Spherical Functions of Mathematical Geosciences: A Scalar, Vectorial, and Tensorial Setup (Geosystems Mathematics) 3662656914, 9783662656914

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Geosystems Mathematics

Willi Freeden Michael Schreiner

Spherical Functions of Mathematical Geosciences A Scalar, Vectorial, and Tensorial Setup Second Edition

Geosystems Mathematics Series Editors Willi Freeden, Mathematics Department, University of Kaiserslautern, Rhineland-Palatinate, Germany M. Zuhair Nashed, Mathematics Department, University of Central Florida, Orlando, FL, USA Editorial Board Hans-Peter Bunge, Geophysics, Ludwig Maximilian University of Munich, München, Bayern, Germany Roussos G. Dimitrakopoulos, Faculty of Engineering, McGill University, Montreal, QC, Canada Yalchin Efendiev, Texas A&M University, College Station, TX, USA Andrew Fowler, University of Limerick, Limerick, Ireland Bulent Karasozen, Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey Jürgen Kusche, Institute of Geodesy and Geoinformation, University of Bonn, Bonn, Nordrhein-Westfalen, Germany Liqui Meng, Aerospace and Geodesy, Technical University of Munich, Munich, Germany Volker Michel, Geomathematics, University of Siegen, Siegen, Nordrhein-Westfalen, Germany Nils Olsen, National Space Institute, Technical University of Denmark, Kongens Lyngby, Denmark Helmut Schaeben, TU Bergakademie Freiberg, Freiberg, Sachsen, Germany Otmar Scherzer, University of Vienna, Wien, Wien, Austria Frederik J. Simons, Department of Geosciences, Princeton University, Princeton, NJ, USA Thomas Sonar, Institut für Analysis, Technische Universität Braunschweig, Braunschweig, Niedersachsen, Germany Peter J. G. Teunissen, Delft University of Technology, DELFT, Zuid-Holland, The Netherlands Johannes Wicht, Max Planck Institute for Solar System Research, Göttingen, Niedersachsen, Germany

Willi Freeden Michael Schreiner •

Spherical Functions of Mathematical Geosciences A Scalar, Vectorial, and Tensorial Setup Second Edition

Willi Freeden Mathematics Department University of Kaiserslautern Kaiserslautern, Germany

Michael Schreiner Institut for Computational Engineering ICE Eastern Switzerland University of Applied Sciences Buchs, Switzerland

ISSN 2510-1544 ISSN 2510-1552 (electronic) Geosystems Mathematics ISBN 978-3-662-65691-4 ISBN 978-3-662-65692-1 (eBook) https://doi.org/10.1007/978-3-662-65692-1 1st edition: © Springer-Verlag Berlin Heidelberg 2009 2nd edition: © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2022 This work is subject to copyright. All rights are solely and exclusively licensed 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. 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. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhäuser-science.com by the registered company Springer-Verlag GmbH, DE The registered company address is: Heidelberger Platz 3, 14197 Berlin, Germany

This book is dedicated to the memory of our parents, Hubertine and Wilhelm Freeden, Hilde and Gerhart Schreiner.

Preface

Special functions applied in Earth’s sciences and geoengineering are confronted with the complexity of a real “potato-like” Earth geometry, that is a striking obstacle particularly for all global purposes of modeling and simulation. The obstacle can only be overcome to some extend in today’s mathematics. The principles to find a way out are found in a suitable transition to a regularly structured geometry for the Earth, viz. a spherical figure. In fact, by modern satellite positioning methods, the maximum deviation of the actual Earth’s surface from the average Earth’s radius (6371 km) can be determined to be less than 0.4%. So, looking at the special functions available in the geoscientific literature today, a spherical shape for purposes of the global Earth is used in almost all publications. Although a mathematical formulation simply using spherical reference geometry amounts to a strong restriction, it is at least acceptable for a large number of problems. This situation explains the strong need for spherical mathematical structures, tools, and methods to handle geoscientifically relevant data sets of high quality within high accuracy and to improve significantly modeling capabilities in Earth system research. Standard spherical functions since the time of A.M. Legendre, P.S. de Laplace, and C.F. Gauss are polynomial trial functions, conventionally called spherical harmonics. Spherical harmonics represent the analogous of trigonometric functions for orthogonal (Fourier) expansions on the sphere. The use of spherical harmonics in diverse areas of geosciences is a well-established procedure, particularly for analyzing scalar potential functions. Nowadays, reference models for the Earth’s gravitational and magnetic potential, for example, are widely known by tables of expansion coefficients, i.e., the frequency constituents of their potentials. In this respect it should be mentioned that vectorial and tensorial potentials – even in a spherical Earth’s reference model – have their own nature. Concerning the mathematical modeling of vector and tensor 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 curlfree, the magnetic field is divergence-free, the equations for incompressible flow, i.e., vii

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Preface

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 like the surface gradient, surface curl gradient, surface divergence, surface curl, etc. So, the types of vector and tensor spherical harmonics as proposed here satisfy adequate requirements of splitting a vectorial or tensorial signal into geoscientifically relevant components, thereby avoiding artificial singularities arising from the use of local coordinates. Basically two transitions are undertaken in our approach to spherical 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 the 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. All in all, 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 global Earth. In addition, spherical harmonics form 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. Our purpose in this work is to present a textbook that allows the reader to concentrate on special fields such as the geosphere, hydrosphere, atmosphere, and also anthroposphere. In other words, the special functions to be discussed vary strongly, dependent on the specific observation and measurement indicators (e.g., in gravitation, magnetics, deformation, climate, fluid flow, etc.) and on the occurring field characteristics (e.g., as potential field, diffusion field, wave field). Obviously, the differential equation under consideration determines the type of special functions that are needed in the desired reduction process. The book is to serve as a self-consistent introductory textbook for (graduate) students of mathematics, geosciences, and geoengineering. In addition, the work should also be a valuable reference for scientists and practitioners facing spherical problems in their professional tasks. The present second edition is based on essential parts of the first edition published in the year 2009 in the Springer series “Advances in Geophysical and Environmental Mechanics and Mathematics” (𝐴𝐺𝐸 𝑀 2 ). Evidently, this first edition forms the fundamental foundation for the present book. However, updates and additions were indispensable. The present version thus claims to present the theory of spherical functions on the basis of today’s essential ideas and concepts, but completely free of specific concrete temporal events of past, present, and future. The result is a work which, while scientifically reflecting the present state of knowledge in a time-related manner, nevertheless claims to be of largely timeless significance for (geo-)mathematical research and teaching. The authors thank M.Z. Nashed, Orlando, for accepting the book for publication in the series “Geosystems Mathematics”. They are obliged to thank H. Nutz, Kaiserslautern, for reading carefully all parts of the book and giving valuable comments.

Preface

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Finally, the authors would like to thank Birkh¨auser, in particular T. Hempfling, M. Peters, and L. Kunz for their obligingness and cooperation.

Kaiserslautern and Buchs, May 2022

Willi Freeden Michael Schreiner

Contents

Part I Introduction 1

Introductional Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 From Spherical Harmonics to Zonal Kernels . . . . . . . . . . . . . . . . . . . . 5 1.2 From Scalar Spherical Harmonics to its Vectorial and Tensorial Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3 Organization of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Part II Background and Nomenclature 2

Basic Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Euclidean Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Euclidean Differential Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Euclidean Integral Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Spherical Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Spherical Differential Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Spherical Integral Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Orthogonal Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 37 40 43 47 52 62

3

Circular Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Homogeneous Harmonic Polynomials and Circular Harmonics . . . . 3.2 Eigenfunctions of the Beltrami Operator . . . . . . . . . . . . . . . . . . . . . . . 3.3 Addition Theorem and Chebyshev Polynomial . . . . . . . . . . . . . . . . . . 3.4 Stereographic Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Closure and Completeness of Circular Harmonics . . . . . . . . . . . . . . . 3.6 Inner/Outer Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 72 72 73 75 77 78

Part III Spherical Harmonics 4

Scalar Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.1 Homogeneous Harmonic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2 Addition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 xi

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4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16

Exact Computation of Homogeneous Harmonic Polynomials . . . . . . 98 Definition of Scalar Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . 108 Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Orthogonal (Fourier) Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Legendre (Spherical) Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Funk-Hecke Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Eigenfunctions of the Beltrami Operator . . . . . . . . . . . . . . . . . . . . . . . 143 Irreducibility of Scalar Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Degree and Order Variances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Associated Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Associated Legendre (Spherical) Harmonics . . . . . . . . . . . . . . . . . . . . 163 Spherical Harmonics in Cartesian Coordinates . . . . . . . . . . . . . . . . . . 171 Spherical Harmonics in Terms of Wigner 𝐷-Matrices . . . . . . . . . . . . 184 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

5

Green’s Functions and Integral Formulas . . . . . . . . . . . . . . . . . . . . . . . . . 193 5.1 Green’s Function with Respect to the Beltrami Operator . . . . . . . . . . 193 5.2 Space Mollifier Green’s Function with Respect to the Beltrami Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 5.3 Frequency Mollifier Green’s Function with Respect to the Beltrami Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 5.4 Modified Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 5.5 Integral Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 5.6 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 5.7 Classical Understanding of Spline Functions . . . . . . . . . . . . . . . . . . . . 218 5.8 Integral Formulas with Respect to Iterated Beltrami Operators . . . . . 224 5.9 Differential Equations Respect to Iterated Beltrami Operators . . . . . . 233 5.10 Pseudodifferential Operators, Green’s Functions, and Splines . . . . . . 236 5.11 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

6

Vector Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 6.1 Normal and Tangential Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 6.2 Helmholtz System of Vector Spherical Harmonics . . . . . . . . . . . . . . . 254 6.3 Orthogonal (Fourier) Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 6.4 Homogeneous Harmonic Vector Polynomials . . . . . . . . . . . . . . . . . . . 271 6.5 Exact Computation of Orthonormal Systems . . . . . . . . . . . . . . . . . . . . 274 6.6 Irreducibility and Orthogonal Invariance of Vector Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 6.7 Vectorial Beltrami Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 6.8 Vectorial Addition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 6.9 Vectorial Funk-Hecke Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 6.10 Counterparts to the Legendre Polynomial . . . . . . . . . . . . . . . . . . . . . . 301 6.11 Degree and Order Variances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 6.12 Hardy-Hodge System of Vector Spherical Harmonics . . . . . . . . . . . . 309 6.13 Orthogonal Expansion Using Vector Legendre Kernels . . . . . . . . . . . 318

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6.14 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 7

Tensor Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 7.1 Some Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 7.2 Normal and Tangential Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 7.3 Integral Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 7.4 Helmholtz System of Tensor Spherical Harmonics . . . . . . . . . . . . . . . 335 7.5 Helmholtz Decomposition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 7.6 Closure and Completeness of Tensor Spherical Harmonics . . . . . . . . 346 7.7 Homogeneous Harmonic Tensor Polynomials . . . . . . . . . . . . . . . . . . . 354 7.8 Tensorial Beltrami Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 7.9 Tensorial Addition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 7.10 Tensorial Funk-Hecke Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 7.11 Counterparts to the Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . 377 7.12 Tensor Spherical Harmonics Related to Tensor Homogeneous Harmonic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 7.13 Hardy-Hodge System of Tensor Spherical Harmonics . . . . . . . . . . . . 383 7.14 Orthogonal Expansion Using Tensor Legendre Kernels . . . . . . . . . . . 389 7.15 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

8

Spin-Weighted Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 8.1 Spin-Weighted Differential Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 8.2 Spin-Weighted Integral Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 8.3 Spin-Weighted Spherical Harmonics in Coordinate Representation . 409 8.4 Spin-Weighted Spherical Harmonics and the Wigner 𝐷-Function . . 423 8.5 Relations to Vector and Tensor Spherical Harmonics . . . . . . . . . . . . . 453 8.6 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

Part IV Zonal Functions 9

Scalar Zonal Kernel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 9.1 Bandlimited/Spacelimited Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 9.2 Zonal Kernel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 9.3 Zonal Kernel Functions in Scalar Context . . . . . . . . . . . . . . . . . . . . . . 466 9.4 Convolutions Involving Scalar Zonal Kernel Functions . . . . . . . . . . . 467 9.5 Classification of Zonal Kernel Functions . . . . . . . . . . . . . . . . . . . . . . . 469 9.6 Localization of Representative Zonal Kernels . . . . . . . . . . . . . . . . . . . 478 9.7 Dirac Families of Zonal Scalar Kernel Functions . . . . . . . . . . . . . . . . 491 9.8 Examples of Dirac Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 9.9 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

