9) t u

10.6 Integral Asymptotics for Lattice Functions

445

For the two-dimensional case (i.e., q D 2), we find the following lemma. Lemma 10.6.3. Suppose that ı 2 .0; 1/. Then, Z S1N \B2ı .y/

j ln.jx  yj/j dS.1/ .x/  

2ı ln.ı/ .N; ı/

(10.6.10)

holds true for all N  1. Proof. If ı 2 .0; 1/, then it follows, for x; y 2 R2 with jx  yj  ı, that  1  jln.jx  yj/j D  ln.jx  yj/ D  ln N 2 C s 2  2N s  : 2

(10.6.11)

From Lemma 10.6.1 we know that   .N; ı/. Thus, we are able to conclude that Z S1N \B2ı .y/

ˇ ˇ ˇ ln.jx  yj/ˇ dS.1/ .x/  N

Z

1

1

.1  t 2 / 2 ln.N 2 C s 2  2N st/ dt:

 .N;ı/

(10.6.12)

We now use, for t 2 Œ.N; ı/; 1, the identity N 2 C s 2  2N st D .s  N t/2 C N 2 .1  t 2 /:

(10.6.13)

This leads us to the relation N 2 C s 2  2N st  N 2 .1  t 2 /  ı 2 : Observing ı 2 .0; 1/, we obtain Z Z jln.jx  yj/j dS.1/.x/  2N ln.ı/ S1N \B2ı .y/



1  .N;ı/

(10.6.14)

t dt p 2 .N; ı/ 1t

2N ln.ı/ ı 2ı ln.ı/ D : .N; ı/ N .N; ı/

This is the result stated in Lemma 10.6.3.

(10.6.15) t u

Next, we evaluate integrals which can be interpreted as the “q-dimensional angles” under which the sphere of radius N around a point y 2 Rq is seen from the origin (see Fig. 10.4). Three cases (illustrated by Fig. 10.4) must be distinguished. Indeed, the three cases require three different calculations: Lemma 10.6.4. For q  2, y 2 Rq and N > 0, Z q1 SN .y/

jx  x j dS.q1/ .x/  kSq1 k; jxjq

where x is the outward pointing normal to the sphere Sq1 in x.

(10.6.16)

10 Lattice Functions in Rq

446

Fig. 10.4 The three cases under consideration in the proof of Lemma 10.6.4

Proof. We introduce polar coordinates x D r. / ; 2 Sq1 , to represent the sphere q1

SN .y/ D fx 2 Rq W jx  yj D N g

(10.6.17)

by its projection onto the unit sphere Sq1 D f 2 Rq W j j D 1g: By means of the q1 polar coordinates x D r. / ; 2 Sq1 , the surface element on SN .y/ can then be represented in the form dS.q1/ .x/ D dS.q1/ .r. / / D .r. //q1 dS.q1/ . /

(10.6.18)

such that jx  x j dS.q1/ .x/ D

r. /  r. / dS.q1/ .r. / / jr. /j

(10.6.19)

D r. /.r. //q1 dS.q1/ . / D .r. //q dS.q1/ . /: q1

This relates the surface element dS.q1/ .x/ on SN .y/ to its projection dS.q1/ . / on Sq1 . We understand Z W Sq1 ! R to be the number of positive solutions of the form r W 7! r. /; 2 Sq1 , of the equation jr. /  yj D N;

(10.6.20)

such that Z q1

SN

.y/

jx  x j dS.q1/ .x/ D jxjq

Z Sq1

Z. / dS.q1/ . /:

(10.6.21)

(i) Case jyj < N (see Fig. 10.4 (left)): We have exactly one positive r to every direction 2 Sq1 such that jr. /  yj2 D .r. /  y  /2  .y  /2 C y 2 D N 2 :

(10.6.22)

10.6 Integral Asymptotics for Lattice Functions

447

Consequently, Z. / D 1 for all 2 Sq1 . This shows that Z q1

SN

.y/

jx  x j dS.q1/ .x/ D kSq1 k: jxjq

(10.6.23)

(ii) Case jyj D N (see Fig. 10.4 (middle)): Now, the equation jr. /  yj2 D .r. /  y  /2  .y  /2 C y 2 D N 2

(10.6.24)

has just one positive solution for y  > 0 and no positive solution for y   0. This leads to Z jx  x j 1 dS.q1/ .x/ D kSq1 k: (10.6.25) q q1 2 SN .y/ jxj (iii) Case jyj > N (see Fig. 10.4 (right)): We have the two positive solutions r1;2 . / D y  ˙

p .y  /2  .y 2  N 2 /

(10.6.26)

p for y   y 2  N 2 . Accordingly, we get in this case, with .N; ı/ of Lemma 10.6.1, Z q1

SN

.y/

jx  x j dS.q1/ .x/ D 2 jxjq

Z

  .N;ı/

2Sq1

dS.q1/ . /:

(10.6.27)

Therefore, this yields the estimate Z q1

SN

.y/

jx  x j dS.q1/ .x/ D 2kSq2 k jxjq

Z

1

.1  t 2 /

q3 2

dt < kSq1 k:

 .N;ı/

(10.6.28)

Altogether, Lemma 10.6.4 follows from (10.6.23), (10.6.25), and (10.6.28).

t u

A change of variables in Lemma 10.6.4 (more precisely, x is substituted by x  y and y is replaced by 0) leads us to the following result. Corollary 10.6.5. For q  2, y 2 Rq , and N > 0, Z q1 SN

j.x  y/  x j dS.q1/ .x/  kSq1 k: jx  yjq

This concludes our potential theoretic excursion (see also Freeden 2011).

(10.6.29)

10 Lattice Functions in Rq

448

10.7 Poisson Summation Formula Next, we apply the multi-dimensional Euler summation formula especially to sums of the spherical type X X : : : D lim ::: : (10.7.1) N !1

g2

q

g2\BN q

Let F be a 2m-times .m 2 N/ continuously differentiable function in BN . Then, we have X

F .g/ D q

g2\BN

Z

Z

1 kF k

q

.m/

F .x/ dV .x/ C

q

BN

BN

G .m I x/ m F .x/ dV .x/ C R.q/ .N /:

(10.7.2) The remainder term is given by .m/ R.q/ .N /

1 D 2

X

F .g/ C

kD0

q1

g2\SN



m1 XZ kD0



q1

G 



m1 XZ

kC1

SN

Ix

q1

SN





 @  kC1  G  I x k F .x/ dS.x/ @

@ k  F .x/ @

 dS.x/:

(10.7.3)

Questions of the convergence as N ! 1 require estimates of the remainder term (10.7.3). The following results play an important part in this respect. Theorem 10.7.1. For all lattices   Rq , the estimates Z jG.I x/j dS.x/ D O.N q1 /; N ! 1; q1

(10.7.4)

SN

and

Z

ˇ ˇ ˇ@ ˇ ˇ G.I x/ˇ dS.x/ D O.N q1 /; ˇ q1 ˇ @

N ! 1;

(10.7.5)

SN

hold true. Proof. The proof is essentially based on the material provided by the potential theoretic preliminaries of Sect. 10.6. We start by remembering that there is a positive constant E dependent on the lattice , such that jg  g 0 j  E holds for all points g; g 0 2  with g ¤ g 0 . We set ı

1 minf1; Eg: 2

(10.7.6)

10.7 Poisson Summation Formula

449

Then, we are able to deduce that there exists a constant C such that the estimates ˇ ˇ jG.I x/j  C ˇ ln jx  gjˇ; q D 2;

(10.7.7)

jG.I x/j  C jx  gj2q ;

(10.7.8)

q3

and ˇ ˇ ˇ ˇ ˇrx G.I x/  1 x  g ˇ  C 1 ; ˇ 2 jx  gj2 ˇ jx  gj ˇ ˇ ˇ 1 x  g ˇˇ 1 ˇrx G.I x/  ; C ˇ ˇ q1 q .q  2/kS k jx  gj jx  gjq2

q D 2;

(10.7.9)

q3

(10.7.10)

q

hold uniformly in Bı .g/. We denote the distance of x 2 Rq to the lattice  by D.xI / D dist.xI / D min jx  gj:

(10.7.11)

g2

It is clear that there is a constant B (depending on ı) such that the estimates jG.I x/j  B;

(10.7.12)

jrx G.I x/j  B

(10.7.13)

are valid for all x 2 Rq with D.xI / D dist.xI /  ı. Moreover, applying a well-known result of lattice point theory due to Gauß (1826) we know that  q # BN Cı

nBqN ı



X

D

  1 D O N q1

(10.7.14)

N ıjgjN Cı g2

for ı (fixed) and N ! 1. Thus, it follows that X

  1 D O N q1 ;

N ! 1:

(10.7.15)

q1

g2\SN

We use the different results for the cases D.xI / > ı and D.xI /  ı to get an estimate for the integrals (10.7.4) and (10.7.5) Z

Z q1

SN

jG.I x/j dS.x/ D

Z q1

x2SN D.xI/>ı

jG.I x/j dS.x/ C

q1

x2SN D.xI/ı

jG.I x/j dS.x/: (10.7.16)

10 Lattice Functions in Rq

450

From (10.7.12) it follows that Z q1 k N q1 : q1 jG.I x/j dS.x/  B kS

(10.7.17)

x2SN D.xI/>ı

Because of the characteristic singularity of G .I  /, for dimensions q  3, the estimate Z (10.7.18) q1 jG.I x/j dS.x/ x2SN D.xI/ı

  Z q q D O # BN Cı n BN ı

q1

SN

q

\Bı .g/

1 dS.q1/ .x/ jx  gjq2



is valid for N ! 1. In connection with (10.7.14) and Lemma 10.6.2 we find Z  q1  (10.7.19) q1 jG.I x/j dS.x/ D O N x2SN D.xI/ı

for q  3 and N ! 1. Note that the case q D 2 can be verified by the same arguments observing the logarithmic singularity (see Lemma 10.6.3). This establishes the proof of the first integral of (10.7.4) listed in Theorem 10.7.1 for q  2: Concerning the second integral (10.7.5) in Theorem 10.7.1 we analogously obtain Z

ˇ ˇ Z ˇ@ ˇ ˇ G.I x/ˇ dS.x/ D ˇ q1 ˇ @

SN

q1

x2SN D.xI/>ı

ˇ ˇ ˇ@ ˇ ˇ G.I x/ˇ dS.x/ ˇ @ ˇ

Z

C

q1

x2SN D.xI/ı

ˇ ˇ ˇ@ ˇ ˇ G.I x/ˇ dS.x/: ˇ @ ˇ

(10.7.20)

Then, by virtue of (10.7.13) and Lemma 10.6.4 we find Z

ˇ ˇ   ˇ@ ˇ   ˇ G.I x/ˇ dS.x/ D O N q1 C O # Bq N Cı ˇ q1 ˇ @

SN

  D O N q1

for N ! 1. Altogether, this is the desired result.

nBqN ı



(10.7.21)

t u  k  Similar results are obtainable for the -lattice functions G  I  , k 2 N. Since   each iteration reduces the order of the singularity by two, G k I  is continuous for k > q2 and continuously differentiable for k > q2 C 1. The estimates

10.7 Poisson Summation Formula

Z q1

451

ˇ  k ˇ   ˇG  I x ˇ dS.x/ D O N q1

(10.7.22)

SN

and

Z

ˇ ˇ ˇ @  k ˇ   ˇ G  I x ˇ dS.x/ D O N q1 ˇ q1 ˇ @

(10.7.23)

SN

for N ! 1, therefore, are obvious for all k > q2 C 1. For the intermediate cases k 2 .1; q2 C 1 we use Lemma 10.6.2 and estimate the integrals Z

Z ::: D q1

SN

Z q1

x2SN D.xI/>ı

::: C

q1

x2SN D.xI/ı

:::

(10.7.24)

in the same way as described above by aid of Lemma 10.6.4. This finally justifies the results of Theorem 10.7.2. For all lattices   Rq and all positive integers k, the -lattice functions for iterated Laplace operators satisfy the asymptotic integral estimates Z q1

ˇ  k ˇ   ˇG  I x ˇ dS.x/ D O N q1

(10.7.25)

SN

and

Z

ˇ ˇ ˇ @  k ˇ   ˇ G  I x ˇ dS.x/ D O N q1 ; ˇ ˇ q1 @

(10.7.26)

SN

for N ! 1. In the following, the properties developed for -lattice functions and iterated operators are used to formulate convergence theorems for multi-dimensional (alternating) series, where the Euler summation formula is the key structure in our context. To this end, we derive three auxiliary results. We begin our discussion of the remainder term (10.7.3) with the boundary terms. Lemma 10.7.3. For given m 2 N, assume that the function F 2 C.2m/ .Rq / satisfies the asymptotic relations   kx F .x/ D o jxj1q and

ˇ ˇ   ˇrx k F .x/ˇ D o jxj1q ; x

(10.7.27)

jxj ! 1;

(10.7.28)

10 Lattice Functions in Rq

452

for k D 0; : : : ; m  1. Then, for N ! 1, we have 1 2

.m/

R.q/ .N / D

X

F .g/ C

kD0

q1

g2\SN



m1 XZ q1



m1 XZ

  G kC1 I x

q1

SN



SN

kD0

 @  kC1  G  I x k F .x/ dS.x/ @

@ k  F .x/ @

 dS.x/

D o.1/:

(10.7.29)

Proof. Under the assumptions imposed on F 2 C.2m/ .Rq / and remembering the lattice point result (10.7.14) due to Gauß (1826) we have 1 2

 F .g/ D o N 1q

X q1

X

1 D o.1/;

N ! 1:

(10.7.30)

q1

g2\SN

g2\SN

Furthermore, we are allowed to conclude 

m1 XZ kD0

q1

SN



  @  kC1   k G  Ix  F .x/ dS.x/ @

m1 XZ kD0

q1

  kC1  G  Ix



SN

  m1 XZ D o N 1q kD0

@ k  F .x/ @

(10.7.31)  dS.x/

ˇ ˇ  ˇ  kC1 ˇ ˇ @  kC1 ˇ ˇG  I x ˇ C ˇ G  I x ˇ dS.x/ ˇ @ ˇ q1

SN

for N ! 1. In connection with Theorem 10.7.2 we find that   m1 XZ @  kC1  G  I x .k F .x// dS.x/ q1 @ SN kD0



m1 XZ kD0

  kC1  G  Ix q1

SN



@ k  F .x/ @

 dS.x/ D o.1/

(10.7.32)

for N ! 1. Collecting all details we get the promised result of Lemma 10.7.3. u t Next, we come to the discussion of the volume integral in (10.7.2) involving the derivative m F . Lemma 10.7.4. For given m 2 N and " > 0, assume that the function F 2 C.2m/ .Rq / satisfies   m F .x/ D O jxj.qC"/ ;

jxj ! 1:

(10.7.33)

10.7 Poisson Summation Formula

Then, the integral

Z Rq

453

G .m I x/ m F .x/ dV .x/

(10.7.34)

is absolutely convergent. Proof. From Theorem 10.7.2 we are immediately able to guarantee, with suitable positive constants M and N , ˇZ ˇ ˇ ˇ m m ˇ ˇ (10.7.35) ˇ q G . I x/  F .x/ dV .x/ˇ BM;N

Z

DO Z DO

N M N M

1 .1 C r/qC" q1

Z

jG .m I x/j dS.x/ q1



 dr

Sr



r dr : .1 C r/qC"

Consequently, the absolute convergence of the integral (10.7.34) is guaranteed.

t u

Combining Lemmas 10.7.3 and 10.7.4, we obtain as a first consequence from (10.7.2) and (10.7.3) Theorem 10.7.5. Let  be an arbitrary lattice in Rq . For given m 2 N, suppose that F 2 C.2m/ .Rq / satisfies the assumptions of Lemmas 10.7.3 and 10.7.4. Then, the limit  X  Z 1 F .g/  F .x/ dV .x/ (10.7.36) lim N !1 kF k BqN q g2\BN

exists and we have the limit relation  X  Z Z   1 lim F .g/ F .x/dV .x/ D G m I x m F .x/ dV .x/: q N !1 kF k x2BN Rq q g2\BN

In order to ensure the convergence of the series limN !1 an additional condition has to come into play.

(10.7.37)

P

q

g2\BN

F .g/, however,

Theorem 10.7.6. Let  be an arbitrary lattice in Rq . For given m 2 N, suppose that F 2 C.2m/ .Rq /, m 2 N, satisfies the following conditions: (i) The asymptotic relations   kx F .x/ D o jxj1q ; ˇ ˇ   ˇrx k F .x/ˇ D o jxj1q ; x hold true for k D 0; : : : ; m  1,

(10.7.38) jxj ! 1;

(10.7.39)

10 Lattice Functions in Rq

454

(ii) There exists " > 0 such that   m F .x/ D O jxj.qC"/ ; (iii) The integral

jxj ! 1;

(10.7.40)

Z Rq

F .x/ dV .x/

(10.7.41)

exists in the (spherical) sense Z

Z Rq

F .x/ dV .x/ D lim

N !1 Bq N

Then, the series

X

F .g/ D lim

N !1

g2

F .x/ dV .x/:

X

F .g/

(10.7.42)

(10.7.43)

q g2\BN

is convergent. More explicitly, we have X

F .g/ D

g2

1 kF k

Z

Z Rq

F .x/ dV .x/ C

Rq

G .m I x/ m F .x/ dV .x/: (10.7.44)

The convergence conditions (in Theorem 10.7.6) enable us to derive multi-dimensional formulations of the Poisson summation formula. For m 2 N with m > q2 , the -lattice function G .m I / permits an absolutely and uniformly convergent Fourier series in Rq , and Lebesgue’s theorem allows us to interchange summation and integration such that Z Rq

G .m I x/ m F .x/ dV .x/

(10.7.45)

Z X 1 1 m F .x/˚h .x/ dV .x/: D p m kF k h21 nf0g .4 2 h2 / Rq Moreover, by observation of the condition .ii/ of Theorem 10.7.6 repeated application of the second Green theorem (Theorem 6.2.2) yields Z Rq

˚h .x/m F .x/ dV .x/ D

Z Rq



 m ˚h .x/ F .x/ dV .x/  2 2 m

D  4 h

(10.7.46)

Z Rq

˚h .x/ F .x/ dV .x/:

10.8 Theta Functions

455

Inserting (10.7.46) into (10.7.45), we find Z

Z X 1 G .m I x/ m F .x/ dV .x/ D p F .x/˚h .x/ dV .x/: kF k h21 nf0g Rq Rq (10.7.47) This finally leads to the multi-dimensional Poisson summation formula. Theorem 10.7.7. Let  be an arbitrary lattice in the Euclidean space Rq . If the function F 2 C.2m/ .Rq /, m > q2 , satisfies the conditions .i /–.iii/ of Theorem 10.7.6, then X Z X 1 F .g/ D p F .x/˚h .x/ dV .x/: (10.7.48) kF k h21 Rq g2 In particular, we have X g2

F .g/ D

1 kF k

Z

1 F .x/ dV .x/ C p kF k Rq

X h21 nf0g

Z Rq

F .x/˚h .x/ dV .x/: (10.7.49)

The sum on the left-hand side of the identity (10.7.48) is not necessarily absolutely convergent in Rq , so the process of summation must be specified. More precisely, following our approach (10.7.1), the convergence of the series on the left-hand side in Theorem 10.7.7 is understood in the spherical sense of (10.7.1).

10.8 Theta Functions The one-dimensional variant of the Theta function (see Definition 9.7.2) is reflected in an adequate way by the following multi-dimensional counterpart. Definition 10.8.1. For arbitrary points x; y 2 Rq and arbitrary lattices   Rq , we .q/ call #n . I x; yI / given by X

#n.q/ .I x; yI / D

2

e jgxj Hn .qI g  x/ e2igy ;

(10.8.1)

g2

where Hn .qI / 2 Harmn .Rq /,  2 C, and Re./ > 0, the Theta function of degree n and dimension q. .q/

Remark 10.8.2. Clearly, #n . I x; yI / is dependent on the homogeneous harmonic polynomial Hn W x 7! Hn .qI x/ D jxjn Yn .qI / ; x 2 Rq , x D jxj ; 2 Sq1 ; of degree n and dimension q, where Yn .qI / is a member of class Harmn .Sq1 /.

10 Lattice Functions in Rq

456

The function F W Rq ! C given by 2

F .z/ D e jzxj Hn .qI z  x/ e2izy ;

z 2 Rq ;

(10.8.2)

 1q  q . is of class C.1/ R .R /: F .z/ together R with all its derivatives is of the order o jzj The integrals Rq F .z/ dV .z/ and Rq jF .z/j dV .z/ are convergent. Thus, we obtain from the Poisson summation formula (Theorem 10.7.7) Z X 1 X F .g/ D F .z/ e2ihz dV .z/: (10.8.3) q kF k R 1 g2 h2

We write out (10.8.3) explicitly. In fact, in the nomenclature of the Theta function (see Definition 10.8.1) we are confronted with the following identity: #n.q/ .I x; yI / D

X

2

e jgxj Hn .qI g  x/ e2igy

(10.8.4)

g2

D

Z 1 X 2ix.yCh/ 2 e e jzxj Hn .qI z  x/e2i.zx/.yCh/ dV .z/: kF k Rq 1 h2

Introducing the polar coordinates z  x D r and y C h D % with ; 2 Sq1 ; we get #n.q/ .I x; yI / D

1 X 2ix.% / e (10.8.5) kF k h21  Z 1 Z 2 2ir% 

Yn .qI /e dS. / e r r nCq1 dr:  0

Sq1

The Funk–Hecke formula (i.e., Corollary 6.5.6) in connection with the integral representation (8.1.4) of the Bessel function gives us Z Sq1

Yn .qI /e2ir%  dS. / D in kSq1 k Jn .qI 2 r%/ Yn .qI /:

(10.8.6)

Thus, we get #n.q/ .I x; yI / D

  yCh in kSq1 k X 2ix.yCh/ (10.8.7) e Yn qI kF k jy C hj h21 Z 1 2  e r Jn .qI 2 r%/r nCq1 dr: 0

The integral on the right-hand side of (10.8.7) can be calculated by technicalities of the theory of Bessel functions.