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10 Vector Zonal Kernel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 10.1 Preparatory Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 10.2 Tensor Zonal Kernel Functions of Rank Two in Vectorial Context . . 525 10.3 Vector Zonal Kernel Functions in Vectorial Context . . . . . . . . . . . . . . 531 10.4 Convolutions Involving Vector Zonal Kernel Functions . . . . . . . . . . . 533 10.5 Dirac Families of Zonal Vector Kernel Functions . . . . . . . . . . . . . . . . 535 10.6 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 11 Tensorial Zonal Kernel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 11.1 Preparatory Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 11.2 Tensor Zonal Kernel Functions of Rank Four in Tensorial Context . . 540 11.3 Convolutions Involving Zonal Tensor Kernel Functions . . . . . . . . . . . 542 11.4 Tensor Zonal Kernel Functions of Rank Two in Tensorial Context . . 544 11.5 Dirac Families of Zonal Tensor Kernel Functions . . . . . . . . . . . . . . . . 548 11.6 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 Part V Geopotentials 12 Potentials in Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 12.1 Newton’s Law of Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 12.2 Volume Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 12.3 Interior/Exterior Green’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 12.4 Volume Potential and Gravitational Field . . . . . . . . . . . . . . . . . . . . . . . 565 12.5 Scalar Inner/Outer Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 12.6 Surface Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 12.7 Boundary-Value Problems for the Laplace Operator . . . . . . . . . . . . . . 578 12.8 Existence and Regularity Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 12.9 Locally and Globally Uniform Approximation . . . . . . . . . . . . . . . . . . 595 12.10Vector Outer Harmonics and the Gravitational Gradient . . . . . . . . . . 612 12.11Satellite-to-Satellite Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 12.12Tensor Outer Harmonics and the Gravitational Tensor . . . . . . . . . . . . 620 12.13Satellite Gravitation Gradiometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 12.14Function Systems for Internal Elastic Fields . . . . . . . . . . . . . . . . . . . . 628 12.15Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 13 Potentials on the Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 13.1 Green’s Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 13.2 Surface Harmonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 13.3 Surface Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 13.4 Curve Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656 13.5 Limit Formulas and Jump Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 656 13.6 Boundary-Value Problems for the Beltrami Operator . . . . . . . . . . . . . 657 13.7 Complete Function Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 13.8 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667

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14 Multiscale Mollifier Potential Systems in Geoapplications . . . . . . . . . . 669 14.1 Surface Mollifier Gradient Fields and Gravitational Deflections of the Vertical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 14.2 Surface Mollifier Curl Gradient Fields and Geostrophic Ocean Flow 680 14.3 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686 Part VI Conclusion 15 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723

Part I

Introduction

Chapter 1

Introductional Remarks

If we are interested in gaining a deeper understanding of a signal (signature) on the sphere, it is advantageous to study its space and frequency contributions. For example, the signal can be obtained from a complete system of polynomials, e.g., spherical harmonics, providing a spectral (frequency) representation. From a mathematical point of view there is an infinite number of ways this can be done. Even more, the idea that a discontinuous signal (function) on the sphere may be expressed as a “frequency expansion” in terms of spherical harmonics turns out to be one of the great innovations in signal analysis.

Fig. 1.1: 𝐿 2 -orthonormal system of spherical harmonics of degree 3 (constituting the addition theorem). Spherical harmonics are the analogues of trigonometric polynomials for (orthogonal) Fourier expansion theory on the sphere (see Fig. 1.1 for an illustration of the degree 3). They were introduced in the 1780s to study gravitational theory (cf. P.S. de Laplace (1785), A.M. Legendre (1785)). Early publications on the theory of spherical harmonics in their original physically motivated meaning as multipoles are, e.g., © The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2022 W. Freeden and M. Schreiner, Spherical Functions of Mathematical Geosciences, Geosystems Mathematics, https://doi.org/10.1007/978-3-662-65692-1_1

3

4

1 Introductional Remarks

due to R.F.A. Clebsch (1861), T. Sylvester (1876), E. Heine (1878), F. Neumann (1887), J.C. Maxwell (1891), and A. Wangerin (1921). A great incentive came from the fact that global geomagnetic data became available in the first half of the 19th century (cf. C.F. Gauss (1838)). Nowadays, reference models for the Earth’s gravitational or magnetic field, for example, are widely known by tables of coefficients of the spherical harmonic orthogonal (Fourier) expansion of their potentials (see, for example, Fig. 1.2). Today, the use of spherical harmonics in diverse procedures is a well-established technique in all geosciences, particularly for the purpose of representing potential functions (see, for example, J. Lense (1954), E.W. Hobson (1955), C. M¨uller (1966), R.T. Seeley (1966), T. McRobert (1967), H. Hochstadt (1971), N.N. Lebedev (1973), W. Freeden, M. Gutting (2013)).

(degree 3 to 15)

(degree 16 to 31)

(degree 3 to 31)

(degree 32 to 63)

(degree 3 to 63)

(degree 64 to 127)

Fig. 1.2: Earth’s gravitational model EGM 2008 (N.K. Pavlis et al. (2008)) in terms of spherical harmonics in m2 s−2 .

1.1 From Spherical Harmonics to Zonal Kernels

5

1.1 From Spherical Harmonics to Zonal Kernels

It is characteristic for the Fourier approach that each spherical harmonic, as an “ansatz function” of polynomial nature, corresponds to exactly one degree, i.e., in the jargon of signal processing to exactly one “frequency”. Thus, the orthogonal expansion in terms of spherical harmonics amounts to the superposition of summands showing an oscillating character determined by the degree (frequency) of the Legendre polynomial. The more spherical harmonics of different degrees are involved in the orthogonal expansion of a signal, the more the oscillations grow in number, and the less are the amplitudes in size. How much energy a spherical signal has is a central problems in geosciences. Signal analysis has been extended to many diverse types of data with different understanding of energy. Today, the usual understanding of the total energy of a signal 𝐹 is achieved by the “continuous summation”, that means integration over all space (i.e., the unit sphere) Ω of the “fractional (pointwise) energy” |𝐹 (𝜉)| 2 , 𝜉 ∈ Ω, in the form ∫ ∥𝐹 ∥ 2L2 (Ω) = |𝐹 (𝜉)| 2 𝑑𝜔(𝜉), (1.1) Ω

where 𝑑𝜔 is the surface element. The space L2 (Ω) of all signals (functions) having finite energy may be appropriately characterized by certain systems of restrictions of homogeneous harmonics polynomials to the sphere, in fact, leading canonically to a spherical harmonics system constituting a Hilbert space. The polynomial structure of spherical harmonics {𝑌𝑛,𝑘 } 𝑛=0,1,...,𝑘=1,...,2𝑛+1 (1.2) in the Hilbert space L2 (Ω) has a tremendous advantage. First, spherical harmonics of different degrees are orthogonal (in the L2 (Ω)-topology implied by (1.1)). Second, the space Harm𝑛 = span 𝑘=1,...,2𝑛+1𝑌𝑛,𝑘 (1.3) of spherical harmonics of degree (frequency) 𝑛 is finite-dimensional. Its dimension is given by dim(Harm𝑛 ) = 2𝑛 + 1, so that Harm0,...,𝑚 = span

𝑛=0,...,𝑚 𝑘=1,...,2𝑛+1

𝑌𝑛,𝑘 =

𝑚 Ê

Harm𝑛

(1.4)

𝑛=0

Í 2 implies dim(Harm0....,𝑚 ) = 𝑚 𝑛=0 (2𝑛+1) = (𝑚+1) (cf. Fig. 1.3). The basis property 2 of {𝑌𝑛,𝑘 } 𝑛=0,1,...,𝑘=1,...,2𝑛+1 in the space L (Ω) of finite energy signals É∞ is equivalently characterized by the completion of the orthogonal direct sum 𝑛=0 Harm𝑛 , i.e., L2 (Ω) =

∞ Ê 𝑛=0

∥ · ∥ L2 (Ω)

Harm𝑛

.

(1.5)

6

1 Introductional Remarks

This is the natural reason why spherical harmonics expansions are essential tools not only in the scalar theory of gravitational and geomagnetic potentials, but also in vectorial research areas of fields, e.g., (geostrophic) ocean circulation, gravitational deflections of the vertical, and elastic geodeformation.

•

• • •

• •

• • •

𝑛 = 4, 𝑘 = 1

𝑛 = 4, 𝑘 = 9

𝑛 = 3, 𝑘 = 1

𝑛 = 3, 𝑘 = 7

𝑛 = 2, 𝑘 = 1

𝑛 = 1, 𝑘 = 1

𝑛 = 0, 𝑘 = 1

𝑛 = 2, 𝑘 = 5

𝑛 = 1, 𝑘 = 3

𝑛 = 0, 𝑘 = 1

.

Fig. 1.3: Spherical harmonics for increasing degrees 𝑛 = 0, . . . , 4 Spectral analysis in terms of spherical harmonics {𝑌𝑛,𝑘 } 𝑛=0,1,...,𝑘=1,...,2𝑛+1 has led to the discovery of basic laws of nature. It allows us to understand the composition and ingredients of features of the Earth (for more details about space and frequency description of signals see, e.g., I. Daubechies (1988, 1990) I. Daubechies, T. Paul (1988), R.N. Zare (1988), L. Cohen (1995), W. Freeden (2015), W. Freeden et al. (2019), and the references therein). All in all, the formalism of a spherical harmonic framework is essentially based on the following principles: (i) The spherical harmonics are obtainable in a twofold way, namely as restrictions of three-dimensional homogeneous harmonic polynomials or intrinsically on the unit sphere Ω as eigenfunctions of, e.g., the Beltrami operator or certain pseudodifferential operators. (ii) The Legendre kernels (Legendre polynomials) are obtainable as the outcome of sums extended over a maximal horizontal orthonormal system of spherical harmonics of degree (i.e., frequency) 𝑛 .

1.1 From Spherical Harmonics to Zonal Kernels

7

(iii) The Legendre kernels are rotation-invariant with respect to orthogonal transformations (leaving one point of the unit sphere Ω fixed). (iv) Each Legendre kernel implies an associated Funk-Hecke formula that determines the constituting features of the convolution (filtering) of a square-integrable field against the Legendre kernel. (v) The orthogonal Fourier expansion of a square-integrable function is the sum of the convolutions of the field against the Legendre kernels being extended over all frequencies. In fact, the theory of spherical harmonics provides a powerful spectral framework to unify, review, and supplement the different approaches in spaces over the unit sphere Ω, where distance and angle are at hand in suitable reference (pre-)Hilbert spaces. The essential tools in these (pre-)Hilbert spaces are the Legendre functions, used in orthogonal (Fourier) expansions and endowed with rotational invariance. The coordinate-free construction yields a number of formulas and theorems that previously were derived only in problem-affected coordinate (more precisely, polar coordinate) representations. As a consequence, any kind of singularity is avoided at points being fixed under orthogonal transformations. Finally, the transition from the scalar to the vectorial as well as the tensorial case opens promising perspectives of constructing important zonal classes of spherical trial functions by summing up Legendre kernel expressions, thereby providing (geo-)physical relevance and increasing local applicability (see, e.g., C. M¨uller (1966, 1998), W. Freeden et al. (1998), W. Freeden (2009), W. Freeden, M. Schreiner (2009), W. Freeden, C. Gerhards (2013), W. Freeden, M. Gutting (2013), W. Freeden, M. Gutting (2018), W. Freeden et al. (2018a), W. Freeden, M. Schreiner (2020a), and the references therein, for a variety of aspects on constructive spherical harmonics approximation). Any signal 𝐹 ∈ L2 (Ω) can be split (cf. Table 1.1) into “orthogonal contributions” involving the Fourier transforms 𝐹 ∧ (𝑛, 𝑘) defined by ∫ 𝐹 ∧ (𝑛, 𝑘) = 𝐹 (𝜉) 𝑌𝑛,𝑘 (𝜉) 𝑑𝜔(𝜉), (1.6) Ω

in terms of L2 (Ω)-orthonormal spherical harmonics {𝑌𝑛,𝑘 } 𝑛=0,1,...,𝑘=1,...,2𝑛+1 . The total energy of a signal should be independent of the method used to calculate it. Hence, ∥𝐹 ∥ 2L2 (Ω) as defined by (1.1) should be the sum of (𝐹 ∧ (𝑛, 𝑘)) 2 over all frequencies. So, Parseval’s identity connects the spatial energy of a signal with the spectral energy, decomposed orthogonally into single frequency contributions ∥𝐹 ∥ 2L2 (Ω) = (𝐹, 𝐹) L2 (Ω) =

∞ 2𝑛+1 ∑︁ ∑︁ 𝑛=0 𝑘=1

𝐹 ∧ (𝑛, 𝑘)


2

.