10.8 Theta Functions

457

Lemma 10.8.3. For % > 0, n 2 N0 , and  2 C with Re./ > 0, we have Z

1 0

2

e r Jn .qI 2 r%/ r nCq1 dr D

%2 %n 1 q   : q q e 2 2  2  nC 2

(10.8.8)

Proof. Using the power series expansion of the Bessel function (see Lemma 8.1.5), we get Z

1 0

2

e r Jn .qI 2 r%/r nCq1 dr

D lim



q Z 2

e

2n

T !1

T

 r 2

(10.8.9) 1 X

r nCq1

0

kD0

.1/k .2 r%/nC2k 22k .k C 1/ .n C k C q2 /

! dr:

2

Because of je r j < 1 the series on the right-hand side of (10.8.9) is uniformly convergent and the members of the series are continuous. Thus, we are allowed to write Z

1

0

2

e r Jn .qI 2 r%/ r nCq1 dr

D lim

T !1



q  2 2n

1 X kD0

(10.8.10)

.1/k .2%/nC2k   2k 2 .k C 1/ n C k C q2

Z

T

2

e r r 2nC2kCq1 dr:

0

The series on the right-hand side of (10.8.10) converges uniformly with respect to T . In fact, we have, for sufficiently large positive M; N with %; , and n fixed, ˇ N ˇ Z T ˇX ˇ .1/k .2%/nC2k 2 ˇ ˇ  r 2nC2kCq1 e r dr ˇ ˇ q ˇ ˇ 22k .k C 1/ .n C k C 2 / 0

(10.8.11)

kDM



N X kDM

.2%/nC2k 2k 2 .k C 1/ .n C k C q2 /

Z

1

e r

2 Re. /

r 2nC2kCq1 dr

0

nCkC 2 1  Z 1 N X 1 r .2%/nC2k r  e dr; 2Re./ 22k .k C 1/ .n C k C q2 / 0 Re./ q

kDM

where

Z

1 0

 q q : eu unCkC 2 1 du D n C k C 2

(10.8.12)

Thus, the sum and the limit on the right-hand side of (10.8.10) may be interchanged

10 Lattice Functions in Rq

458

Z

1 0

2

e r Jn .qI 2 r%/ r nCq1 dr

(10.8.13)

1 . q2 / X .1/k .2%/nC2k   D 2n 22k .k C 1/ n C k C q2

Z

kD0

1

2

e r r 2nC2kCq1 dr:

0

Together with (10.8.12) this implies Z

1 0

2

e r Jn .qI 2 r%/ r nCq1 dr

(10.8.14)

 q  2 k 1 1  q 1 2  % n X .1/k % D : 2 2   .k C 1/  kD0

Summing up the last exponential series, we arrive at the identity Z

1 0

2

e r Jn .qI 2 r%/r nCq1 dr D

%2 1 q %n   : q q e 2 2  2  nC 2

(10.8.15) t u

This is the desired result of Lemma 10.8.3. Therefore, we obtain from Lemma 10.8.3 that X 2 e jgxj Hn .qI g  x/ e2igy #n.q/ .I x; yI / D

(10.8.16)

g2

    X in Sq1  q2  2 2ixy D e  jhCyj Hn .qI h C y/ e2ihx : q q e 2kF k  2  nC 2 1 h2

 1 q  Together with kSq1 k D 2 2 q2 , this yields the functional equation of the Theta function of degree n and dimension q. .q/

Theorem 10.8.4. For all  2 C with Re./ > 0, the Theta function #n .I x; yI / is holomorphic and we have #n.q/ .I x; yI / D

in 2ixy n q .q/ 2 # e  n kF k



 1 I y; xI 1 : 

(10.8.17)

The functional equation allows a palette of specializations or advancements. For example, it helps us to describe some properties for the lattice  D Zq and halfvalued vectors, i.e., x 2 Rq of the form q

xD

 1X 1 1 k1 " C : : : C kq "q ; kj "j D 2 j D1 2

(10.8.18)

10.8 Theta Functions

459

where k1 ; : : : ; kq are integers (note that x 2 D Theorem 10.8.4 q

#n.q/ .I x; xI Zq / D in  n 2 q

D in  n 2

1 4

Pq

2 j D1 kj ).

In this case we get from

  gCx  2 e2i.gCx/x e  jgCxj jg C xjn Yn qI jg C xj g2Zq X

  gx  2 e2i.gx/x : e  jgxj jg  xjn Yn qI jg  xj g2Zq X

(10.8.19)

It should be noted that a translation by 2x D

q X

kj "j D k1 "1 C : : : C kq "q 2 Zq

j D1

leaves the Theta series unchanged. Consequently, we find, for the half-valued vectors x 2 Rq of the form (10.8.18), that   q X gx .q/ q n 2ix 2 n 2   jgxj2 n #n .I x; xI Z / D i e  e jg  xj Yn qI e2igx jg  xj q g2Z

n 2ijxj2

Di e

2



q

n 2

#n.q/ q

2

D in .i/k1 C:::Ckq  n 2



 1 q I x; xI Z    1 .q/ q #n I x; xI Z : 

(10.8.20)

Summarizing our results, we obtain q Lemma 10.8.5.  1Let  2 C with  Re./ > 0, n 2 N0 . For all points x 2 R of the 1 q form x D 2 k1 " C : : : C kq " , kj 2 Z, j D 1; : : : ; q, we have   q 1 .q/ q n 2ijxj2 n 2 .q/ q I x; xI Z : (10.8.21)  #n #n .I x; xI Z / D i e 

If n C k12 C : : : C kq2 D n C 4x 2 is an odd number, then the expression n

q

(10.8.22)

./ D  2 C 4 #n.q/ .I x; xI Zq / 1 satisfies the property ./ D ˙i  . The substitution  D e provides a holomorphic function Q W  7! ./ Q D .e /,  2 < Im./ < 2 , such that ./ Q D ˙i ./. Q .m/ m Q (cf. Knopp 1971) which yields that Therefore, we find Q .0/ D ˙.1/ i .0/

Q .m/ .0/ D 0 for m 2 N0 . This means Q D 0, hence, D 0. Lemma 10.8.6. If n C 4x 2 is an odd number, where the point x 2 Rq is of the form x D 12 .k1 "1 C : : : C kq "q /, kj 2 Z, j D 1; : : : ; q, then #n.q/ .I x; xI Zq / D 0

(10.8.23)

10 Lattice Functions in Rq

460

independently of the choice of the spherical harmonic Yn .qI / 2 Harmn .Sq1 / of degree n and dimension q. An immediate consequence is the following remarkable result. Theorem 10.8.7. Let n C 4x 2 be an odd number, where the point x 2 Rq is of the form x D 12 .k1 "1 C : : : C kq "q /, kj 2 Z, j D 1; : : : ; q. If l 2 N is given in such a way that there exist elements g 2 Zq with jg  xj2 D 4l , then X jgxj2 D 4l g2Zq

  gx e2igx D 0 Yn qI jg  xj

(10.8.24)

independently of the choice of the spherical harmonic Yn .qI / 2 Harmn .Sq1 / of degree n and dimension q. Remark 10.8.8 (Classical Theta Functions). The classical Theta functions can easily be expressed in terms of the Theta function (Definition 10.8.1). More concretely, for Im./  0, #0 .s; / D

X

.1/ 

2

ei n ei n.2sC1/ D #0

   iI 0; s C 12 "1 I Z ;

(10.8.25)

n2Z

#1 .s; / D i

X 1 2 1 X 1 2 1 .1/n ei .n 2 / ei.2n1/s D ei .sC 2 / ei .n 2 / e2i n.sC 2 / n2Z

 .1/  #0 iI 12 "1 ; s



n2Z

 D ie C " IZ ; X X 1 2 1 2 #2 .s; / D ei .nC 2 / ei.2nC1/s D eis ei .nC 2 / e2i ns is

n2Z

D #3 .s; / D

1 2

 .1/  eis #0 iI  12 "1 ; s"1 I Z ; X

e

i n2

e2i ns

1

(10.8.26)

n2Z

 .1/  D #0 iI 0; s"1 I Z :

(10.8.27) (10.8.28)

n2Z

For more details the reader is referred to, e.g., Magnus et al. (1966) and the cited literature therein.

10.9 Exercises (Algebraic, Periodic, and Spherical Splines, Lattice Point Sums, Lattice Point Distributions) As we pointed out in our introduction (see Chap. 1), every phenomenon, e.g., in Earth’s sciences, can be virtually described in terms of differential or integral equations based on the laws of physics, relating various quantities of relevance

10.9 Exercises

461

and interest. While the derivation of the governing equations is usually not unduly difficult, their solution by analytical methods is a formidable task. In such cases, approximate methods provide alternative means of finding solutions. Among these the spline method plays an important role in modern numerics. Indeed, whenever problems of interpolation, smoothing, or best approximation of discrete data information occur, splines that are adapted to the specific features of the problem are applicable. The reason is both physically and numerically motivated. Splines provide approximation under variational constraints to suppress artificial oscillations between the discrete data points. In consequence, the spline method is endowed with two basic features which make it appealing for computation. First, the operator which is involved in the differential and/or integral equation (i.e., pseudodifferential equation) under consideration, is reflected by the use of an associated Green functions at the finite number of preselected nodes. Second, smooth approximation is derived by minimizing a (linearized) “curvature norm” such that the “oscillation energy” is minimal. In the following, we discuss the spline method starting from the classical one-dimensional algebraic and periodic cases and going over to the multi-variate spherical as well as periodic cases. It turns out that the constituting ingredients (differential operator, associated Green function) and construction principles (interpolation, smoothing, best approximation) can be handled analogously in all cases.

Algebraically Polynomial Splines Definition 10.9.1. By ˘k we denote the space of polynomials of degree lower than k or equal to k in one variable (see Definition 3.0.8) and by xC the truncated power function ( k xC

D

xk

;

x > 0;

0

;

x  0:

(10.9.1)

Now, let x1 ; x2 ; : : : ; xn be points in R. We say a scalar function S of class C.k1/ .R/ is a spline of degree k with knots x1 ; : : : ; xn if there exist constants c1 ; : : : ; cn 2 R and a polynomial P 2 ˘k such that S.x/ D P .x/ C

n X

cj .x  xj /kC ;

x 2 R:

(10.9.2)

j D1

The set containing all splines of degree k with knots x1 ; : : : ; xn is denoted by Sk .x1 ; : : : ; xn /.

10 Lattice Functions in Rq

462

Definition 10.9.2. Let x1 ; x2 ; : : : ; xn 2 R and k  1. A spline S of degree 2k  1 with knots x1 ; : : : ; xn is called a natural spline if S.x/ D P .x/ C

n X

cj .x  xj /2k1 ; C

x2R

(10.9.3)

j D1

for a polynomial P 2 ˘k1 and constants c1 ; : : : ; cn 2 R with n X

cj xj` D 0;

` D 0; : : : ; k  1:

(10.9.4)

j D1

By Spline2k1 .x1 ; : : : ; xn / we denote the set of all natural splines of degree 2k  1 with knots x1 ; : : : ; xn . Exercise 10.9.3 (Spline Integration Formula). Assume that a  x1 < x2 < : : : < xn  b:

(10.9.5)

Let S 2 S2k1 .x1 ; : : : ; xn / be a spline of degree 2k  1. Show that if a function F W Œa; b ! R has the properties (i) F 2 C.k1/ .Œa; b/ and F .k/ is continuous on each interval .a; x1 /, .xn ; b/ and .xj ; xj C1 /, j D 1; : : : ; n  1, (ii) F .k`1/ .x/S .kC`/ .x/ D 0, ` D 0; : : : ; k  2, where x D a or x D b, (iii) (using the notation G.x˙/ D lim G.x ˙ h/ for the one-sided limits of a function G)

h!0C

F .a/S .2k1/ .a/ D F .b/S .2k1/ .bC/ D 0;

(10.9.6)

then the following spline integration formula holds true: Z

b

F

.k/

.x/S

.k/

k

.x/ dx D .1/ .2k  1/Š

a

n X

cj F .xj /:

(10.9.7)

j D1

Exercise 10.9.4 (Spline Integration Formula). Suppose that (10.9.5) is given. Prove that if F 2 C.k1/ .Œa; b/ and F .k/ is continuous on each interval .a; x1 /, .xn ; b/ and .xj ; xj C1 /, j D 1; : : : ; n  1, then we find the following spline integration formula for every natural spline S 2 Spline2k1 .x1 ; : : : ; xn / of degree 2k  1: Z

b a

F .k/ .x/S .k/ .x/ dx D .1/k .2k  1/Š

n X

cj F .xj /:

(10.9.8)

j D1

Exercise 10.9.5 (Uniqueness of Spline Interpolation). Assume that the abscissas x1 < x2 < : : : < xn as well as y1 ; : : : ; yn 2 R and 1  k  n are given. Show