8

1 Introductional Remarks

Table 1.1 Orthogonal (Fourier) expansion of scalar square-integrable functions on the unit sphere Ω.

Weierstrass approximation theorem: use of homogeneous polynomials

↓

(geo-)physical constraint of harmonicity

spherical harmonics 𝑌𝑛, 𝑗 as restrictions of homogeneous harmonic polynomials 𝐻𝑛, 𝑗 to the unit sphere Ω ⊂ R3 orthonormality and orthogonal invariance

↓

addition theorem

one-dimensional Legendre polynomial 𝑃𝑛 satisfying 𝑃𝑛 ( 𝜉 · 𝜂) =

2𝑛+1 4 𝜋 ∑︁ 𝑌𝑛, 𝑗 ( 𝜉 )𝑌𝑛, 𝑗 ( 𝜂) , 𝜉 , 𝜂 ∈ Ω 2𝑛 + 1 𝑗=1

convolution against the Legendre kernel

↓

Funk-Hecke formula

Legendre transform of 𝐹: ∫ 2𝑛 + 1 ( 𝑃𝑛 ∗ 𝐹 ) ( 𝜉 ) = 𝑃𝑛 ( 𝜉 · 𝜂) 𝐹 ( 𝜂) 𝑑 𝜔 ( 𝜂) , 𝜉 ∈ Ω 4𝜋 Ω superposition over frequencies

↓

orthogonal (Fourier) series expansion

Fourier series of 𝐹 ∈ L2 (Ω): 𝐹(𝜉) =

∫ ∞ ∑︁ 2𝑛 + 1 𝑃𝑛 ( 𝜉 · 𝜂) 𝐹 ( 𝜂) 𝑑 𝜔 ( 𝜂), 𝜉 ∈ Ω 4𝜋 Ω 𝑛=0

This explains why the (global) geosciences work more often with the frequency energy, i.e., amplitude spectrum {𝐹 ∧ (𝑛, 𝑘)} 𝑛=0,1,...,𝑘=1,...,2𝑛+1 than with the original space signal 𝐹 ∈ L2 (Ω) (see, e.g., N.K. Pavlis et al. (2008) for the gravitational field, S. Macmillan et al. (2003) for the geomagnetic field, A. Dziewonski, D.L. Anderson (1981) for the Earth’s interior density distribution). As a consequence, the “inverse Fourier transform” ∞ 2𝑛+1 ∑︁ ∑︁ 𝐹= 𝐹 ∧ (𝑛, 𝑘)𝑌𝑛,𝑘 (1.7) 𝑛=0 𝑘=1

allows the geoscientists to think of the function (signal) 𝐹 as weighted superpositions of “wave functions” 𝑌𝑛,𝑘 corresponding to different frequencies.

1.1 From Spherical Harmonics to Zonal Kernels

9

There is a way of mathematically expressing the impossibility of simultaneous confinement of a function to space and frequency (more accurately, angular momentum), namely the uncertainty principle. If we consider |𝐹 (𝜉)| 2 , 𝜉 ∈ Ω, as a density in space (1) space so that ∥𝐹 ∥ 2L2 (Ω) = 1, the average space (expectation) 𝑔𝐹 (later designated by 𝑔𝐹𝑜 ) can be defined in the usual way any average is understood: ∫ space 𝑔𝐹 = 𝜉 (𝐹 (𝜉)) 2 𝑑𝜔(𝜉). (1.8) Ω

The reason for introducing an average is that it may give a gross characterization of the density. Moreover, it may indicate where the density is concentrated around the average. Various measures can be used to make certain whether the density is concentrated around the average. The most common measure is the standard (1) space deviation, 𝜎𝐹 (later designated in operator form by 𝜎𝐹𝑜 ), given by ∫ space space (𝜎𝐹 ) 2 = (𝜉 − (𝑔𝐹 ))(𝐹 (𝜉)) 2 𝑑𝜔(𝜉). (1.9) Ω

The standard deviation is an indication of the space localization of the signal. If the standard deviation is small, then most of the signal is concentrated around the average space. If (𝐹 ∧ (𝑛, 𝑗)) 2 represents the density in frequency, then we may use it to calculate averages, the motivation being the same as in space domain. It also gives a rough idea of the main characteristics of the spectral density. The average frequency (fre(3) frequency quency expectation), 𝑔𝐹 (later designated by 𝑔𝐹𝑜 ), and its standard deviation, (3) frequency 𝜎𝐹 (later designated by 𝜎𝐹𝑜 ), (also understandable as bandwith) are given by frequency

𝑔𝐹

=

∞ 2𝑛+1 ∑︁ ∑︁

𝑛(𝑛 + 1) 𝐹 ∧ (𝑛, 𝑗)


2

(1.10)

𝑛=0 𝑗=1

and frequency 2

(𝜎𝐹

) =

∞ 2𝑛+1 ∑︁ ∑︁ 

frequency

𝑛(𝑛 + 1) − 𝑔𝐹

2

𝐹 ∧ (𝑛, 𝑗)


2

.

(1.11)

𝑛=0 𝑗=1

Note that, for reasons of consequency with the theory of spherical harmonics we chose 𝑛(𝑛 + 1) instead of the 1D standard choice 𝑛 (see, e.g., L. Cohen (1995)). The discovery of the uncertainty relation by W. Heisenberg (1927) is one of the great achievements of the last century. For signal analysis it roughly states the fact that a narrow “spatial waveform” implies a wide frequency spectrum, and a wide “spatial waveform” yields a narrow spectrum. Both the spatial waveform and the frequency spectrum cannot be made arbitrarily small simultaneously.

10

1 Introductional Remarks

Space localization is at the cost of frequency localization, and vice versa. Expressed in formulas it means that space frequency Δ𝐹 Δ𝐹 ≥ 1, (1.12) space

where the so-called uncertainties Δ𝐹

frequency

, Δ𝐹

are given by

space

space Δ𝐹

and

frequency

Δ𝐹

=

𝜎𝐹

space

|𝑔𝐹

(1.13)

|

frequency 1/2

= (𝑔𝐹

)

.

(1.14)

The details of this spherical variant of the uncertainty principle on the unit sphere can be found in Chapter 9 of this work. Altogether, the uncertainty principle describes the trade-off between two “spreads” of the zonal kernels, one for the space and the other for the frequency (more precisely, angular momentum). The main statement is that sharp localization of zonal kernels in space and in frequency is mutually exclusive. The reason for the validity of the uncertainty relation is that the aforementioned operators 𝑜 (1) and 𝑜 (3) do not commute. Thus, 𝑜 (1) and 𝑜 (3) cannot be sharply defined simultaneously. As already mentioned, extremal members in the space/frequency relation are the Legendre kernels and the Dirac kernels (see Table 1.2). More explicitly, the uncertainty principle allows us to give a quantitative classification in the form of a canonically defined hierarchy of the space/frequency localization properties of zonal kernel functions, be they of scalar, vectorial, or tensorial nature. As a consequence, the uncertainty principle enables us to understand the transition from the theory of spherical harmonics through zonal kernel functions to the Dirac kernel. To this end we have to realize the relative advantages of the classical Fourier expansion method by means of spherical harmonics, and this is not only in the spectral (frequency) domain, but also in the space domain. It is a characteristic for Fourier techniques that the spherical harmonics as polynomial trial functions admit no localization in space domain, while in the frequency domain, they always correspond to exactly one degree, i.e., frequency, and therefore, are said to show ideal frequency localization. Because of the ideal frequency localization and the simultaneous absence of space localization local changes of fields (signals) in the space domain affect the whole table of orthogonal (Fourier) coefficients. This, in turn, causes global changes of the corresponding (truncated) Fourier series in the space domain. Nevertheless, ideal frequency localization is often helpful for meaningful physical interpretations by relating the different observables of a geopotential to each other at a fixed frequency, e.g., the Meissl scheme in Physical Geodesy (see P.A. Meissl (1971a,b, 1976), H. Nutz (2002), R. Rummel (1997), E.W. Grafarend (2001), W. Freeden, H. Nutz (2018), W. Freeden et al. (2018a), and the references therein). Taking these aspects on spherical harmonics modeling by Fourier series

1.1 From Spherical Harmonics to Zonal Kernels

11

into account, trial functions which simultaneously show ideal frequency localization as well as ideal space localization would be a desirable choice. In fact, such an ideal system of trial functions would admit models of highest spatial resolution which were expressible in terms of single frequencies. However, once more from the uncertainty principle, space and frequency localization are mutually exclusive. So, Fourier expansion methods are well suited to resolve low and medium frequency phenomena, i.e., the “trends” of a signal, while their application to obtain high resolution in global or local models is critical. This difficulty is also well known to theoretical physics, e.g., when describing monochromatic electromagnetic waves or considering the quantum-mechanical treatment of free particles. There, plane waves with fixed frequencies (ideal frequency localization, no space localization) are the solutions of the corresponding differential equations, but they do certainly not reflect the physical reality. As a remedy, plane waves of different frequencies are superposed into “wave-packages” that gain a certain amount of space localization, while losing their ideal spectral localization. In a similar way, we are confronted with the following situation: Any kernel function 𝐾 : Ω × Ω → R that is characterized by the property that there exists a function 𝐾˜ : [0, 2] → R such that  √︁ 𝐾 (𝜉, 𝜂) = 𝐾˜ (|𝜉 − 𝜂|) = 𝐾˜ 2 − 2𝜉 · 𝜂 = 𝐾ˆ (𝜉 · 𝜂), 𝜉, 𝜂 ∈ Ω, (1.15) is called a (spherical) radial basis function (at least in the theory of constructive approximation). The application of a rotation (i.e., a 3 × 3 orthogonal matrix t with tT = t−1 ) leads to 𝐾 (t𝜉, t𝜂) = 𝐾 (𝜉, 𝜂). (1.16) In particular, a rotation around the axis 𝜉 ∈ Ω (i.e., we have t𝜉 = 𝜉) yields 𝐾 (𝜉, 𝜂) = 𝐾 (𝜉, t𝜂) for all 𝜂 ∈ Ω. Hence, 𝐾 (𝜉, ·) possesses rotational symmetry with respect to the axis 𝜉. 𝐾ˆ : Ω × Ω → R satisfying 𝐾ˆ (𝜉 · 𝜂) = 𝐾ˆ (t𝜉 · t𝜂), 𝜉, 𝜂 ∈ Ω, for all orthogonal transformations t is known as a zonal kernel function. To highlight the reducibility of 𝐾ˆ to a function defined on the 1D interval [−1, 1], the notation (𝜉, 𝜂) ↦→ 𝐾ˆ (𝜉 · 𝜂), (𝜉, 𝜂) ∈ Ω × Ω, is used. A suitable superposition of spherical harmonics (cf. Fig. 1.5) leads to kernel functions with a reduced frequency, but increased space localization. From the theory of spherical harmonics we obtain a representation of any L2 (Ω)– zonal kernel function 𝐾 in terms of a Legendre expansion (note that we simply write 𝐾 (𝜉·) instead of 𝐾 (𝜉 · ·)) 𝐾 (𝜉·) =

∞ ∑︁ 2𝑛 + 1 𝑛=0

4𝜋

𝐾 ∧ (𝑛)𝑃𝑛 (𝜉·)

(1.17)

12

1 Introductional Remarks

(in the ∥ · ∥ L2 (Ω) –sense), where the sequence {𝐾 ∧ (𝑛)} 𝑛∈N0 given by 𝐾 ∧ (𝑛) = 2𝜋

1

∫

𝐾 (𝑡)𝑃𝑛 (𝑡) 𝑑𝑡

(1.18)

−1

is called the Legendre symbol of the zonal kernel 𝐾 (𝜉·). A simple but extreme example (with optimal frequency localization and no space localization) is the Legendre kernel, where 𝐾 ∧ (𝑛) = 1 for one particular 𝑛 and 𝐾 ∧ (𝑚) = 0 for 𝑚 ≠ 𝑛, i.e., the Legendre kernel is given by Ω × Ω ∋ (𝜉, 𝜂) ↦→

2𝑛 + 1 𝑃𝑛 (𝜉 · 𝜂). 4𝜋

(1.19)

Additive clustering of weighted Legendre kernels generates zonal kernel functions. We distinguish bandlimited kernels (i.e., 𝐾 ∧ (𝑛) = 0 for all 𝑛 > 𝑁) and nonbandlimited ones, for which infinitely many numbers 𝐾 ∧ (𝑛) do not vanish.