10.9 Exercises

463

that there exists a uniquely determined natural spline S 2 Spline2k1 .x1 ; : : : ; xn / interpolating .x1 ; y1 /; : : : ; .xn ; yn /, i.e., S.xj / D yj ;

j D 1; : : : ; n:

(10.9.9)

Exercise 10.9.6 (Smoothest Interpolation Problem). Assume that (10.9.5) is given. Let there be known y1 ; : : : ; yn 2 R for 1  k  n. Show the following assertions: .k/

.k/

(a) Let F1 ; F2 2 C.k1/ .Œa; b/ with F1 ; F2 Z

b a

.k/

piecewise continuous and

.k/

F1 .x/F2 .x/ dx D 0;

(10.9.10)

then we have, for F D F1 C F2 ; .k/

kF1 k2L2 .Œa;b/  kF .k/ k2L2 .Œa;b/ :

(10.9.11)

If we further assume that F2 vanishes in k or more distinct points of Œa; b, then equality holds in (10.9.11) if and only if F D F1 . (b) Let S 2 Spline2k1 .x1 ; : : : ; xn / be the unique natural spline interpolating the points .x1 ; y1 /; : : : ; .xn ; yn /. Furthermore, let F 2 C.k1/ .Œa; b/ be any other function with F .k/ continuous on each interval .a; x1 /, .xn ; b/, and .xj ; xj C1 / for j D 1; : : : ; n  1, that is also interpolating .x1 ; y1 /; : : : ; .xn ; yn /. Then, we have kS .k/ k2L2 .Œa;b/  kF .k/ k2L2 .Œa;b/ : (10.9.12) Note that a transition to the Sobolev space structure of H .2/ .Œa; b/ can be avoided in Exercises 10.9.3–10.9.6. Exercise 10.9.7 (Spline Exact Integration). Assume that (10.9.5), m D k  1, and 1  k  n are given. Let Z

b

I.F / D

F .x/ dx

(10.9.13)

a

for F 2 C.m/ .Œa; b/. Prove the following assertions: (a) There exists a unique sum Q of the form Q.F / D

n X

bj F .xj /

(10.9.14)

j D1

satisfying I.F / D Q.F / for all F 2 Spline2k1 .x1 ; : : : ; xn /.

(10.9.15)

10 Lattice Functions in Rq

464

(b) For a given F 2 C.m/ .Œa; b/ let SF 2 Spline2k1 .x1 ; : : : ; xn / denote the unique natural spline interpolating the points .x1 ; F .x1 //; : : : ; .xn ; F .xn //. Then, there exists a uniquely defined Q of the form (10.9.14) with the property Q.F / D I.SF /

(10.9.16)

for every F 2 C.m/ .Œa; b/. (c) There is a Q of form (10.9.14) uniquely determined by the conditions that Q.F / D I.F /

(10.9.17)

for every F 2 ˘k1 and that there exists a T 2 ˘k1 such that

D T .xj /; R .x  xj /2k1 C

j D 1; : : : ; n;

(10.9.18)

where R D I  Q. Hint: Show (c) first and then use it to show the existence of Q in the cases (a) and (b). Exercise 10.9.8 (Peano’s Theorem). Let F 2 C.mC1/ .Œa; b/ and let L be a linear functional of the form Z L .F / D

m bX a kD0

ak .x/F .k/ .x/ dx C

jk m X X

bi;k F .xi;k /

(10.9.19)

kD0 i D1

with the property that L P D 0 for all P 2 ˘m . Prove that Z b L .F / D K.t/F .mC1/ .t/ dt;

(10.9.20)

a

where

1 Lx .x  t/m (10.9.21) C : mŠ The function K given by (10.9.21) is called the Peano kernel of the functional L . K.t/ D

Exercise 10.9.9 (Sch¨onberg’s Theorem). Prove the following theorem which is known as Sch¨onberg’s Theorem: Assume that a  x1 < x2 < : : : < xn  b. Let I be of the form (10.9.13) with m D k  1 < n. Let Q0 be any approximation of I of the form (10.9.14) exact for degree k  1, and let Q be the unique approximation from Exercise 10.9.7. If K is the Peano kernel of I  Q and K 0 the Peano kernel of I  Q0 , we get Z

b

jK.t/j2 dt 

a

with equality only if Q0 D Q.

Z a

b

jK 0 .t/j2 dt

(10.9.22)

10.9 Exercises

465

Exercise 10.9.10. Suppose that a D x1 < x2 < : : : < xn D b are equidistant knots with  D x2  x1 and F1 ; : : : ; Fn 2 R. Furthermore, let S be a spline of class S3 .x1 ; : : : ; xn / (observe that S 2 C.2/ .R/) with S.xj / D Fj ;

j D 1; : : : ; n;

S .2/ .a/ D S .2/ .b/ D 0;

(10.9.23) (10.9.24)

and let mj D S .2/ .xj /, j D 1; : : : ; n. Show that S possesses the representation S.x/ D

.xj C1  x/3 mj C .x  xj /3 mj C1 .xj C1  x/Fj C .x  xj /Fj C1 C 6   1    .xj C1  x/mj C .x  xj /mj C1 (10.9.25) 6

for x 2 .xj ; xj C1 /, and that the mj are given by 2

4 61 6 6 6 6 6 4

1 4 1 :: :: :: : : : :: :: : : 0 1

m1 D mn D 0; 32 3 2 3 m2 0 F1  2F2 C F3 7 6 m3 7 6 F2  2F3 C F4 7 76 7 6 7 7 6 :: 7 6 7 :: 6 76 : 7 D 6 7: : 76 7 6 7 2  6 76 : 7 7 :: : 4 5 5 4 5 1 : : 4 mn1 Fn2  2Fn1 C Fn

(10.9.26)

(10.9.27)

Remark 10.9.11. The case k D 2 is of particular significance. The integral Zb

jF 00 .x/j dx

(10.9.28)

a

may be physically interpreted (at least in linearized sense) as the potential energy of a statically deflected thin beam which is indeed proportional to the integral taken over the square of the (linearized) curvature of the elastics of the beam. For more details the reader is referred to, e.g., Greville (1969).

Periodic Polynomial Splines Definition 10.9.12. Suppose that XN D fx1 ; : : : ; xN g  F is a set of mutually distinct points. Then, we denote any function S of the form S.x/ D C C

N X j D1

aj G. I x  xj /;

x2R

(10.9.29)

10 Lattice Functions in Rq

466

with

N X

aj D 0

(10.9.30)

j D1

as Z-periodic spline function (of order 0) relative to the point set XN . Exercise 10.9.13 (Spline Integration Formula). Suppose that F is of class .2/ HZ .R/ as introduced in Definition 9.8.12. Use the Euler summation formula (Theorem 9.4.1) to show that N X

Z aj F .xj / D

j D1

S.x/F .x/ dx:

(10.9.31)

F

Exercise 10.9.14 (Uniqueness of Spline Interpolation). Given .xj ; yj / 2 XN  R. Then, there exists a unique SN of the form (10.9.29) which fulfills (10.9.30) and the interpolation property by SN .xj / D yj ;

j D 1; : : : ; N:

(10.9.32)

Exercise 10.9.15 (Smoothest Interpolation Problem). Let the pointset .xj ; yj / 2 XN  R; j D 1; : : : ; N , be given. Show that Z

Z

2

F

jSN .x/j dx 

jF .x/j2 dx

(10.9.33)

F

.2/

holds true for all F 2 HZ .R/ satisfying the interpolation conditions F .xj / D yj ;

j D 1; : : : ; N;

(10.9.34)

with equality in (10.9.33) if and only if F D SN . Exercise 10.9.16 (Smoothing). Let ı be a positive constant and ˇ1 ; : : : ; ˇN > 0 positive coefficients. Prove that there exists a uniquely determined spline of the form (10.9.29) such that  N  X S.xj /  yj 2 j D1

ˇj

Z

.S.x//2 dx 

Cı F

 N  X F .xj /  yj 2 j D1

ˇj

Z

.F .x//2 dx

Cı F

(10.9.35) .2/ holds for all F 2 HZ .R/ with equality in (10.9.35) if and only if F D S . Show that S satisfies the linear equations S.xj / C ıˇj2 aj D yj ;

j D 1; : : : ; N

(10.9.36)

10.9 Exercises

467

with .xj ; yj / 2 XN  R, j D 1; : : : ; N , being the point set that represents the data under consideration. Remark 10.9.17. The set ˇ1 ; : : : ; ˇN are given positive weights, which should be adapted to the standard derivation of the measurements y1 ; : : : ; yN and ı is a constant chosen by the user to control the “smoothness” of S . Exercise 10.9.18 (Peano’s Theorem). Prove the following theorem which is known as Peano’s theorem: .2/ Let L be a bounded functional on HZ .R/, e.g., of one of the types L .F / DF .x/; 0

L .F / DF .x/; .2/

for F 2 HZ .R/ which fulfills

R F

x 2 F;

(10.9.37)

x 2 F;

(10.9.38)

F .x/ dx D 0. Then, Z

L .F / D

K.y/F .y/ dy

(10.9.39)

F

.2/

holds for all F 2 HZ .R/, where the Peano kernel K is given by K.y/ D Lx ŒG.I x  y/:

(10.9.40)

Note that Lx means that the linear functional L is applied to the variable x. Exercise 10.9.19 (Approximation of Linear Functionals). Let L be a bounded .2/ linear functional on HZ .R/. Then, ˇ ˇ 0 ˇ ˇ N N X X ˇ ˇ



2 ˇL .F /  ˇ @ aj F .xj /ˇ  Lx Ly G. I x  y/  2 Lx G.2 I x  xj / ˇ ˇ ˇ j D1 j D1 C

N N X X j D1 kD1

1 12 aj ak G.2 I xj ; xk /A

Z

jF .x/j2 dx

 12

F

(10.9.41) holds for all selections of coefficients a1 ; : : : ; aN satisfying (10.9.30).

Spherical Spline Interpolation One-dimensional periodic splines show an extensive study in the literature (see, e.g., Schoenberg 1964; Schumaker 1981 and the references therein). In the form as they are developed here from the Euler summation formula, canonical generalizations

10 Lattice Functions in Rq

468

to the q-dimensional spherical as well as periodic cases become accessible. More concretely, our exercises on q-dimensional spherical splines are concerned with iterated Beltrami operators and an approximation order 0, while the q-dimensional periodic spline theory is based on the iterated Laplace operator and on arbitrary lattices. Clearly, more general (pseudo-)differential operators may be included (see Freeden 1981, 1984; Freeden and Hermann 1986; Freeden et al. 1997, 1998 for the spherical theory and Freeden 1988; Freeden and Reuter 1981 for the periodic case). Exercise 10.9.20. Let 1 ; : : : ; N 2 Sq1 be given nodes. Derive from the integral formula for the iterated Beltrami operator . /m (see Theorem 6.3.6) the spline integration formula Z Sq1

. /m S. /. /m F . / dS. / D

N X

aj F . j /

(10.9.42)

j D1

for all F 2 C.2m/ .Sq1 /, m 

q1 4 ,

and all coefficients a1 ; : : : ; aN 2 R with N X

aj D 0;

(10.9.43)

j D1

where the spherical spline S with respect to . /m of order 0 is given by S. / D C0;1 Y0;1 .qI / C

N X

  aj G . /m I ; j ;

2 Sq1 :

(10.9.44)

j D1

Exercise 10.9.21. Show for all splines S of the form (10.9.44) that Z Sq1

. /m S. /. /m S. / dS. / D

N X

aj S. j /

(10.9.45)

j D1

holds true for all coefficients a1 ; : : : ; aN 2 R satisfying (10.9.43). Exercise 10.9.22. Let ˛1 ; : : : ; ˛N 2 R be given. Show that there exists one and only one function S of type (10.9.44) satisfying S. j / D ˛j , j D 1; : : : ; N . We denote this interpolating spline by SN . Exercise 10.9.23. Verify that, for all F 2 C.2m/ .Sq1 /, m > ˛j , j D 1; : : : ; N , Z Sq1



 m

. / F . /

2

Z dS. / D

Sq1

with F . j / D

  m 2 . / SN . / dS. /

Z

C

q1 4 ,

Sq1



. /m SN . /  . /m F . /

(10.9.46) 2

dS. /:

In analogy to Definition 9.8.12 we introduce Sobolev spaces on the sphere.