•

• • •

• •

• • •

𝑛 = 4, 𝑘 = 1

𝑛 = 4, 𝑘 = 9

𝑛 = 3, 𝑘 = 1

𝑛 = 3, 𝑘 = 7

𝑛 = 2, 𝑘 = 1

𝑛 = 1, 𝑘 = 1

𝑛 = 0, 𝑘 = 1

𝑛 = 2, 𝑘 = 5

𝑛 = 1, 𝑘 = 3

𝑛 = 0, 𝑘 = 1

Fig. 1.4: Spacelimited zonal kernel generated as weighted infinite sum over spherical harmonics.

1.1 From Spherical Harmonics to Zonal Kernels

13

Non-bandlimited kernels show a much stronger space localization than their bandlimited counterparts. Empirically, if 𝐾 ∧ (𝑛) ≈ 𝐾 ∧ (𝑛 + 1) ≈ 1 for many successive large integers 𝑛, then the support of the series (1.17) in the space domain is small, i.e., the kernel is spacelimited (i.e., in the jargon of approximation theory “locally supported”). This leads to the other extremal kernel (in contrast to the Legendre kernel) which is the Dirac kernel with optimal space localization but no frequency localization and 𝐾 ∧ (𝑛) = 1 for all 𝑛. However, the Dirac kernel does not exist as a classical function in mathematical sense, it is a generalized function (i.e., distribution). Nevertheless, it is well known that, if we have a family of kernels {𝐾 𝐽 } 𝐽=0,1,... where lim 𝐽→∞ 𝐾 𝐽∧ (𝑛) = 1 for each 𝑛 and an additional (technical) condition holds, then {𝐾 𝐽 } 𝐽=0,1,... is an approximate identity, i.e., 𝐾 𝐽 ∗ 𝐹 tends to 𝐹 in the sense of L2 (Ω) for all 𝐹 ∈ L2 (Ω), or to 𝐹 for all 𝐹 ∈ C (0) (Ω), respectively. Assuming lim𝑛→∞ 𝐾 ∧ (𝑛) = 0 we are led to the assertion that the slower the sequence {𝐾 ∧ (𝑛)} 𝑛=0,1,... converges to zero, the lower is the frequency localization, and the higher is the space localization. A unifying scheme can be found in Table 1.2. Table 1.2 From Legendre kernels via zonal kernels to the Dirac kernel.

Legendre kernels

zonal kernels

Dirac kernel

general case bandlimited

spacelimited

All in all, kernels exist as bandlimited and non-bandlimited functions. Every bandlimited kernel refers to a cluster of a finite number of polynomials i.e., spherical harmonics, hence, it corresponds to a certain band of frequencies. In contrast to a single polynomial which is localized in frequency but not in space, a bandlimited kernel such as the Shannon kernel already shows a certain amount of space localization. The Shannon kernel tends to have high oscillations, i.e., sign changes of its slope. A particularly interesting bandlimited kernel is the Bernstein kernel (cf. M.J. Fengler et al. (2006a)). On the one side it avoids oscillations, on the other side it constitutes a family that minimizes the uncertainty product in the limit. So, the Bernstein family becomes more and more balanced with respect to space as well as frequency. If we move from bandlimited to non-bandlimited kernels the frequency localization decreases and the space localization increases in accordance with the relationship provided by the uncertainty principle. The Gauss-Weierstrass kernel is known to be a non-bandlimited minimizer of the uncertainty principle (cf. W. Freeden, U. Windheuser (1996, 1997)). Kernel function approximation exists in spline and wavelet specifications, respectively, naturally based on certain fixed and

14

1 Introductional Remarks

“zooming” realizations in frequency and space localization. Obviously, if a certain accuracy should be guaranteed in kernel function approximation, adaptive sample point grids are required for the resulting extension of the spatial area determined by the kernels under investigation. In parallel, our considerations lead to the following characterization of trial functions in constructive approximation: Fourier expansion methods with polynomial ansatz functions offer the canonical “trend-approximation” of low-frequent phenomena (for global modeling), while bandlimited kernels can be used for the transition from long-wavelength to short-wavelength phenomena (global to local modeling). Because of their excellent localization properties in the space domain, the nonbandlimited kernels can be used for the modeling of short-wavelength phenomena (local modeling).

Fig. 1.5: Illustration of locally supported (i.e., spacelimited) kernels. In case of so-called scaling functions, the width of the corresponding frequency bands and, consequently, the amount of space localization is controlled (in continuous and/or discrete way) using a so-called scale parameter, such that the Dirac kernel acts as limit kernel as the scale parameter takes its limit. Typically, the generating kernels of scaling functions have the characteristics of lowpass filters, i.e., the zonal kernels involved in the convolution of the field against the Legendre kernels are significantly based on low frequencies, while the higher frequencies are attenuated or even completely left out in the summation. Conventionally, the difference between successive members in a scaling function is called a wavelet function. Clearly, it is again a zonal kernel. In consequence, wavelet functions have the typical properties of bandpass filters, i.e., the weighted Legendre kernels of low and high frequency within the wavelet kernel are attenuated or even completely left out. According to their particular construction, wavelet-techniques provide a decomposition of the reference space into a sequence of approximating subspaces – the scale spaces – corresponding to the scale parameter. In each scale space, a filtered version of a spherical field under consideration is calculated as a convolution of the field against the respective member of the scaling function and, thus, leading to an approximation of the field at certain resolutions. For increasing scales, the approximation improves and the information obtained on coarse levels is usually contained in foregoing levels. The difference between two successive bandpass filtered version of the signal

1.1 From Spherical Harmonics to Zonal Kernels

15

is called the detail information and is collected in the so-called detail space. The wavelets constitute the basis functions of the detail spaces and, summarizing our excursion to multiscale modeling, every element of the reference space can be represented as a structured linear combination of scaling functions and wavelets corresponding to different scales and at different positions. That is, using scaling functions und wavelets at different scales, the corresponding multiscale technique can be constructed as to be suitable for the specific local field structure. Consequently, although most fields show a correlation in space as well as in frequency, the zonal kernel functions with their simultaneous space and frequency localization allow for the efficient detection and approximation of essential features by only using fractions of the original information (decorrelation). Decorrelation enables us to specify transitional phenomena and delimitations in signatures. Table 1.3 brings together, into a unified nomenclature, the formalisms for zonal kernel function theory based on the following principles: • Weighted Legendre kernels are the constituting summands of zonal kernel functions. • The only zonal kernel that is both band- and spacelimited is the trivial kernel; the Legendre kernel is ideal in frequency localization, the Dirac kernel is ideal in space localization. • The convolution of a field (signal) against a zonal kernel function provides a filtered version of the original. • Scaling kernels, i.e., certain sequences of (parameter-dependent) zonal kernels tending to the Dirac kernel, provide better and better approximating lowpass filtered versions of the field (signal) under consideration. To summarize, the theory of zonal kernels as presented in this book (see Chapters 9, 10, and 11) is a unifying attempt of reviewing, clarifying and supplementing the different additive clusters of weighted Legendre kernels. The kernels exist as bandlimited and non-bandlimited, spacelimited, and non-spacelimited variants. The uncertainty principle determines the frequency/space window for approximation. A fixed space window is used for the windowed Fourier transform of fields (signals), where the approximation is still taken over the frequencies. The power of the scaling function lies in the fact that zonal kernels with a variable (space localizing) support come into use. The multiscale transform using scaling (kernel) functions is a space-reflected replacement of the Fourier transform, however, giving the dynamical space-varying frequency distribution of a field. Due to the possibility that variable kernel functions (i.e., scaling functions as sequential space localizing reductions)

16

1 Introductional Remarks

Table 1.3 Interrelations between space and frequency localization, kernel type, correlation. space localization

✛

✲

no space localization

ideal space localization frequency localization

✛

✲

ideal frequency localization

no frequency localization kernel type

✛ Legendre kernel

✲ bandlimited

locally supported

Dirac kernel

correlation

✛ ideal correlation

✲ no correlation

are being applied, a substantial better modeling of the high-frequency “short wavelength” part of a field (signal) is possible. This finally amounts to the transition from global to (scale dependent) local approximation (including multiresolution by spherical wavelets).

1.2 From Scalar Spherical Harmonics to its Vectorial and Tensorial Extensions

Concerning the mathematical representation of spherical vector and tensor fields in applied sciences, one is usually not interested in their separation into their (scalar) Cartesian component functions. Instead, we have to observe inherent physical constraints. For example, the external gravitational field is curl-free, the magnetic field is divergence-free, and the equations for incompressible Navier-Stokes equations in meteorological applications or the geostrophic formulation of ocean circulation include divergence-free vector solutions. In many cases, certain quantities are related to each other in an obvious manner by vector operators like the surface gradient or the surface curl gradient. In this respect, the gravity field, the magnetic field,

1.2 From Scalar Spherical Harmonics to its Vectorial and Tensorial Extensions

17

the wind field, the field of oceanic currents, or electromagnetic waves generated by surface currents should be mentioned as important examples. In addition, spherical modeling in terms of spherical harmonics arises naturally in the analysis of the elastic-gravitational free oscillations of a spherically symmetric, non-rotating Earth. Altogether, vector/tensor spherical harmonics are used throughout mathematics, theoretical physics, geo- and astrophysics, and engineering – indeed, wherever one deals with physically based fields. In the second half of the last century, a physically motivated approach for the decomposition of spherical vector and tensor fields was presented based on a spherical variant of the Helmholtz theorem (see, e.g., P.M. Morse, H. Feshbach (1953), E.H. Hill (1954), K.S. Thorne (1980), G.E. Backus (1966, 1967, 1986)). Following this concept, e.g., the tangential part of a spherical vector field is split up into a curlfree and a divergence-free field by use of two differential operators, viz. the already mentioned surface gradient and the surface curl gradient. Of course, an analogous splitting is valid in tensor theory.

Table 1.4 Twofold transition. scalar Legendre kernels

↓

scalar zonal kernels

↓

scalar Dirac kernel

→ → →

vector Legendre kernels

↓

vector zonal kernels

↓

vector Dirac kernel

→ → →

tensor Legendre kernels

↓

tensor zonal kernels

↓

tensor Dirac kernel

In subsequent publications during the second half of the last century, however, the vector spherical harmonic theory was usually written in local coordinate expressions that make mathematical formulations lengthy and hard to read. Tensor spherical harmonic settings are even more difficult to understand. In addition, when using local coordinates within a global spherical concept, differential geometry tells us that there is no representation of vector and tensor spherical harmonics which is free of singularities. In consequence, the mathematical arrangement involving vector and

18

1 Introductional Remarks

tensor spherical harmonics has led to an inadequately complex and less consistent literature, yet. Coordinate free explicit formulas on vector and/or tensor variants of the Legendre polynomials could not be found in the literature. As an immediate result, the orthogonal invariance based on specific vector /tensor extensions of the Legendre polynomials was not worked out suitably in a unifying scalar/vector/tensor framework. Even more, the concept of zonal (kernel) functions was not generalized adequately to the spherical vector/tensor case. All these new structures concerning spherical functions in mathematical (geo-)physics are developed and analyzed in this work. Basically two transitions are undertaken in our approach, namely the transition from spherical harmonics via zonal kernel functions to the Dirac kernels on the one hand and the transition from scalar to vector and tensor theory on the other hand (see Table 1.4). Table 1.5 Legendre kernel functions. scalar Legendre polynomial 𝑃𝑛 =

↓↑

application of 𝑜 (𝑖) vector Legendre kernel 𝑝𝑛(𝑖) =

𝑜 (𝑖) 𝑃𝑛 ( 𝜇𝑛(𝑖) ) 1/2

=

𝑂 (𝑖) 𝑂 (𝑖) p𝑛(𝑖,𝑖) 𝜇𝑛(𝑖)

application 𝑂 (𝑖)

𝑂 (𝑖) 𝑣 p𝑛(𝑖,𝑖)

↓↑

=

𝑜 (𝑖) 𝑝𝑛(𝑖) ( 𝜇𝑛(𝑖) ) 1/2

=

vectorial context

𝑜 (𝑖) 𝑜 (𝑖) 𝑃𝑛 𝜇𝑛(𝑖)

𝑂 (𝑖,𝑘) 𝑂 (𝑖,𝑘) P𝑛(𝑖,𝑘) 𝜇𝑛(𝑖,𝑘)

↓↑

application application of 𝑂 (𝑖,𝑘) 𝑜 (𝑖,𝑘) tensor Legendre kernel (order 2) 𝑡 (𝑖,𝑘) p𝑛

( 𝜇𝑛(𝑖) ) 1/2

application application of 𝑂 (𝑖) 𝑜 (𝑖) tensor Legendre kernel (order 2) 𝑣 (𝑖,𝑖) p𝑛

of

=

of

=

𝑜 (𝑖,𝑘) 𝑃𝑛 ( 𝜇𝑛(𝑖,𝑘) ) 1/2

=

𝑂 (𝑖,𝑘) P𝑛(𝑖,𝑘) ( 𝜇𝑛(𝑖,𝑘) ) 1/2

↓↑

application application of 𝑂 (𝑖,𝑘) 𝑜 (𝑖,𝑘) tensor Legendre kernel (order 4) P𝑛(𝑖,𝑘,𝑖,𝑘) =

𝑜 (𝑖,𝑘) 𝑡 p𝑛(𝑖,𝑘) ( 𝜇𝑛(𝑖,𝑘) ) 1/2

=

of

of

𝑜 (𝑖,𝑘) 𝑜 (𝑖,𝑘) 𝑃𝑛 𝜇𝑛(𝑖,𝑘)

tensorial context

To explain the transition from the theory of scalar spherical harmonics to its vectorial and tensorial extensions (see Chapters 5, 6, and 7 for details), our work starts from physically motivated dual pairs of operators (the reference space being always the space of signals with finite energy, i.e., the space of square-integrable fields). The pairs 𝑜 (𝑖) , 𝑂 (𝑖) , 𝑖 ∈ {1, 2, 3}, are originated in the constituting ingredients of the Helmholtz decomposition and subsequently the Hardy-Hodge decomposition of a vector field (see Section 6.2), while 𝑜 (𝑖,𝑘 ) , 𝑂 (𝑖,𝑘 ) , 𝑖, 𝑘 ∈ {1, 2, 3}, take the analogous role for the Helmholtz and Hardy-Hodge decomposition of tensor fields

1.2 From Scalar Spherical Harmonics to its Vectorial and Tensorial Extensions

19

(see Section 7.4). For example, in vector theory, 𝑜 (1) 𝐹 is assumed to be the normal (2) 𝐹 is the surface gradient field 𝜉 ↦→ field 𝜉 ↦→ 𝑜 (1) 𝜉 𝐹 (𝜉) = 𝐹 (𝜉)𝜉, 𝜉 ∈ Ω, 𝑜 ∗ (3) 𝐹 is the surface curl gradient field 𝜉 ↦→ 𝑜 (2) 𝜉 𝐹 (𝜉) = ∇ 𝜉 𝐹 (𝜉), 𝜉 ∈ Ω, and 𝑜 ∗ ∗ ∗ 𝑜 (3) 𝜉 𝐹 (𝜉) = 𝐿 𝜉 𝐹 (𝜉), 𝜉 ∈ Ω, with 𝐿 𝜉 = 𝜉 ∧ ∇ 𝜉 applied to a scalar-valued function

𝐹, while 𝑂 (1) 𝑓 is the normal component 𝜉 ↦→ 𝑂 (1) 𝜉 𝑓 (𝜉) = 𝑓 (𝜉) · 𝜉, 𝜉 ∈ Ω, ∗ 𝑂 (2) 𝑓 is the negative surface divergence 𝜉 ↦→ 𝑂 (2) 𝜉 𝑓 (𝜉) = −∇ 𝜉 · 𝑓 (𝜉), 𝜉 ∈ Ω, ∗ and 𝑂 (3) 𝑓 is the negative surface curl 𝜉 ↦→ 𝑂 (3) 𝜉 𝑓 (𝜉) = −𝐿 · 𝑓 (𝜉), 𝜉 ∈ Ω, taken over a vector-valued function 𝑓 . Clearly, the operators 𝑜 (𝑖,𝑘 ) , 𝑂 (𝑖,𝑘 ) are also definable in orientation to the tensor Helmholtz decomposition theorem (for reasons of simplicity, however, their explicit description is omitted here). It should be noted that, in vector as well as tensor theory, the connecting link from the operators to the Helmholtz decomposition is the Green’s function with respect to the (scalar) Beltrami operator and its iterations (for more details, the reader is referred to Chapter 5 of this work). The pairs 𝑜 (𝑖) , 𝑂 (𝑖) and 𝑜 (𝑖,𝑖) , 𝑂 (𝑖,𝑖) of dual operators lead to an associated palette of Legendre kernel functions, all of them generated by the classical onedimensional Legendre polynomial 𝑃𝑛 of degree 𝑛. To be more concrete, three types of Legendre kernels occur in the vectorial as well as tensorial context (see Table 1.5). The Legendre kernels 𝑜 (𝑖) 𝑃𝑛 , 𝑜 (𝑖) 𝑜 (𝑖) 𝑃𝑛 , 𝑖 = 1, 2, 3, are of concern for the vector approach to spherical harmonics, whereas 𝑜 (𝑖,𝑖) 𝑃𝑛 , 𝑜 (𝑖,𝑖) 𝑜 (𝑖,𝑖) 𝑃𝑛 , 𝑖 = 1, 2, 3, form the analogues in tensorial theory. Corresponding to each Legendre kernel, we are led to two variants for representing square-integrable fields by orthogonal (Fourier) expansion, where the reconstruction – as in the scalar case – is undertaken by superposition over all frequencies.

Tables 1.5, 1.6, and 1.7 bring together – into a single unified notation – the formalisms for the vector/tensor spherical harmonic theory based on the following principles: • The vector/tensor spherical harmonics involving the 𝑜 (𝑖) , 𝑜 (𝑖,𝑖) -operators, respectively, are obtainable as restrictions of three-dimensional homogeneous harmonic vector/tensor polynomials, respectively, that are computable exactly exclusively by integer operations. • The vector/tensor Legendre kernels are obtainable as the outcome of sums extended over a maximal orthonormal system of vector/tensor spherical harmonics of degree (frequency) 𝑛, respectively. • The vector/tensor Legendre kernels are zonal kernel functions, i.e., they are orthogonally invariant (in vector/tensor sense, respectively) with respect to orthogonal transformations. • Spherical harmonics of degree (frequency) 𝑛 form an irreducible subspace of the reference space of (square-integrable) fields on Ω.

20

1 Introductional Remarks

Table 1.6 Fourier expansion of (square-integrable) vector fields 𝑓 . vector spherical harmonics (𝑖) (𝑖) −1/2 (𝑖) 𝑦𝑛, 𝑜 𝑌𝑛, 𝑗 𝑗 = ( 𝜇𝑛 )

addition theorem

↓

𝑣 (𝑖,𝑖) p𝑛 ( 𝜉 , 𝜂) 2𝑛+1 ∑︁ (𝑖) = 𝑦𝑛, 𝑗(𝜉) 𝑗=1

addition theorem

vectorial variant

(𝑖) ⊗ 𝑦𝑛, 𝑗 ( 𝜂)

𝑗=1

↓

2𝑛 + 1 4∫𝜋

𝑣 (𝑖,𝑖) p𝑛 ( 𝜉 ,

↓

Funk-Hecke vectorial variant formula Legendre transform 2𝑛 + 1 (𝑖) −1/2 ( 𝜇𝑛 ) 4∫𝜋

𝜂) 𝑓 ( 𝜂) 𝑑 𝜔 ( 𝜂)

𝑝𝑛(𝑖) ( 𝜉 , 𝜂)𝑂 𝜂(𝑖) 𝑓 ( 𝜂) 𝑑 𝜔 ( 𝜂)

×

Ω

Ω

superposition

𝑓 (𝜉) =

↓

superposition

over frequencies

3 ∑︁ ∞ ∑︁ 2𝑛 + 1 4𝜋 𝑖=1 𝑛=0

𝑓 (𝜉) =

𝑖

∫ ×

tensorial variant

𝑝𝑛(𝑖) ( 𝜉 , 𝜂) 2𝑛+1 ∑︁ (𝑖) = 𝑦𝑛, 𝑗 ( 𝜉 )𝑌𝑛, 𝑗 ( 𝜂)

Funk-Hecke tensorial variant formula Legendre transform

×

↓

𝑣 (𝑖,𝑖) p𝑛 ( 𝜉 ,

𝜂) 𝑓 ( 𝜂) 𝑑 𝜔 ( 𝜂)

over frequencies

3 ∑︁ ∞ ∑︁ 2𝑛 + 1 (𝑖) −1/2 ( 𝜇𝑛 ) 4𝜋 𝑖=1 𝑛=0 𝑖

∫

Ω

↓

×

𝑝𝑛(𝑖) ( 𝜉 ,

𝜂)𝑂 𝜂(𝑖) 𝑓 ( 𝜂) 𝑑 𝜔 ( 𝜂)

Ω

rank–2 tensorial approach

vectorial approach

• Each Legendre kernel implies an associated Funk-Hecke formula that determines the constituting features of the convolution of a square-integrable field against the Legendre kernel. • The orthogonal Fourier expansion of a square-integrable field is the sum of the convolutions of the field against the Legendre kernels being extended over all frequences. In Fig. 1.6 some vector spherical harmonics generated by 𝑜 (𝑖) , 𝑖 = 2, 3, are depicted to give an impression of the different characteristic behavior of surface gradient and surface curl gradient fields.

1.2 From Scalar Spherical Harmonics to its Vectorial and Tensorial Extensions

21

Table 1.7 Fourier expansion of square-integrable tensor fields f. tensor spherical harmonics (𝑖,𝑘) (𝑖,𝑘) −1/2 (𝑖,𝑘) y𝑛, ) o 𝑌𝑛, 𝑗 𝑗 = ( 𝜇𝑛

addition theorem

↓

rank-4 tensorial variant

addition theorem

P𝑛(𝑖,𝑘,𝑖,𝑘) ( 𝜉 , 𝜂) 2𝑛+1 ∑︁ (𝑖,𝑘) (𝑖,𝑘) = y𝑛, 𝑗 ( 𝜉 ) ⊗ y𝑛, 𝑗 ( 𝜂)

↓

𝑡 (𝑖,𝑘) p𝑛 ( 𝜉 , 𝜂) 2𝑛+1 ∑︁ (𝑖,𝑘) = y𝑛, 𝑗 ( 𝜉 )𝑌𝑛, 𝑗 ( 𝜂) 𝑗=1

𝑗=1

↓

↓

rank-4 tensorial Funk-Hecke variant formula Legendre transform

rank-2 tensorial Funk-Hecke variant formula Legendre transform

2𝑛 + 1 4∫𝜋

2𝑛 + 1 (𝑖,𝑘) −1/2 ( 𝜇𝑛 ) 4∫𝜋

×

×

P𝑛(𝑖,𝑘,𝑖,𝑘) ( 𝜉 , 𝜂)f ( 𝜂) 𝑑 𝜔 ( 𝜂)

Ω

𝑡 (𝑖,𝑘) p𝑛 ( 𝜉 ,

f(𝜉) =

↓

superposition

over frequencies

3 ∞ ∑︁ ∑︁ 2𝑛 + 1 4𝜋 𝑖,𝑘=1 𝑛=0

f(𝜉) =

𝑖𝑘

×

𝜂)𝑂 𝜂(𝑖,𝑘) f ( 𝜂) 𝑑 𝜔 ( 𝜂)

Ω

superposition

∫

rank-2 tensorial variant

P𝑛(𝑖,𝑘,𝑖,𝑘) ( 𝜉 ,

∫ 𝜂)f ( 𝜂) 𝑑 𝜔 ( 𝜂)

Ω

×

↓

over frequencies

3 ∞ ∑︁ ∑︁ 2𝑛 + 1 1 (𝑖,𝑘) 1/2 4 𝜋 ( 𝜇𝑛 ) 𝑖,𝑘=1 𝑛=0𝑖𝑘

𝑡 (𝑖,𝑘) p𝑛 ( 𝜉 ,

𝜂)𝑂 𝜂(𝑖,𝑘) f ( 𝜂) 𝑑 𝜔 ( 𝜂)

Ω

rank-4 tensorial approach

rank-2 tensorial approach

Unfortunately, the vector spherical harmonics generated by the operators 𝑜 (𝑖) , 𝑂 (𝑖) , 𝑖 = 1, 2, 3, do not constitute eigenfunctions with respect to the Beltrami operator. However, the resulting Hardy-Hodge decomposition enables us to introduce certain operators 𝑜˜ (𝑖) , 𝑖 = 1, 2, 3, in terms of the operators 𝑜 (𝑖) , 𝑖 = 1, 2, 3, which define alternative classes of vector spherical harmonics that represent eigensolutions to the Beltrami operator. The price to be paid is that the separation of spherical vector fields into normal and tangential parts is lost. More precisely, the operators 𝑜˜ (𝑖) , 𝑖 ∈ {1, 2}, generate so-called spheroidal fields, while 𝑜˜ (3) generates poloidal fields. In fact, all statements involving orthogonal (Fourier) expansion of spherical fields remain valid for this new class of operators. Moreover, analogous classes of tensor spherical harmonics can be introduced by operators o˜ (𝑖,𝑘 ) , 𝑂˜ (𝑖,𝑘 ) , 𝑖, 𝑘 = 1, 2, 3, in close analogy to the vector case. In addition, it should be noted that the spherical harmonics based

22

1 Introductional Remarks

(𝑖) Fig. 1.6: Vector spherical harmonics 𝑦 𝑛,𝑘 = 𝑜 (𝑖) 𝑌𝑛,𝑘 of (surface curl-free) type 𝑖 = 2 (left column) and of (surface divergence-free) type 𝑖 = 3 (right column), the degree 𝑛 = 8 in each case, the order 𝑘 = 0 in the top row, 𝑘 = 4 in the middle row, and 𝑘 = 8 in the bottom row (the colors illustrate the absolute values and the arrows show the directions of the vector fields).

on the 𝑜˜ (𝑖) , 𝑂˜ (𝑖) , o˜ (𝑖,𝑘 ) , 𝑂˜ (𝑖,𝑘 ) -operators play a particular role whenever the Laplace operator comes into play, e.g., for representing any kind of harmonic fields.