10.9 Exercises

469

Definition 10.9.24. The Sobolev-space H .2m/ .Sq1 /, m >

q1 , 4

is defined by

H .2m/ .Sq1 / D n

(10.9.47)

F 2 C.1/ .Sq1 / W kF k2L2 .Sq1 / C k. /m F . /k2L2 .Sq1 / < 1

okkL2 .Sq1 /

:

Exercise 10.9.25. Show that spherical spline interpolation is well-posed in the sense that SN exists in H .2m/ .Sq1 /, is unique, and obeys Z Sq1

.. /m SN . // dS. / D min F 2Y

Z Sq1

.. /m F . //2 dS. /;

(10.9.48)

where Y  H .2m/ .Sq1 / consists of all functions F 2 H .2m/ .Sq1 / satisfying F . j / D ˛j , j D 1; : : : ; N . Exercise 10.9.26. Let L be a bounded linear functional on H .2m/ .Sq1 /. Suppose that N is given by N X aj F . j / (10.9.49) N .F / D j D1 .2m/

q1

for F 2 H .S /. Prove the following assertions: (a) There exists a unique functional N of the type (10.9.49) satisfying (i) L ŒY0;1 .qI / D N ŒY0;1 .qI /, (ii) L ŒG.. /2m I ; k / D N ŒG.. /2m I  k /, k D 1; : : : ; N , where Y0;1 .qI / is the member of the orthonormal basis of spherical harmonics that has degree 0. (b) Let F 2 H .2m/ .Sq1 / be given. Suppose that SF denotes the unique spline function of type (10.9.44) satisfying F . j / D SF . j /, j D 1; : : : ; N . Then, N of the form (10.9.49) is also uniquely determined by the property that N .F / D L .SF / for every F 2 H .2m/ .Sq1 /, where L is again a linear functional of any one of the types in Exercise 10.9.18. (c) N is uniquely determined by the requirement that L .S / D N .S /, whenever S is of the form (10.9.44). Remark 10.9.27. For m D 1 and q D 3,Rthe function G.. /2m I ; / is explicitly known (see Remark 4.6.6). In this case S2 j F . /j2 dS. / may be physically interpreted (in a linearized sense) as the bending energy of a (thin) membrane spanned wholly over S2 . F denotes the deflection normal to the rest position supposed, of course, to be spherical. Remark 10.9.28. More details can be found in Freeden (1981, 1984), Freeden and Hermann (1986), and Freeden et al. (1997, 1998).

10 Lattice Functions in Rq

470

Multi-periodic Monosplines Exercise 10.9.29. Prove the following assertion: Let  be an arbitrary lattice in Rq . Let G  Rq be a regular region. Suppose that F is a member of class C.2m/ .G /, m 2 N. Then, for every x 2 F , X0

F .g C x/ D

gCx2G g2

1 kF k Z

C @G

Z

Z F .y/ dV.q/ .y/ C

G

G

G.m I x  y/m F .y/ dV.q/ .y/

ŒF .y/; G.m I x  y/ dS.q1/ .y/;

(10.9.50)

where Z @G

ŒF .y/; H.y/ dS.q1/ .y/ D





m1 XZ rD0 @G m1 XZ



rD0 @G

 @ m.rC1/ H.y/ .r F .y// dS.q1/ .y/  @

m.rC1/ H.y/

  @ r F .y/ dS.q1/ .y/ @

(10.9.51) and

P0

is defined in (10.4.4).

Exercise 10.9.30. Show that G.m I x/ D

1 .4 2 /m .m/

X ˚h .x/ m 2 .m; h /  jhj2m m

h2nf0g

C

2m 32

(10.9.52)

! X . 3  m; jg  xj2 / 2 ; jg  xj2m3 g2

where  D Z3 , m > 1, and .x/ D .v; x/ C .v; x/ with the incomplete Gamma functions of Definition 2.5.1. Remark 10.9.31. The sum (10.9.52) in this form holds for x 2 F n f0g. For the case x D 0, the second sum on the right-hand side should be taken over all g 2 Z3 with g ¤ 0 and the term  m .m  32 / should be added (see Nijboer and de Wette 1957). Exercise 10.9.32. Show by use of Exercise 10.9.29 that the following result holds true: For given W 2 C.0/ .G /, given XN D fx1 ; : : : ; xN g  F and weights a1 ; : : : ; aN , let K designate a monospline of the form

10.9 Exercises

471

K.x/ D V .x/ C S.x/

(10.9.53)

with N X

S.x/ D C0 ˚0 .x/ C

aj G.2m I x  xj /;

C0 D const;

(10.9.54)

j D1

where V 2 C.2m/ .G / satisfies

m V D W

(10.9.55)

in G . Then, for each F 2 C.2m/ .G /, N X

X0

aj

j D1

Z F .g C xj / D

gCxj 2G g2

G

@G

N X

Z aj

G

j D1

Z

ŒF .y/; K.y/ dS.q1/ .y/ C

@G

R

kF k

Z C

where

1

W .y/F .y/ dV.q/ .y/ C

G

F .y/ dV.q/ .y/

K.y/m F .y/ dV.q/ .y/;

(10.9.56)

ŒF .y/; K.y/ dS.q1/ .y/ is defined by (10.9.51) and the remainder term Z R.F / D G

K.y/m F .y/ dV.q/ .y/

(10.9.57)

vanishes for all functions F with m F D 0 in G . Exercise 10.9.33. Show that the following remainder estimates for R.F / with F of class C.2m/ .G / and K as in Exercise 10.9.32 are valid: (a) For m  b q2 c C 1; Z

m

jR.F /j  sup j F .x/j G

x2G

jK.y/j dV.q/ .y/:

(10.9.58)

(b) For m  12 .b q2 c C 1/; Z

m

jR.F /j  G

2

j F .y/j dV.q/ .y/

 12 Z

2

G

jK.y/j dV.q/ .y/

 12

: (10.9.59)

Exercise 10.9.34 (Optimal Rules for Monosplines). For m  12 .b q2 c C 1/, let for given XN D fx1 ; : : : ; xN g  F the function  .a1 ; : : : ; aN / depending on the weights a1 ; : : : ; aN be defined by Z  .a1 ; : : : ; aN / D

2

F

jK.y/j dV.q/ .y/

 12

;

(10.9.60)

10 Lattice Functions in Rq

472

where K, V are defined as in (10.9.53) and (10.9.54) of Exercise 10.9.32, but V is .2m/ now of class C .Rq /, i.e., a -periodic function. Show the following assertions: .2m/

(a) For all F 2 C .Rq /; it holds (see (10.9.51) for the definition of this expression) that Z ŒF .y/; K.y/ dS.q1/ .y/ D 0: (10.9.61) @F

ı

ı

(b) For fixed xj , j D 1; : : : ; N , let  .a1 ; : : : ; aN / be given by ı



ı

 .a1 ; : : : ; a N / D min  .a1 ; : : : ; aN / W aj 2 R;

N X

aj D 0 :

(10.9.62)

j D1 ı

Formulate the linear system determining the coefficients aj , j D 1; : : : ; N of R the optimal cubature rule for F W .y/F .y/ dV .y/, i.e., N X

aj

j D1

X0

Z F .g C xj / D

gCxj 2G g2

G

W .y/F .y/ dV.q/ .y/ C R.F /

(10.9.63)

where the remainder term R.F / given by (10.9.57) is minimized. .2m/

Remark 10.9.35. Note that the transition to the Sobolev space H .Rq / is .s/ obvious (cf. Definition 9.8.12), where H .Rq / is understood as the closure of .1/ C .Rq / with respect to the norm 0 kF kH .s/ .Rq / D @jF ^ .0/j2 C 

X

1 12 jF ^ .h/j2 .4 2 h2 / A ;

(10.9.64)

h2nf0g

i.e., .s/

s 2

.1/

H .Rq / D C .Rq /

kk

.s/

H .Rq /

:

(10.9.65)

For more details see Freeden (1981, 1982, 1988) and Freeden and Fleck (1987).

Lattice Points and Lattice Balls Inside Balls The branch of analytic theory of numbers concerned with lattice point summation has a long history which reaches back to Euler (1736a,b) and Gauß (1801, 1826). Enlightening accounts of the development on asymptotic expansions of lattice points in balls are due to Erd¨os et al. (1989), Hardy (1915), Hardy and Landau (1924), Kr¨atzel (1988), Landau (1915, 1927, 1962), and Walfisz (1960) to mention just a few. While exact asymptotic relations on lattice points in balls are known for

10.9 Exercises

473

dimensions q  4, the cases q D 2 and q D 3 are still a challenge for future work. The reason is the alternating character of lattice point sums caused by the occurring Bessel functions such that these sums cannot be handled in the same way as, e.g., by the Leibniz criterion known from one-dimensional analysis. Survey publications concerning the lattice point theory in dimensions q  2 are, e.g., the contributions by Freeden (2011) and Fricker (1975, 1982). Exercise 10.9.36. Show that Z J1 .qI 2jwjR/ q R e2iwz dV .z/ D kSq1 k q 2jwjR BR .0/

(10.9.66)

holds for all R > 0 and all w 2 Rq . Exercise 10.9.37. Use the Euler summation formula (Theorem 10.4.1) to verify the lattice point identity X0

q

1D

q g2BN \

kBN k C kF k

Z q1 SN

@ G.I x/ dS.x/: @

(10.9.67)

Exercise 10.9.38. For  2 .0; infx2@F jxj/, let G .I / be defined by 1 q kB k

G .I x/ D

Z q

G.I x  y/ dV .y/;

B

Prove that x G .I x/ D where Bq C .x/ D 

1 q B

x 2 Rq n :

 q  kB k ; Bq C .x/   kF k

8 0 in form of an alternating series in terms of Bessel functions of dimension q D 2, namely X0

1 D lim R N !1

g2B2R \Z2

Note that

X0

1 D

g2B2R \Z2

X h2B2N \Z2

X

1 C

g2B2R \Z2

J1 .2I 2jhjR/ : jhj

1 2

X

1

(10.9.75)

(10.9.76)

g2@B2R \Z2

as elaborated in Theorem 10.4.1 (also see (6.2.33) for the definition of the solid angle in Rq ). Remark 10.9.43. For the one-dimensional lattice Z the reader is referred to (9.6.10). In the following, our exercises provide a proof of the Hardy–Landau identity, however, by use of the two-dimensional Euler summation formula with respect to Helmholtz operator  C 4 2 R2 and for arbitrary lattices . Definition 10.9.44. The function G. C I  / W R2 n  ! R,  2 R fixed, is called -lattice function for the Helmholtz operator  C  in R2 if it satisfies the following properties:

10.9 Exercises

475

(i) (Periodicity) For all x 2 R2 n  and g 2 , G. C I x C g/ D G. C I x/:

(10.9.77)

(ii) (Differential equation) G. C I  / is twice continuously differentiable for all x … . For  … Spect ./; . C /G. C I x/ D 0:

(10.9.78)

For  2 Spect ./; X 1 ˚h .x/: . C /G. C I x/ D  p kF k 4 2 h2 ¤

(10.9.79)

h21

(iii) (Characteristic singularity) The function x 7! G. C I x/ 

1 ln jxj 2

is continuously differentiable for all x 2 F . (iv) (Normalization) For all h 2 1 with 4 2 h2 D ; Z G. C I x/˚h .x/ dV.2/ .x/ D 0:

(10.9.80)

(10.9.81)

F

Exercise 10.9.45. Prove by virtue of the second Green theorem (Theorem 6.2.2) that for h 2 1 with 4 2 h2 D ; Z 1 1 G. C I x/˚h .x/ dV.2/ .x/ D : (10.9.82) kF k   4 2 h2 F Show also that G. C I  / is uniquely defined by the four constituting properties of Definition 10.9.44. Hint: Observe that the formal Fourier series of G. C I  / is X 1 ˚h p :   4 2 h2 kF k 4 2 h2 ¤

(10.9.83)

h21

Exercise 10.9.46. Prove the two-dimensional Euler summation formula for the Helmholtz operator  C ,  2 R, and a regular region G  R2 :

10 Lattice Functions in Rq

476

X0

F .g/ D p

g2G \

X

1 kF k Z

C Z C

G

Z

4 2 h2 D h21

G

F .x/˚h .x/ dV.2/ .x/

(10.9.84)

G. C I x/. C /F .x/ dV.2/ .x/

@G

     @ @F F .x/ G. C I x/  G. C I x/ .x/ dS.1/ .x/; @ @

where, by convention, the sum X Z 4 2 h2 D h21

G

F .x/˚h .x/ dV.2/ .x/

(10.9.85)

is understood to be zero if there exists no lattice point h 2 1 which fulfills 4 2 h2 D . Exercise 10.9.47. Verify that x

J2 .2I 2jxjR/ J1 .2I 2jxjR/ J1 .2I 2jxjR/ C 4 2 R2 D 4R jxj jxj jxj2

(10.9.86)

holds for all x 2 R2 . Exercise 10.9.48. Apply the asymptotic expansion of Bessel functions (see Lemma 8.3.4) to guarantee ˇ ˇ   ˇ J1 .2I 2jxjR/ ˇ 1 ˇ ˇDO ; ˇ ˇ jxj jxj3=2 ˇ ˇ   ˇ J1 .2I 2jxjR/ ˇ 1 ˇrx ˇDO ; ˇ ˇ jxj jxj3=2 ˇ ˇ   ˇ ˇ 1 ˇ. C 4 2 R2 / J1 .2I 2jxjR/ ˇ D O ; ˇ ˇ jxj jxj5=2

jxj ! 1;

(10.9.87)

jxj ! 1;

(10.9.88)

jxj ! 1:

(10.9.89)

Exercise 10.9.49. Use the Euler summation formula of Exercise 10.9.46 for the value  D 4 2 R2 and the asymptotic expansions (10.9.87)–(10.9.89) of Exercise 10.9.48 to establish the limit relation  lim

N !1

X g2B2N \

Z

J1 .2I 2jgjR/ 2  jgj kF k 1

D 4R 0

J2 .2I 2 rR/ r2

Z S1r

X h2S1R \1

Z

N 0

 J1 .2I 2 rR/J0 .2I 2jhjr/ dr

G. C 4 2 R2 I x/ dS.1/ .x/ dr:

(10.9.90)

10.9 Exercises

477

Exercise 10.9.50. Show that Z 1 Z J2 .2I 2 rR/ 4R G. C 4 2 R2 I x/ dS.1/ .x/ dr (10.9.91) r2 S1r 0 Z 1 X 2 4R J2 .2I 2 rR/ J0 .2I 2jhjR/ dr D 2 R 2  4 2 h2 kF k 4 r 0 1 1 h2

D

2 kF k

nSR

X

Z

1 0

h21 nS1R

J1 .2I 2 rR/J0 .2I 2jhjr/ dr:

(10.9.92)

Exercise 10.9.51. Use Exercise 7.4.2 on discontinuous integrals involving Bessel functions of dimension q D 2 to verify the limit X

lim

N !1

g2B2N \

1 J1 .2I 2jgjR/ D jgj kF kR

X0

1:

(10.9.93)

h2B2R \1

Remark 10.9.52. Replacing   R2 by its inverse lattice 1  R2 we finally obtain the Hardy–Landau identity for an arbitrary lattice   R2 , i.e., X0

R N !1 kF k

X

1 D lim

g2B2R \

h2B2N \1

J1 .2I 2jhjR/ : jhj

(10.9.94)

Exercise 10.9.53. Show that, for all R > 0 and all lattices   R2 ; X0

1D

g2B2R \

R R2 C lim kF k N !1 kF k

X h2B2N nf0g\1

J1 .2I 2jhjR/ : jhj

(10.9.95)

Exercise 10.9.54. Show that, for R ! 1, R N !1 kF k lim

X h2B2N nf0g\1

J1 .2I 2jhjR/ jhj

p R D lim N !1 kF k

X h2B2N nf0g\1

(10.9.96)   cos 2jhjR  34  C O.R1=2 /: jhj3=2

Hint: Use the asymptotic relations for Bessel functions of dimension q D 2.

10 Lattice Functions in Rq

478

Non-uniform Distribution of Lattice Points in R2 Besides the Hardy–Landau series (10.9.94), sums of the form X h2B2N \Z2

J .2I 2jhjR/ jhj

(10.9.97)

with R > 0,  2 . 12 ; 1/, have attracted attention in geometric number theory (cf. Dressler 1972; M¨uller and Dressler 1972). In what follows we discuss the alternating series (10.9.97) in more detail based on the results for the two-dimensional Bessel functions (see Exercises 7.4.1–7.4.3). Exercise 10.9.55. Prove that, for N ! 1, X Z N X J .2I 2jhjR/ D 2 J .2I 2 rR/J0 .2I 2jhjr/ r C1 dr jhj 0 1 2 g2SR \Z

h2B2N \Z2

C

 1 C2 2R ./

X

  2 1 1  jgj C o.1/: R

g2B2R \Z2

(10.9.98) Hint: Use the Euler summation formula for  C  in R2 of Exercise 10.9.46 corresponding to an appropriately given  2 R. Exercise 10.9.56. Show that the sum on the left-hand side of (10.9.98) is convergent if the radius R is chosen in such a way that jgj ¤ R for all g 2 Z2 . Exercise 10.9.57. Show that the sum on the left-hand side of (10.9.98) is divergent to C1 if the radius R is chosen in such a way that there exist g 2 Z2 with jgj D R. Remark 10.9.58. Exercises 10.9.56 and 10.9.57 admit the following geometric interpretation: For certain radii R, there must be more lattice points in circular rings, where J .2I 2 rR/ of (10.9.98) is of positive sign, than in circular rings, where J .2I 2 rR/ of (10.9.98) is negative. This observation is the keystone of the theory of non-uniform radial distribution of lattice points in the plane (for more details see Freeden 2011 and the references therein).

Epstein Zeta Function Lattice sums arise in a variety of problems in mathematics, (geo-)physics, biology, and chemistry. Examples are electromagnetic scattering by periodic arrays of obstacles, seismic exploration, evaluation of the lattice energy of crystals, analysis of thermodynamics and structural properties of electrolytes, as well as the calculation of periodic solutions of partial differential equations.

10.9 Exercises

479

Next, we consider lattice point sums which can be understood as generalizations of the Zeta function as discussed by Epstein (1903, 1907), Freeden (2011), and Wienkamp (1958): nq .sI a; yI ˛I / D

X g2nfyg

 e2iag Pn qI ˛  s jg  yj

gy jgyj



;

(10.9.99)

where Re.s/ > q, n 2 N0 , ˛ 2 Sq1 , a; y 2 Rq , and  is an arbitrary lattice in Rq . As particular cases, Coulombic lattice point sums occurring in physics are included (see, e.g. Berman and Greengard 1994; Borwein et al. 1989; Ewald 1921; Nijboer and de Wette 1957; Schmidt and Lee 1991). Exercise 10.9.59. Formulate (in analogy to Definition 10.3.1) the defining properties (periodicity, differential equations, characteristic singularity, normalization) for the -lattice function G.D.a/ C I / D G. C 4ia  r C I / for the operator D.a/ C  D  C 4ia  r C ,  2 R, a 2 Rq fixed, in Rq . Note that we use the abbreviations D.a/ D  C 4ia  r and D.a/ D   4ia  r here. Exercise 10.9.60. Show that Z 1 1 G.D.a/ C I x/˚h .x/ dV.q/ .x/ D p 2 kF k   4 ..h C a/2  a2 / F (10.9.100)   for all h 2 1 with 4 2 .h C a/2  a2 ¤ . Exercise 10.9.61. Prove the Euler summation formula for the differential operator D.a/ C  D  C 4ia  r C , a 2 Rq ,  2 R: Let  be an arbitrary lattice in Rq . Suppose that G  Rq is a regular region. Let F be of class C.2/ .G /. Then, X0

1

X Z

F .g/ D p F .x/˚h .x/ dV.q/ .x/ kF k h2A.a;/ G Z C G.D.a/ C I x/.D.a/ C /F .x/ dV.q/ .x/

(10.9.101)

g2G \

Z

G

C Z

@G



 @ C 4ia   G.D.a/ C I x/ dS.q1/ .x/ F .x/ @ 

G.D.a/ C I x/ @G

@ F .x/ dS.q1/ .x/; @

where A.a; / D fh 2 1 W 4 2 ..h C a/2  a2 / D g: Note that D.a/ D   4ia  r is the complex conjugate of D.a/.

(10.9.102)

10 Lattice Functions in Rq

480

Definition 10.9.62. The -lattice functions of the iterated operators .D.a/ C /m , a 2 Rq , m 2 N, are defined by G ..D.a/ C /m I x/ D

Z F

  G .D.a/ C /m1 I y G.D.a/ C I x  y/ dV.q/ .y/; (10.9.103)

for m  2 and for m D 1; G ..D.a/ C /m I x/ D G .D.a/ C I x/ :

(10.9.104)

Exercise 10.9.63. Prove that the bilinear expansion X

G ..D.a/ C /m I x  y/ D

h2nA.a;/

is valid for all m >

q 2

˚h .x/˚h .y/ .  4 2 ..h C a/2  a2 //m

(10.9.105)

and x; y 2 Rq .

Exercise 10.9.64. Prove the Euler summation formula for .D.a/C/m with a 2 Rq and m 2 N: Let  be an arbitrary lattice in Rq . Suppose that G  Rq is regular. Let F be of class C.2m/ .G /. Then, X0

1

F .g/ D p kF k h2A.a;/

g2G \

Z

C G

C



m1 XZ @G

m1 XZ kD0

G

F .x/˚h .x/ dV.q/ .x/

(10.9.106)

G ..D.a/ C /m I x/ .D.a/ C /m F .x/ dV.q/ .x/

kD0



X Z

@G

   @ C 4ia   G .D.a/ C /kC1 I x .D.a/ C /k F .x/ dS.q1/ .x/ @

  @ .D.a/ C /k F .x/ dS.q1/ .x/; G .D.a/ C /kC1 I x @

where A.a; / is defined by (10.9.102) in Exercise 10.9.61. Exercise 10.9.65. Prove the following properties of the q-dimensional (Epstein) Zeta function (see also Theorem 10.5.3): q

(a) For n > 0, the Zeta function n .I a; yI ˛I / as defined by (10.9.99) admits a holomorphic continuation such that the continuation represents an entire function in C. (b) For n D 0 and a … 1 the continuation also represents an entire function in C.