To summarize, the theory of spherical harmonics as presented in this book is a unifying attempt of consolidating, reviewing and supplementing the different approaches in real scalar, vector, and tensor theory. The essential tools are the Legendre kernels which are shown to be explicitly available and tremendously significant in rotational invariance and in orthogonal Fourier expansions. The work is self-contained: the reader is told how to derive all equations occuring in due course. Most importantly, our coordinate-free setup yields a number of formulas and theorems that previously were derived only in coordinate representation (such as polar coordinates). In doing so, any kind of singularities is avoided at the poles. Finally, our philosophy opens new promising perspectives of constructing important, i.e., zonal classes of spher-

1.3 Organization of the Work

23

Table 1.8 Basic constituents of the book. addition theorem

tensorial

vectorial

scalar orthogonal invariance

closure, completeness

ical trial functions by summing up Legendre kernel expressions, thereby providing (geo-)physical relevance and increasing local applicability.

1.3 Organization of the Work

The layout of this work is as follows: • The introductory remarks of Chapter 1 explain a twofold transition: first the uncertainty principle enables us to understand the transition from the theory of spherical harmonics through zonal kernel functions to the Dirac kernel; second decomposition theorems provide the transition from the theory of scalar spherical harmonics to its vectorial and tensorial extensions. • Chapter ?? gives an introduction into spherical nomenclature and settings. Fundamental results of spherical vector analysis are recapitulated. Orthogonal invariance is explained within the scalar, vectorial, and tensorial concept (see Table 1.8). • Chapter 3 deals with circular harmonics and their properties. This chapter is meant as 2D-preparation of the 3D-theory of spherical harmonics.

24

1 Introductional Remarks

• In Chapter 4, the scalar theory of spherical harmonics is formulated based on the work of C. M¨uller (1952, 1966) and W. Freeden (1979, 1980b). Important ingredients are the addition theorem of spherical harmonics and the Funk-Hecke formula. The closure and completeness of scalar spherical harmonics in the space of square-integrable functions is shown by Bernstein and/or Abel-Poisson summability. Exact generation of linearly independent systems of homogeneous harmonic polynomials only by integer operations is investigated briefly. Fourier (orthogonal) expansions are discussed, (the energy of) a square-integrable function (signal) is split into degree variances in terms of spherical harmonics. The scalar spherical harmonics are recognized to be eigenfunctions of the scalar Beltrami operator on the (unit) sphere. The Legendre polynomials are identified as the only scalar spherical harmonics invariant under orthogonal transformations. Zonal, tesseral, and sectorial spherical harmonics, i.e., associated Legendre harmonics, are introduced by use of associated Legendre functions. Scalar angular derivatives are seen to produce anisotropic operators within the scalar framework. • Chapter 5 presents the theory of Green’s functions with respect to the scalar Beltrami operator (as proposed by W. Freeden (1979, 1980b, 1981a)). Its definition is given by formulating four constituting properties, i.e., the Beltrami differential equation relating the Green function to the Dirac function(al), the characteristic logarithmic singularity, the rotational symmetry, and a certain normalization condition to assure uniqueness. Space and frequency mollifiers of the Green’s function with respect to the scalar Beltrami operator are used to solve the surface gradient and surface curl gradient equation. Integral formulas are formulated that enable us to estimate the error between a (sufficiently smooth) function and its truncated orthogonal expansion in terms of scalar spherical harmonics. Integral expressions are deduced which act as solutions of the equations involving surface gradient, surface curl gradient, and (iterated) Beltrami differential operators. The results on Green’s functions yield decisive material for the decomposition theorems of sphericals vector and tensor fields, respectively, in accordance with the Helmholtz approach. Iterated Beltrami equations are solved by integral expressions involving Green’s functions. The chapter ends with the generalization of the concept of Green’s functions to scalar pseudodifferential operators (cf. W. Freeden et al. (1998)). • In Chapter 6, the vector theory of spherical harmonics is developed in consistency with its scalar counterpart (based on the work W. Freeden et al. (1998)). A particular role is played by the Helmholtz decomposition theorem which separates a spherical vector field into three field components, namely a radial, a tangential divergence-free, and a tangential curl-free part. As already pointed out, an essential tool for representing a spherical vector field is the Green’s function with respect to the Beltrami operator. The physical background for the Helmholtz decomposition is based on well-known facts of surface vector analysis, viz. the existence of surface potentials and stream functions, and the characterization of tangential vector fields such as surface (curl) gradient fields. To be more concrete,

1.3 Organization of the Work

25

the surface gradient field on the sphere is seen to be generated by a potential function, while the surface curl gradient field is canonically related to a stream function. Vectorial analogues of the Legendre polynomials are introduced, their properties are analyzed in detail. Outstanding keystones in the vectorial framework of vector spherical harmonics are the addition theorem and the formulas of Funk and Hecke. The Helmholtz decomposition serves as essential tool to deduce the Hardy-Hodge decompositions to deliver vector spherical harmonics as eigensolutions of the Beltrami operator. The closure and completeness of vector spherical harmonics for the space of square-integrable vector fields is shown via Bernstein summability. Two different ways of expanding square-integrable fields in terms of (an orthonormal system of) vector spherical harmonics are described alternatively based on a (one-step) tensor-vector multiplication or on a consecutive (two-step) vector-scalar and scalar-vector multiplication. • Chapter 7 deals with the theory of tensor spherical harmonics (in close orientation to M. Schreiner (1994), W. Freeden et al. (1994, 1998)). All essential results known from the scalar and vectorial approach are extended to the tensor case. Orthonormal tensor spherical harmonics are introduced in the space of squareintegrable tensor fields on the unit sphere. In particular, the addition theorem for tensor spherical harmonics is formulated and the decomposition theorem for spherical tensor fields is verified by use of the Green’s function with respect to iterations of the Beltrami operator. The tensor spherical harmonics are characterized as eigenfunctions of a tensorial analogue of the Beltrami operator. Tensorial versions of the Funk-Hecke formula are described in more detail. • Chapter 8 presents the theory of spin-weighted spherical harmonics. The definition is given in terms of spherical coordinates. Wigner 𝐷-matrices are introduced to provide the core of the theory of spin-weighted spherical harmonics (i.e., spin-weighted Legendre polynomials, addition theorem, completeness properties, eigenfunctions of the spin-weighted Beltrami operator). Relations to vector and tensor spherical harmonics are pointed out. • Chapter 9 presents the mathematical classification of zonal kernel functions. The verification and interpretation of an uncertainty principle for fields (with second distributional derivatives) on the the unit sphere is the essential tool for the classification. Frequency as well as space localization are formulated by means of the expectation value and the variance of the surface curl gradient and the radial projection operator, respectively. The results obtained by certain tools of spherical vector analysis are used for a large class of bandlimited and non-spacelimited as well as non-bandlimited and spacelimited zonal kernel functions. The particular role of the Legendre kernel and the Dirac kernel is pointed out. The series expansions of vector/tensor zonal kernel functions in terms of (zonal) Legendre kernels are indicated by the specification of their symbols. All representations are coordinate-free.

26

1 Introductional Remarks

• Chapter 10 considers two different ways of generating vectorial and tensorial zonal kernel functions, in accordance with the Helmholtz and Hardy-Hodge decomposition of vector fields. In particular, scale dependent bandlimited and nonbandlimited zonal kernel functions are listed such that the scale parameter acts as regulation for the amount of space/frequency localization. The Funk-Hecke formulas enable us to establish filtered versions of spherical fields by forming convolutions. The sequences of zonal kernel functions tending to the Dirac kernel, i.e., the so-called scaling functions, provide a “zooming-in” approximation of square-integrable fields from global to local features under (geophysical) constraints. • Chapter 11 presents the concept of tensorial zonal kernel functions. Their description is given in parallel to the vectorial case. Particular emphasis is laid on tensor scaling functions. • Chapters 12 and 13 constitude links of the spherically oriented special function theory to geomathematically relevant research. The essential goal is to present the concepts, structures, and tools of spherically based function systems in parallel for balls and spheres in R3 to understand potential data. Vector and tensor outer harmonic zonal kernels are shown to be the adequate means for “downward continuation” of vectorial and tensorial gravitational data from satellite orbits to the Earth’s surface. • Chapter 14 presents the decorrelation of the terrestrial Earth’s gravitational anomaly field and the geostrophic surface flow of oceans by means of vectorial (geophysically adapted) mollifier wavelets. • Finally, the concluding remarks of Chapter 15 point out that only the joint use of mathematical technology and highly accurate sensors combining (globally available) spaceborne data with local airborne and/or terrestrial observations will contribute to a deeper knowledge of the Earth system and, in turn, to the development of sustainable strategies to safeguard the human habitat for future generations. In this respect, the spherically oriented structures, methods and procedures such as polynomial (spherical harmonic), spline, and wavelet approximations as presented in the book by W. Freeden et al. (2018a) in the “Geosystems Mathematics” series form an essential step for handling terrestrial, airborne, and spaceborne data under relevant physical, approximation theoretic as well as numerical assumptions.

1.3 Organization of the Work

27

A brief view over the contents of this book is given in Table 1.9. Table 1.9 Contents (in brief). Scalar framework

Vector framework

Tensor framework

Basic settings (differential operators, orthogonal invariance, circular harmonics)

Chapter 2

Chapter 2

Chapter 2

Green’s functions, integral theorems

Chapter 5

Spherical harmonics (definition, Legendre functions, addition theorems, Funk-Hecke formulas)

Chapters 3, 4, and 8

Chapters 6 and 8

Chapters 7 and 8

Zonal kernel functions (definition, classification, scaling functions, Dirac kernel)

Chapter 9

Chapter 10

Chapter 11

Geoscientific function systems (potentials, closure, geoapplications, etc.)

Chapters 12, 13, and 14

Chapters 12, 13, and 14

Chapters 12, 13, and 14

Part II

Background and Nomenclature

Chapter 2

Basic Settings

In this chapter we start with some notation in the three-dimensional Euclidean space R3 . The most important differential operators in R3 to be needed throughout the work are listed. We give the representation of the gradient and the Laplace operator and split them into their radial and angular parts. Certain differential operators on the unit sphere Ω in R3 are introduced, including the surface gradient, the surface curl gradient, the surface divergence, the surface curl, and the Beltrami operator. Although we rely on coordinate-free representations throughout this book (to avoid coordinate-implied singularities on the (global) sphere Ω), these operators will be discussed, for the convenience of the reader, in the particular system of spherical coordinates. Function spaces of scalar- and vectorvalued functions on the unit sphere are characterized. Basic theorems on vector analysis are recapitulated in spherical language. We continue with basic results on spherical symmetry and orthogonal invariance in the scalar, vector, and tensor context, respectively.