10.9 Exercises

481

(c) For n D 0, the continuation is a meromorphic function possessing the single pole kSq1 k 1 : (10.9.107) kF k s  q q

(d) For all n 2 N0 , n .I a; yI ˛I / satisfies the functional equation nq .sI a; yI ˛I / D

nCps q . / 2iay q in 2  s 2 n .q  sI y; aI 1 /: e sCn kF k . 2 / (10.9.108)

Hint: Write out the Euler summation formula for the operator .D.a/ C /m , m > q2 , q

q

q

Re.s/ > q, G D B%;N .y/ D BN .y/ n B% .y/, % sufficiently small,  D 4 2 a2 , and the function x 7! F .x/ D

 e2iag qI ˛  P n jx  yjs

xy jxyj



;

x 2 Rq n fyg:

(10.9.109)

Let N tend to infinity and % tend to zero as in the proof of Theorem 10.5.3.

Fast Lattice Point Summation There are several techniques available for calculating lattice sums. The best-known method is due to Ewald (1921) (and its extensions, e.g., due to Berman and Greengard 1994; Borwein et al. 1989; Nijboer and de Wette 1957; Schmidt and Lee 1991). Usually the method works by separating a lattice sum into two parts X g2

F .g/ D

X g2

F .g/˚.g/ C

X

F .g/.1  ˚.g//;

(10.9.110)

g2

where ˚ is chosen such that the first sum on the right-hand side of (10.9.110) converges rapidly in “space domain”, while the second sum decays rapidly in Fourier domain (see Exercise 10.9.30 based on the Ewald approach following Nijboer and de Wette 1957). The function ˚ is commonly chosen to be a variant of the Gaussian kernel. An example is realized in (10.9.52). Restricting our attention to the function (10.9.109) we are able to base our fast lattice point summation technique on the Poisson summation formula and conclusions known from the theory of the (Epstein) Zeta functions. We start with the verification of the Poisson summation formula by virtue of the functional equation for the Theta function of Theorem 10.8.4. Exercise 10.9.66. Let F be given in the form (10.9.109). Prove that, for each y 2 Rq ;

10 Lattice Functions in Rq

482 q

./ 2

XZ q

q g2 R nB% .y/

D

e

jxgj2 

F .x/ dV .x/

(10.9.111)

Z 1 X   2 h2 e F .x/e2ihx dV .x/ q q kF k R nB% .y/ 1 h2

holds true for all  > 0, % 2 .0; infx2F jxj/ and each y 2 Rq . Exercise 10.9.67. Under the assumptions of Exercise 10.9.66 prove that X g2nfyg

 e2iag qI ˛  P n jg  yjs q

D lim ./ 2  !0C

gy jgyj

XZ q

q g2 R nB% .y/

(10.9.112)

e

jxgj2 

 e2iag Pn qI ˛  s jg  yj

gy jgyj



dV .x/:

In conclusion, by (10.9.112), the lattice point sum (10.9.111) can be rewritten in the form X g2nfyg

 e2iag qI ˛  P n jg  yjs

gy jgyj

(10.9.113)

Z 1 X   2 h2 e2i.ah/x  e Pn qI ˛  s q  !0C kF k Rq nB% .y/ jx  yj 1

D lim

xy jxyj



dV .x/:

h2

Exercise 10.9.68. Show that X g2nfyg

 e2iag qI ˛  P n jg  yjs

gy jgyj

(10.9.114)

q

D lim

 !0C

in  s 2 . nsCq / X 2 kF k . nCs / 2

h21

e 

2 h2

 e2i.ah/y P qI ˛  n ja  hjqs

ah jahj



:

In other words, fast methods for calculating lattice sums such as Coulombic lattice sums in dimension q become available from techniques known in the theory of (Epstein) Zeta functions. Applications of our calculation technique in analytic number theory can be found in Freeden (2011) (see also the references therein).

Chapter 11

Concluding Remarks

In this textbook on special functions, we have focused on theoretical structures and methods with geomathematical background and applications such as modeling of the gravitational or magnetic field, deformation analysis, and fluid flow in the atmosphere. In a canonical way, we have been led to the concept of spheriodization of the Earth so that spherical harmonics for scalar as well as for vectorial fields have come into play. Their necessary foundation on orthogonal polynomials has been laid in a preparatory chapter, followed by generalization to arbitrary dimensions has also been included. Even in the spherical simplification the image of the Earth as a “potato drenched by rainfall” (which is sometimes drawn by oceanographers) is not geometrically false. The humid layer on this potato, maybe only a fraction of a millimeter in thickness, is made up of the oceans. The entire atmosphere that hosts weather and climate events is only a little bit thicker. Flat bumps protruding from the humid layer form the continents. Human life in its entirety exists in a very narrow region of the outer layer, comprising only a few kilometers of the vertical extension of 6,371 km. However, the basically excellent comparison of the Earth with a huge potato does not give explicit information about the essential physical ingredients of the Earth system. Gravitation, magnetic field, deformation, wind distribution, ocean currents, internal structures, etc. are not reflected by this image. Unfortunately, if we want to include these physical components in a perfectly realistic non-spherical geometry, today’s modeling procedures and simulation methods confront us with a huge complexity. This is the reason why, nowadays, formulations of theories and models can only be realized for regularly structured geometries (cf. Freeden et al. 2010). An ellipsoidal model would be a good choice within which to observe the flattening of the Earth (note that an excellent overview of ellipsoidal approaches can be found, e.g., in Grafarend et al. 2010). However, physical theory and numerical efficiency should be able to master this complexity. At the current geoscientific stage in numerics, the authors doubt that an “ellipsoidal potato” is the proper choice. In order to take advantage of the rotational invariance and the one-dimensional nature of zonal functions, a “spherical potato” seems to be the better choice—at least in W. Freeden and M. Gutting, Special Functions of Mathematical (Geo-)Physics, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-0348-0563-6 11, © Springer Basel 2013

483

484

11 Concluding Remarks

today’s numerics. Our work is a more suitable introduction for students who can afterwards delve into the theory of ellipsoidal special functions, while having the spherical background at hand. As a result, the concept of spheriodization in this book helps to understand the necessary structures and function systems that are both physically relevant and economically computable as seen from the point of view of today’s geomathematics. Nevertheless, there is no doubt that the dramatic change in the observational situation of geometric and physical information will require new components of mathematical thinking, numerical implementation and data handling such that geophysically relevant “realistic potatoes” can be addressed. This is a great challenge for future work. Furthermore, the concept of periodization has been explored leading to the theory of lattice functions and periodic polynomials. In addition, the relationship between the summation formulas of Euler and Poisson becomes obvious and multidimensional series can be investigated in convergence acceleration. All in all, the core of this textbook obviously consists of the theory of spherical as well as periodic functions with regard to geoscientifically relevant applications which are reflected in numerically oriented exercises.

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Index

Addition theorem 2D homogeneous harmonic polynomials, 315 2D spherical harmonics, 314, 315 3D harmonic polynomials, 135, 138 3D homogeneous polynomials, 128 3D spherical harmonics, 140 complex-valued spherical harmonics, 142 homogeneous polynomials satisfying the Navier equation, 261 qD Bessel functions, 365 qD harmonic polynomials, 307 qD homogeneous polynomials, 304, 306 qD spherical harmonics, 312 vector spherical harmonics, 239, 240 Angular function, 289 Anharmonic, 184 Approximate identity, 176 Area 2D sphere, 31 3D sphere, 31 qD sphere, 32 Associated Legendre function, 142, 323 Beltrami integral formula, 162, 164, 300 Helmholtz, 335 iterated, 301 iterated Helmholtz, 336 Beltrami operator, 116, 166, 289 3D eigenspectrum, 140 Helmholtz, 331 Laplace, 298, 331 qD eigenspectrum, 337 vectorial, 236, 238 Bernoulli function, 397, 411 of degree 1, 403 of degree 2, 401

Bernoulli numbers, 396, 415 representations, 411 Bernoulli polynomials, 396 of degree 1,2,3,4, 396 recurrence relations, 395 Bernstein kernel, 148, 224, 270, 318 Bernstein summability, 148, 224, 228 Bessel functions, 13, 347 2D half odd integer order, 370 2D theory, 370 differential equation, 349 generating function, 355 Hankel functions, 352 integral representations, 357 Neumann functions, 352 of first kind, 350 of the second kind, 352 of the third kind, 352 orthogonality relations, 354 qD addition theorem, 365 qD asymptotic behavior, 368, 375 qD differential equation, 367, 369 qD generating function, 365, 369 qD Hankel functions, 373 qD integral representation, 364 qD Kelvin function, 379 qD modified, 371 qD modified Hankel function, 379 qD Neumann functions, 378 qD recurrence relations, 367, 369 qD regular, 364 qD series representation, 366 qD theory, 363 recurrence relations, 358 with integer index, 355 Beta distribution, 46

W. Freeden and M. Gutting, Special Functions of Mathematical (Geo-)Physics, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-3-0348-0563-6, © Springer Basel 2013

495

496 Cauchy–Riemann operator, 204 Christoffel–Darboux formula monic orthogonal polynomials, 64 orthonormal polynomials, 63 Clenshaw algorithm, 104 modified, 105 Clifford algebra, 202 Closure, 50 3D spherical harmonics, 148, 152 qD periodic polynomials, 431 qD spherical harmonics, 320 vector spherical harmonics, 230, 235 Completeness, 50 3D spherical harmonics, 153 qD periodic polynomials, 431 qD spherical harmonics, 321 vector spherical harmonics, 231, 235 Convergence theorems 1D Poisson summation formula, 418 qD Poisson summation formula, 454 Coordinates polar, 114, 288 Coupling integrals, 275 advection related terms, 279, 283 Coriolis related terms, 279, 282

Differential equation Bessel, 349 Legendre, 93 of the Helmholtz operator, 12, 348, 363, 385 of the Laplace–Beltrami operator, 165 Poisson, 2, 90, 296, 333, 435 qD Bessel, 367 qD Legendre, 309, 337 Dirac operator, 204 Dirichlet problem, 327 Discontinuous integrals of Weber-Schlafheitlin type, 360 Discrepancy, 171 Duplication formula, 36

Eigenspectrum 1D Laplace operator, 399 3D Beltrami operator, 140 qD Beltrami operator, 332 qD Laplace operator, 431 Elasticity, 16, 254 Cauchy stress tensor, 18 Cauchy–Navier equation, 20, 254 elasticity tensor, 18 Lam´e parameters, 19, 254

Index Navier equation, 20, 254 Poisson’s ratio, 20 vector spherical harmonics, 262 Equidistribution, 171 Error functions, 44 Euler angles, 191 Euler summation formula 1D Laplace operator, 404, 407 1D ordinary, 404 1D shifted, 407 qD extended, 438 qD for the operator D.a/ C , 479 qD for the operator D.a/ C , iterated, 480 qD Laplace operator, 8, 436, 437 Euler’s Beta function, 29 Beta distribution, 46 incomplete, 44 relation to the Gamma function, 30 Euler’s constant, 40, 406, 407 Expectation value frequency domain, 265 space domain, 264

Fast multipole method, 109, 186, 191, 196 Fourier coefficients, 48, 147, 228, 432 Bessel’s equality, 49 Bessel’s inequality, 49 Functional equation 1D Riemann Zeta function, 414 1D Theta function, 419, 420 qD Theta function, 458 qD Zeta function, 442 Fundamental cell 1D lattice, 398 qD inverse lattice, 429 qD lattice, 428 Fundamental solution Laplace operator, 292, 382 Funk–Hecke formula, 157, 246, 249, 250, 316

Gamma distribution, 45 Gamma function, 25 definition, 26 derivative, 27 duplication formula, 36 extended Stirling’s formula, 405 extension to the complex plane, 38 functional equation, 27 Gamma distribution, 45 incomplete, 43 multiplication formula, 38 product representation, 41