2.1 Euclidean Nomenclature

As usual, the letters N, N0 , Z, Q, R, and C denote the set of positive, non-negative integers, integers, rational numbers, real numbers, and complex numbers, respectively. Let us use 𝑥, 𝑦, . . . to represent the elements of the Euclidean space R3 . For all 𝑥 ∈ R3 , 𝑥 = (𝑥1 , 𝑥2 , 𝑥3 ) 𝑇 , different from the origin, we have √︃ 𝑥 = 𝑟𝜉, 𝑟 = |𝑥| = 𝑥12 + 𝑥22 + 𝑥 32 , (2.1)

© The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2022 W. Freeden and M. Schreiner, Spherical Functions of Mathematical Geosciences, Geosystems Mathematics, https://doi.org/10.1007/978-3-662-65692-1_2

31

32

2 Basic Settings

where 𝜉 = (𝜉1 , 𝜉2 , 𝜉3 ) 𝑇 is the uniquely determined directional unit vector of 𝑥 ∈ R3 . The unit sphere in R3 is denoted by Ω,  Ω = 𝜉 ∈ R3 |𝜉 | = 1 . If the vectors 𝜀 1 , 𝜀 2 , 𝜀 3 form the canonical orthonormal basis in R3 1 © ª 𝜀1 = ­ 0 ® , «0¬

0 © ª 𝜀2 = ­ 1 ® , «0¬

0 © ª 𝜀3 = ­ 0 ® , «1¬

(2.2)

we may represent the points 𝑥 ∈ R3 in Cartesian coordinates 𝑥𝑖 = 𝑥 · 𝜀 𝑖 , 𝑖 = 1, 2, 3, by 3 3 ∑︁ ∑︁ 𝑥= (𝑥 · 𝜀 𝑖 )𝜀 𝑖 = 𝑥𝑖 𝜀𝑖 . (2.3) 𝑖=1

𝑖=1

The inner (scalar), vector, and dyadic (tensor) product of two elements 𝑥, 𝑦 ∈ R3 , are defined by 𝑥 · 𝑦 = 𝑥𝑇 𝑦 =

3 ∑︁

(2.4)

𝑥𝑖 𝑦 𝑖 ,

𝑖=1

𝑥 ∧ 𝑦 = (𝑥2 𝑦 3 − 𝑥 3 𝑦 2 , 𝑥3 𝑦 1 − 𝑥 1 𝑦 3 , 𝑥1 𝑦 2 − 𝑥2 𝑦 1 ) 𝑇 ,

(2.5)

𝑥1 𝑦 1 𝑥1 𝑦 2 𝑥1 𝑦 3 © ª 𝑥 ⊗ 𝑦 = 𝑥𝑦 𝑇 = ­ 𝑥 2 𝑦 1 𝑥2 𝑦 2 𝑥2 𝑦 3 ® , « 𝑥3 𝑦 1 𝑥3 𝑦 2 𝑥3 𝑦 3 ¬

(2.6)

respectively. Clearly, 𝑥 2 = |𝑥| 2 = 𝑥 · 𝑥 = 𝑥 𝑇 𝑥, 𝑥 ∈ R3 . Moreover, for 𝑥, 𝑦 ∈ R3 , we have the Cauchy-Schwarz inequality |𝑥 · 𝑦| ≤ |𝑥| · |𝑦|

(2.7)

||𝑥| − |𝑦|| ≤ |𝑥 ± 𝑦| ≤ |𝑥| + |𝑦|.

(2.8)

and the triangle inequality

With the alternator (Levi-Civit`a alternating symbol)

𝜀

𝑖 𝑗𝑘

   1, (𝑖, 𝑗, 𝑘) is an even permutation of (1, 2, 3),  = −1, (𝑖, 𝑗, 𝑘) is an odd permutation of (1, 2, 3),   0, (𝑖, 𝑗, 𝑘) is not a permutation of (1, 2, 3), 

we obtain (𝑥 ∧ 𝑦) · 𝜀 𝑖 = (𝑥 ∧ 𝑦)𝑖 =

3 ∑︁ 3 ∑︁ 𝑗=1 𝑘=1

𝜀𝑖 𝑗 𝑘 𝑥 𝑗 𝑦 𝑘 .

(2.9)

(2.10)

2.1 Euclidean Nomenclature

33

Moreover, we have 3 ∑︁

𝜀 𝑖 𝑗 𝑘 𝜀 𝑖 𝑝𝑞 = 𝛿 𝑗 𝑝 𝛿 𝑘𝑞 − 𝛿 𝑗𝑞 𝛿 𝑘 𝑝 ,

(2.11)

𝑖=1

where 𝛿𝑖 𝑗 is the Kronecker delta  𝛿𝑖 𝑗 =

0, 𝑖 ≠ 𝑗, 1, 𝑖 = 𝑗 .

(2.12)

As usual, a tensor x ∈ R3 ⊗ R3 of second rank (of rank 2 or second order) is understood to be a linear mapping that assigns to each 𝑥 ∈ R3 a vector 𝑦 ∈ R3 : 𝑦 = x𝑥. The (Cartesian) components x𝑖 𝑗 of x are defined by x𝑖 𝑗 = 𝜀 𝑖 · (x𝜀 𝑗 ) = (𝜀 𝑖 ) 𝑇 (x𝜀 𝑗 ),

(2.13)

so that 𝑦 = x𝑥 is equivalent to 𝑦𝑖 = 𝑦 · 𝜀𝑖 =

3 ∑︁

x𝑖 𝑗 (𝑥 · 𝜀 𝑗 ) =

𝑗=1

3 ∑︁

x𝑖 𝑗 𝑥 𝑗 .

(2.14)

𝑗=1

The inner product x · y of two rank-2 tensors x, y ∈ R3 ⊗ R3 (also known as double dot product x : y) is defined by x · y = tr(x𝑇 y) =

3 ∑︁ 3 ∑︁

x𝑖 𝑗 y𝑖 𝑗 ,

(2.15)

𝑖=1 𝑗=1

while |x| = (x · x) 1/2 is called the norm of x ∈

R3

⊗

(2.16)

R3 .

Given any tensor x and any pair 𝑥, 𝑦 ∈ R3 , we have 𝑥 · (x𝑦) = x · (𝑥 ⊗ 𝑦).

(2.17)

In connection with (2.17) it is easy to see that (𝜀 𝑖 ⊗ 𝜀 𝑗 ) · (𝜀 𝑘 ⊗ 𝜀 𝑙 ) = 𝛿𝑖𝑘 𝛿 𝑗𝑙 ,

(2.18)

so that the nine tensors 𝜀 𝑖 ⊗ 𝜀 𝑗 are orthonormal. Moreover, it follows that 3 ∑︁ 3 ∑︁ 𝑖=1 𝑗=1

3 ∑︁ 3 ∑︁
x𝑖 𝑗 𝜀 𝑖 ⊗ 𝜀 𝑗 𝑥 = x𝑖 𝑗 (𝑥 · 𝜀 𝑗 )𝜀 𝑖 = x𝑥. 𝑖=1 𝑗=1

Thus, x ∈ R3 ⊗ R3 can be written in the form

(2.19)

34

2 Basic Settings

x=

3 ∑︁ 3 ∑︁

x𝑖 𝑗 𝜀 𝑖 ⊗ 𝜀 𝑗 .

(2.20)

𝜀𝑖 ⊗ 𝜀𝑖 .

(2.21)

𝑖=1 𝑗=1

The identity tensor i is given by i=

3 ∑︁ 𝑖=1

Moreover, we write tr(x) for the trace of x and det(x) for the determinant of x. It is not hard to see that tr(𝑥 ⊗ 𝑦) = 𝑥 · 𝑦, 𝑥, 𝑦 ∈ R3 . (2.22) Furthermore,     x · (yz) = y𝑇 x · z = xz𝑇 · y,

x, y, z ∈ R3 ⊗ R3 .

We write x𝑇 for the transpose of x; it is the unique tensor satisfying   (x𝑦) · 𝑥 = 𝑦 · x𝑇 𝑥 ,

(2.23)

(2.24)

for all 𝑥, 𝑦 ∈ R3 . We call x symmetric if x = x𝑇 , and skew if x = −x𝑇 . Every tensor x admits the unique decomposition x = sym x + skw x,

(2.25)

into the symmetric part sym x and the skew part skw x. More explicitly, sym x =

 1 x + x𝑇 , 2

skw x =

 1 x − x𝑇 . 2

(2.26)

It should be noted that there is a one-to-one correspondence between vectors and skew tensors: Given any skew tensor w, there exists a unique vector 𝑤 such that w𝑥 = 𝑤 ∧ 𝑥 for every 𝑥 ∈ R3 ; indeed, 3

𝑤𝑖 = −

3

1 ∑︁ ∑︁ 𝑖 𝑗 𝑘 𝜀 w𝑗𝑘. 2 𝑗=1 𝑘=1

(2.27)

We call 𝑤 the axial vector corresponding to w. Conversely, given a vector 𝑤, there exists a unique skew tensor w such that the above relation holds; in fact, w𝑖 𝑗 = −

3 ∑︁ 𝑘=1

𝜀𝑖 𝑗 𝑘 𝑤𝑘 .

(2.28)

2.1 Euclidean Nomenclature

35

The dyadic (tensor) product 𝑥 ⊗ 𝑦 of two elements 𝑥, 𝑦 ∈ R3 (see (2.6)) is the tensor that assigns to each 𝑢 ∈ R3 the vector (𝑦 · 𝑢)𝑥. More explicitly, (𝑥 ⊗ 𝑦)𝑢 = (𝑦 · 𝑢)𝑥

(2.29)

for every 𝑢 ∈ R3 . By use of the canonical orthonormal basis {𝜀 1 , 𝜀 2 , 𝜀 3 } of R3 , a tensor F of rank 𝑘 is written in the form F=

3 ∑︁

𝐹𝑖1 ...𝑖𝑘 𝜀 𝑖1 ⊗ . . . ⊗ 𝜀 𝑖𝑘 ,

𝐹𝑖1 ...𝑖𝑘 ∈ R,

(2.30)

𝑖1 ,...,𝑖𝑘 =1

and the set {𝜀 𝑖1 ⊗ . . . ⊗ 𝜀 𝑖𝑘 }𝑖1 ,...,𝑖𝑘 ∈ {1,2,3} is an orthonormal basis of the linear space of all tensors of rank 𝑘. The scalar product F · G of two tensors of rank 𝑘 is defined by 3 ∑︁

F·G=

(2.31)

𝐹𝑖1 ...𝑖𝑘 𝐺 𝑖1 ...𝑖𝑘 ,

𝑖1 ,...,𝑖𝑘 =1

and the Euclidean norm is |F| = (F · F) 1/2 . Í3

(2.32) Í3

If F = 𝑖1 ,...,𝑖𝑘 =1 𝐹𝑖1 ...𝑖𝑘 ⊗ ... ⊗ and G = 𝑖1 ,...,𝑖𝑙 𝐺 𝑖1 ...𝑖𝑘 ⊗ . . . ⊗ 𝜀 𝑖𝑙 are tensors of rank 𝑘 and 𝑙, respectively, then F ⊗ G is the tensor of rank 𝑘 + 𝑙 defined by F⊗G=

𝜀 𝑖1

3 ∑︁

3 ∑︁

𝜀 𝑖𝑘

𝜀 𝑖1

F𝑖1 ...𝑖𝑘 G 𝑗1 ... 𝑗𝑙 𝜀 𝑖1 ⊗ . . . ⊗ 𝜀 𝑖𝑘 ⊗ 𝜀 𝑗1 ⊗ . . . ⊗ 𝜀 𝑗𝑙 .

(2.33)

𝑖1 ,...,𝑖𝑘 =1 𝑗1 ,..., 𝑗𝑙 =1

A tensor of rank two, f=

3 ∑︁

𝐹𝑖𝑘 𝜀 𝑖 ⊗ 𝜀 𝑘 ,

(2.34)

𝑖,𝑘=1

can be viewed as a linear operator on vectors (tensors of rank one) 𝑔 = the sense of 3 ∑︁ f𝑔 = 𝐹𝑖𝑘 𝐺 𝑘 𝜀 𝑖 .