Index reciprocal, 38 Stirling’s formula, 33, 405 Gauß theorem, 290 surface, 116 Gegenbauer polynomials, 72, 80, 308 derivative, 81 differential equation, 81 estimates, 107 explicit representation, 81, 84 generating function, 86 hypergeometric representation, 81–83 L2 -norm, 82 recurrence relations, 89 Rodrigues’ representation, 81 three-term recurrence, 82 Geomagnetic field, 9 Gravitational field, 1 Green’s function 1D Z-lattice Laplace operator, 400 1D Z-periodic, 400 1D Legendre operator, 101 3D sphere, iterated Laplace–Beltrami, 163 3D sphere, Laplace–Beltrami, 159, 220, 224 qD iterated Laplace operator, 435 qD sphere, Helmholtz–Beltrami, 331 qD sphere, iterated Helmholtz–Beltrami, 335 qD sphere, iterated Laplace–Beltrami, 301 qD sphere, Laplace–Beltrami, 298 qD Z-lattice Laplace operator, 432 qD Z-periodic, 432 Green’s surface theorem first, 117, 298 second, 118, 298 third, 161, 162, 164, 300 Green’s theorem extended second, 291 first, 290 second, 290 third, 295

Haar kernel, 176, 269 Hankel functions, 352 2D theory, 384 qD asymptotic behavior, 375, 376 qD characteristic singularity, 380 qD definition, 373 qD modified, 379 qD theory, 372 Hankel transform, 359 Hardy–Landau identity 1D interval, 417

497 2D ball, 474, 477 Harmonic, 131, 291 Helmholtz decomposition theorem, 220 Helmholtz equation entire solutions, 390 expansion for the inner space of a ball, 386 expansion for the outer space of a ball, 388 Helmholtz–Beltrami operator, 331 Hermite polynomials, 95, 98 differential equation, 96 explicit representation, 95 Rodrigues’ representation, 95 three-term recurrence, 95 Holomorphic H-holomorphic, 205 H-left holomorphic, 205 H-right holomorphic, 205 Homogeneous harmonic polynomials 2D addition theorem, 315 2D theory, 313, 314 3D addition theorem, 135 3D theory, 131 qD addition theorem, 307 qD theory, 302 vectorial, 231 Homogeneous polynomials, 125, 302 satisfying the Navier equation, 256, 257 Hydrogen atom, 165 Hypergeometric function, 76, 82, 83, 106

Incomplete Beta function, 44 Incomplete Gamma function, 43 relation to error functions, 44 Inner harmonics, 187 H-holomorphic, 209 translation theorem, 189 Integral formula 3D Laplace–Beltrami, 162, 164 qD Helmholtz–Beltrami, 335 qD iterated Helmholtz–Beltrami, 336 qD iterated Laplace–Beltrami, 301 qD Laplace–Beltrami, 300 Invariance orthogonal, 119, 120, 247, 258, 260, 315 with respect to reflections, 121 with respect to rotations, 119, 121, 156, 247 Inverse lattice, 429 Irreducible, 121, 122 Harmn , 156 harmn , 247

498 Jacobi polynomials, 70 derivative, 77 differential equation, 73, 74 explicit representation, 76 hypergeometric representation, 76 L2 -norm, 78 Rodrigues’ formula, 77 special cases, 72 three-term recurrence, 79

Kelvin functions 2D theory, 384 differential equation, 383 qD asymptotic behavior, 379, 382 qD characteristic singularity, 380 qD definition, 379 qD theory, 378 recurrence relations, 383

Laguerre polynomials, 98, 100, 168 differential equation, 99 explicit representation, 98 Rodrigues’ representation, 98 three-term recurrence, 99 Laplace–Beltrami operator, 116, 289, 298, 331 Lattice 1D integer, 398 qD inverse, 429 qD periodic, 5, 427 Lattice balls inside balls, 472 Lattice function 1D bilinear expansion, 402 1D explicit representation, 402 1D Fourier expansion, 402 1D iterated Laplacian, 408, 410, 420, 421 1D Laplace operator, 400 1D Z-periodic, 400 2D Helmholtz operator, 474 integral asymptotics, 443, 448 qD for the operator D.a/ C , 479 qD for the operator D.a/ C , iterated, 480 qD iterated Laplace operator, 435 qD Laplace operator, 432 qD Z-periodic, 432 uniqueness, 401 Lattice points inside balls, 472 inside circles, 474 non-uniform distribution in the plane, 478 Legendre coefficient, 157 Legendre differential equation, 93

Index Legendre functions 3D associated, 142 qD associated, 323 Legendre harmonic, 154 Legendre operator, 93, 116 Green’s function, 101 qD, 309, 337 Legendre polynomials, 73, 93, 135 3D Maxwell representation, 144 Laplace’s integral representation, 316 qD, 90, 308 qD asymptotic estimates, 317 qD differential equation, 309, 337 qD generating function, 311 qD integral relations, 310 qD Maxwell representation, 311 qD orthogonality, 308 qD recurrence relations, 309 qD Rodrigues formula, 309 qD theory, 308, 315 vectorial counterpart, 251 Legendre rank-2 tensor kernel of type .i; k/ with respect to o.i/ ; O .i/ , 241, 243, 251 with respect to o.i/ ; O .i/ , 241 Legendre vector field, 253 Legendre vector field of type i , 253 Lipschitz continuity, 176, 297 Low discrepancy method, 170

Maxwell representation, 144, 311 Maxwell’s equations full system, 9 pre-Maxwell equations, 11, 90, 204 Metaharmonic, 292 Metric tensor, 274 Modulus of continuity, 297 Moving frame, 114, 123 Multi-dimensional angle, 445 Multi-indices, 287 Multiplication formula, 38 Multiresolution analysis, 182

Navier–Stokes equation, 13, 15 incompressible, 15 incompressible, spherical, 16 Neumann functions, 352 qD definition, 378 Newton integral, 5 Newton potential, 2 Normal vector field, 213 Numerical integration

Index Gauß quadrature, 67, 101, 424 interpolatory quadrature, 66 low discrepancy method, 170 partioning the sphere, 170 trapezoidal rule, 422

Orthogonal group, 120 Orthogonal invariance, 119, 120, 247, 258, 260, 315 scalar context, 122 vectorial context, 123 Orthogonal polynomials, 47 monic orthogonal polynomials, 54 symmetry, 56 three-term recurrence, 59 weighted Hilbert spaces, 48 zeros, 57, 65 Outer harmonics, 187 translation theorem, 189

Peano kernel, 464, 467 Periodic -lattice, 430 Z-lattice, 398 Periodic convolution, 408, 420, 435 Periodic polynomials, 398 Pochhammer factorial, 38 Point set boundary, 286 closure, 286 region, 286 Pointwise expansion theorem spherical harmonics, 324, 327, 330 Poisson differential equation, 2, 90, 296, 333, 435 Poisson integral formula, 145, 325 Poisson summation formula 1D, 417 1D Laplace operator, 416 1D interval, 416 qD Laplace operator, 6, 448, 454, 455 Polar coordinates, 114, 288 Poloidal vector field, 258 Polynomial Bernoulli, 396 Chebyshev, 72, 82, 314 Gegenbauer, 72, 80, 308 harmonic, 304 Hermite, 95 homogeneous, 125, 302 homogeneous harmonic, 131, 304 homogeneous harmonic vectorial, 231

499 Jacobi, 70 Laguerre, 98, 168 Legendre, 73, 308 monic, 54 monic orthogonal, 54 periodic, 398 -periodic, 430 satisfying the Navier equation, 256, 257 ultraspherical, 72, 80, 308 Potential function, 216 Quadrature rule, 66 Gauß quadrature, 67, 101 Gauß–Lobatto rule, 70 Gauß–Radau rule, 70 interpolatory, 66 Quantum-mechanical oscillator, 97 Quaternions, 200 conjugate, 201 imaginary part, 201 real part, 201 real quaternionic algebra, 200 Radial function, 289 Recurrence relations Bessel function, 358 cylinder functions, 378 Gegenbauer polynomials, 89 Kelvin function, 383 modified Bessel function, 371 qD Bessel function, 367, 369 qD Legendre polynomials, 309 ultraspherical polynomials, 89 Reducible, 120 Reflection, 120 Region, 286 regular, 116, 286, 290 Reproducing kernel, 128, 141, 251, 313 Riemann Zeta function, 411, 422 Rodrigues’ rule, 310 Rotation, 120, 191, 246 Scaling function Haar, 179 smoothed Haar, 177 up function, 182 Sectorial spherical harmonics, 143 Sinc-function, 370, 417 Sobolev space 1D periodic, 423 qD periodic, 472 qD spherical, 469

500 Solid angle, 295, 437 Special orthogonal group, 120 Sphere function 3D Laplace–Beltrami, 159, 220, 224 qD Helmholz–Beltrami, 331 qD iterated Helmholtz–Beltrami, 335 qD iterated Laplace–Beltrami, 301 qD Laplace–Beltrami, 298 Spherical convolution, 163, 224, 335 infinite, 181 Spherical harmonics 2D addition theorem, 314, 315 3D addition theorem, 140 3D closure, 148, 152 3D completeness, 153 3D complex-valued, 141 3D definition, 138 3D eigenfunctions, 140 3D Funk–Hecke formula, 157 3D real-valued, 142 gravitational field, 4 H-spherical harmonics, 200, 206 in geomagnetism, 11 irreducible, 156 of degree n and order j , 140–142 qD addition theorem, 312 qD asymptotic relations, 328 qD closure, 320 qD completeness, 321 qD definition, 312 qD eigenfunctions, 330, 336 qD expansion theorem, 324, 327 qD Funk–Hecke formula, 315 qD Laplace representation, 316 qD orthogonal coefficients, 328 qD theory, 285, 312 quaternionic representation, 200 sectorial, 143 tesseral, 143 zonal, 143 Spherical summation, 453 Spheroidal vector field, 222 Splines, 461 algebraically polynomial, 461 monosplines, 470 multi-periodic, 470 natural, 462 periodic polynomial, 466 spherical, 467, 468 Stirling’s formula, 33 extended, 405, 407 limit form, 36 Stream function, 216 Summation formula

Index 1D Euler, 405 1D Poisson, 415, 417 qD Euler, 436 qD Poisson, 455 Surface curl, 115 Surface curl gradient, 115, 214 Surface divergence, 115, 289 Surface gradient, 115, 214, 289 Surface identity tensor field, 239 Surface rotation tensor field, 239

Tangential vector field, 213 Tesseral spherical harmonics, 143 Theta function, 420, 455 1D functional equation, 419 classical, 460 qD functional equation, 458 qD properties, 458 qD theory, 455 Three-term recurrence, 59, 79, 82, 95, 99 Toroidal vector field, 223, 258 Translation theorem, 189 for coefficients, 190

Ultraspherical polynomials, 72, 80, 308 derivative, 81 differential equation, 81 estimates, 107 explicit representation, 81, 84 generating function, 86 hypergeometric representation, 81–83 L2 -norm, 82 recurrence relations, 89 Rodrigues’ representation, 81 three-term recurrence, 82 Uncertainty principle, 264 Up function, 179, 181

Variance frequency domain, 265 space domain, 264 Vector spherical harmonics, 211 addition theorem, 239, 240 alternate system, 235 by Edmonds, 276 closure, 230, 235 completeness, 231, 235 definition, 219 eigenfunctions, 238 Funk–Hecke formula, 246, 249, 250 in elasticity, 21, 262

Index in geomagnetism, 12 in Navier–Stokes equation, 16 irreducibility, 247 Vectorial Beltrami operator, 236, 238 Volume of the qD ball, 33

Wavelets Haar, 179 locally supported, 176 smoothed Haar, 176, 178 spherical, 178 up function, 183 Wigner matrices, 191 Wigner symbols, 271

501 3j , 272 6j , 274 9j , 275

Zeta function 1D functional equation, 414 1D Riemann, 411 fast lattice point summation, 481 qD Epstein, 439, 478, 481 qD functional equation, 443 qD integral representation, 413 Zonal function, 118, 298 Bernstein, 148, 151, 224, 318 Zonal spherical harmonics, 143