Í3

𝑖=1 𝐺 𝑖 𝜀

𝑖

in

(2.35)

𝑖,𝑘=1

Interpreting a tensor of rank four as a linear operator on tensors of rank two, we define 3 ∑︁ Fg = 𝐹𝑖 𝑗 𝑘𝑙 𝐺 𝑘𝑙 𝜀 𝑖 ⊗ 𝜀 𝑗 , (2.36) 𝑖, 𝑗,𝑘,𝑙=1

where

36

2 Basic Settings

F=

3 ∑︁

𝐹𝑖 𝑗 𝑘𝑙 𝜀 𝑖 ⊗ 𝜀 𝑗 ⊗ 𝜀 𝑘 ⊗ 𝜀 𝑙

(2.37)

𝑖, 𝑗,𝑘,𝑙=1

is a tensor of rank four (also called rank-4 tensor) and 3 ∑︁

g=

𝐺 𝑘𝑙 𝜀 𝑘 ⊗ 𝜀 𝑙

(2.38)

𝑘,𝑙=1

is a tensor of rank two (i.e., rank-2 tensor). As usual, we define the product of two tensors of rank two, f = Í 𝐹𝑖𝑘 𝜀 𝑖 ⊗ 𝜀 𝑘 and g = 3𝑘, 𝑗=1 𝐺 𝑘 𝑗 𝜀 𝑘 ⊗ 𝜀 𝑗 , by 3 ∑︁

fg =

𝐹𝑖𝑘 𝐺 𝑘 𝑗 𝜀 𝑖 ⊗ 𝜀 𝑗 .

Í3 𝑖,𝑘=1

(2.39)

𝑖, 𝑗,𝑘=1

Furthermore, the product of two rank-4 tensors F=

3 ∑︁

𝐹𝑖 𝑗 𝑘𝑙 𝜀 𝑖 ⊗ 𝜀 𝑗 ⊗ 𝜀 𝑚 ⊗ 𝜀 𝑛

(2.40)

𝐺 𝑚𝑛𝑘𝑙 𝜀 𝑚 ⊗ 𝜀 𝑛 ⊗ 𝜀 𝑘 ⊗ 𝜀 𝑙

(2.41)

𝐹𝑖 𝑗𝑚𝑛 𝐺 𝑚𝑛𝑘𝑙 𝜀 𝑖 ⊗ 𝜀 𝑗 ⊗ 𝜀 𝑘 ⊗ 𝜀 𝑙 .

(2.42)

𝑖, 𝑗,𝑚,𝑛=1

and G=

3 ∑︁ 𝑚,𝑛,𝑘,𝑙=1

is analogously defined by FG=

3 ∑︁ 𝑖, 𝑗,𝑘,𝑙,𝑚,𝑛=1

If G is a set of points in R3 , 𝜕G denotes its boundary. The set G = G ∪ 𝜕G is called the closure of G. The complement of G in R3 is denoted by G 𝑐 . A set G ⊂ R3 is called a region if and only if it is open and connected. An open set A ⊂ R3 is said to be totally contained in the open set G ⊂ R3 (in brief, A ⋐ G), if A ⊂ G and dist(A, 𝜕G) > 0. An open set A ⊂ R3 is said to be compactly contained in the open set G ⊂ R3 , if A is bounded and A ⋐ G. By a scalar, vector, or tensor function (field) on a region G ⊂ R3 , we mean a function that assigns to each point of G, a scalar, vectorial, or tensorial function value, respectively. Unless otherwise specified, all fields are assumed to be realvalued throughout this book. It will be of advantage to use the following general scheme of notations:

2.2 Euclidean Differential Calculus

capital letters F, G lower-case letters f , g boldface lower-case letters f, g boldface capital letters F, G

37

: : : :

scalar functions, vector fields, tensor fields of second rank, tensor fields of fourth rank.

The restriction of a scalar-valued function 𝐹, a vector-valued function 𝑓 , or a tensorvalued function f to a subset 𝑀 of its domain is denoted by 𝐹 |𝑀, 𝑓 |𝑀, or f|𝑀, respectively. For a set 𝑆 of functions, we set 𝑆|𝑀 = {𝐹 |𝑀 𝐹 ∈ 𝑆}.

2.2 Euclidean Differential Calculus Let G ⊂ R3 be a region. Suppose that 𝐹 : G → R is differentiable. ∇𝐹 : 𝑥 ↦→ (∇𝐹)(𝑥), 𝑥 ∈ G, denotes the gradient of 𝐹 on G. The partial derivatives of 𝐹 at 𝑥 ∈ G, briefly written 𝐹|𝑖 , 𝑖 ∈ {1, 2, 3}, are given by 𝐹|𝑖 (𝑥) =

𝜕𝐹 (𝑥) = (∇𝐹) (𝑥) · 𝜀 𝑖 = ((∇𝐹) (𝑥)) 𝑖 . 𝜕𝑥𝑖

(2.43)

𝐿𝐹 : 𝑥 ↦→ 𝐿𝐹 (𝑥) = 𝑥 ∧ (∇𝐹)(𝑥), 𝑥 ∈ G, is called the curl gradient of 𝐹 on G. We say that the scalar function 𝐹 : G → R, the vector function 𝑓 : G → R3 , and the tensor function f : G → R3 ⊗ R3 , respectively, is of class C (1) on G, c (1) on G, and c (1) on G, if 𝐹, 𝑓 , f, respectively, is differentiable at every point of G and ∇𝐹, ∇ 𝑓 , ∇f, respectively, is continuous on G. The gradient of ∇𝐹, ∇ 𝑓 , ∇f is denoted by ∇ (2) 𝐹, ∇ (2) 𝑓 , ∇ (2) f. Continuing in this manner, we say that 𝐹, 𝑓 , f, respectively, is of class C (𝑛) , c (𝑛) , c (𝑛) on G, 𝑛 ≥ 1 (briefly, 𝐹 ∈ C (𝑛) (G), 𝑓 ∈ c (𝑛) (G), f ∈ c (𝑛) (G)) if it is of class C (𝑛−1) , c (𝑛−1) , c (𝑛−1) and its (𝑛 − 1)st gradient ∇ (𝑛−1) 𝐹, ∇ (𝑛−1) 𝑓 ,∇ (𝑛−1) f, respectively, is continuously differentiable (note that we usually write C, c, c instead of C (0) , c (0) , c (0) , respectively). A function 𝐹 : G → R is said to be Lipschitz-continuous in G if there exists a (Lipschitz) constant 𝐶𝐹 > 0 such that |𝐹 (𝑥) − 𝐹 (𝑦)| ≤ 𝐶𝐹 |𝑥 − 𝑦| holds for all 𝑥, 𝑦 ∈ G. The class of all Lipschitz-continuous functions in G is denoted by Lip(G). Clearly, C (1) (G) ⊂ Lip(G). Obviously, the gradient of a differentiable scalar field is a vector field, while the gradient of a differentiable vector field is a tensor field, etc. We say that 𝐹 is of class C (𝑛) on G, G = G ∪ 𝜕G (briefly, 𝐹 ∈ C (𝑛) (G)), if 𝐹 is of class C (𝑛) on G and, for each 𝑘 ∈ {0, . . . , 𝑛}, ∇ (𝑘 ) 𝐹 has a continuous extension to G (in this case, we also

38

2 Basic Settings

write ∇ (𝑛) 𝐹 for the extended function). Analogous definitions can be given for the vectorial and tensorial cases. Let 𝑢 : G → R3 be a vector field, and suppose that 𝑢 is differentiable at a point 𝑥 ∈ G. The partial derivatives of 𝑢 at 𝑥 ∈ G are given by 𝑢 𝑖 | 𝑗 (𝑥) =

𝜕𝑢 𝑖 (𝑥) = 𝜀 𝑖 · (∇𝑢)(𝑥)𝜀 𝑗 . 𝜕𝑥 𝑗

(2.44)

Then, the divergence of 𝑢 at 𝑥 ∈ G is the scalar value ∇ · 𝑢(𝑥) = div 𝑢(𝑥) = tr(∇𝑢(𝑥)).

(2.45)

Thus we have the identity ∇ · 𝑢(𝑥) = div 𝑢(𝑥) =

3 ∑︁

𝑢 𝑖 |𝑖 (𝑥).

(2.46)

𝑖=1

The curl of 𝑢 at 𝑥 ∈ G, denoted by 𝐿 · 𝑢(𝑥) = curl 𝑢(𝑥), is the unique vector with the property   (∇𝑢) (𝑥) − (∇𝑢) (𝑥) 𝑇 𝑎 = (curl 𝑢(𝑥)) ∧ 𝑎 = (𝐿 · 𝑢(𝑥)) ∧ 𝑎

(2.47)

for every 𝑎 ∈ R3 . In components, we have (𝐿 · 𝑢(𝑥)) · 𝜀 𝑖 = curl 𝑢(𝑥) · 𝜀 𝑖 =

3 ∑︁ 3 ∑︁

𝜀 𝑖 𝑗 𝑘 𝑢 𝑘 | 𝑗 (𝑥).

(2.48)

𝑗=1 𝑘=1

We write ( b ∇𝑢)(𝑥) for the symmetric gradient of 𝑢 given by (b ∇𝑢)(𝑥) = sym (∇𝑢)(𝑥) =

 1 (∇𝑢)(𝑥) + (∇𝑢)(𝑥) 𝑇 . 2

(2.49)

Let f : G → R3 ⊗ R3 be a tensor field of second order, and suppose that f is differentiable at 𝑥 ∈ G. The partial derivatives of f at 𝑥 ∈ G are given by f𝑖 𝑗 | 𝑘 (𝑥) =

  𝜕f𝑖 𝑗 (𝑥) = 𝜀 𝑖 · (∇f) (𝑥)𝜀 𝑘 𝜀 𝑗 , 𝜕𝑥 𝑘

(2.50)

Then the tensor field f 𝑇 : 𝑥 ↦→ (f (𝑥)) 𝑇 , 𝑥 ∈ Γ, is also differentiable at 𝑥 ∈ G. The divergence of f at 𝑥, written by ∇ · f (𝑥) = div f (𝑥), is the unique vector with the property

2.2 Euclidean Differential Calculus

39

(∇ · f (𝑥)) · 𝑎 = div f (𝑥) · 𝑎     = div f 𝑇 (𝑥)𝑎 = ∇ · f 𝑇 (𝑥)𝑎

(2.51)

for every (fixed) vector 𝑎 ∈ R3 . In the same manner, we define the curl of f at 𝑥, written by 𝐿 · f (𝑥) = curl f (𝑥), to be the unique tensor with the property     (𝐿 · f (𝑥)) 𝑎 = curl f (𝑥)𝑎 = curl f 𝑇 (𝑥)𝑎 = 𝐿 · f 𝑇 (𝑥)𝑎 (2.52) for every (fixed) vector 𝑎 ∈ R3 . Clearly, (∇ · f (𝑥)) 𝑖 = div f (𝑥) · 𝜀 𝑖 =

3 ∑︁

f𝑖 𝑗 | 𝑗 (𝑥),

(2.53)

𝑗=1 3 ∑︁ 3 ∑︁
𝜀 𝑖 · 𝐿 · f (𝑥)𝜀 𝑗 = 𝜀 𝑖 · curl f (𝑥)𝜀 𝑗 = 𝜀 𝑖 𝑝𝑞 f 𝑗𝑞 | 𝑝 (𝑥).

(2.54)

𝑝=1 𝑞=1

Let 𝐹 : G → R be a differentiable scalar field, and suppose that ∇𝐹 is differentiable at 𝑥 ∈ G. Then we introduce the Laplace operator (Laplacian) of 𝐹 at 𝑥 ∈ G by Δ𝐹 (𝑥) = div ((∇𝐹)(𝑥)) = ∇ · ((∇𝐹) (𝑥)) .

(2.55)

Analogously, we define the Laplacian of a vector field 𝑓 : G → R3 (with ∇ 𝑓 being differentiable at 𝑥 ∈ G) by Δ 𝑓 (𝑥) = div ((∇ 𝑓 ) (𝑥)) = ∇ · ((∇ 𝑓 ) (𝑥)) .

(2.56)

Clearly, for sufficiently often differentiable 𝐹, 𝑓 , Δ𝐹 (𝑥) =

3 ∑︁

𝐹|𝑖 |𝑖 (𝑥),

(2.57)

𝑓𝑖 | 𝑗 | 𝑗 (𝑥).

(2.58)

𝑖=1

Δ 𝑓 (𝑥) · 𝜀 𝑖 =

3 ∑︁ 𝑗=1

Finally, the Laplacian Δf (𝑥) of a sufficiently smooth tensor field f is the unique tensor (of second order) with the property (Δf) (𝑥)𝑎 = Δ (f (𝑥)𝑎) for every fixed 𝑎 ∈ R3 . In components,

(2.59)

40

2 Basic Settings

𝜀 𝑖 · (Δf) (𝑥)𝜀 𝑗 =

3 ∑︁

f𝑖 𝑗 |𝑞 |𝑞 (𝑥).

(2.60)

𝑞=1

For the convenience of the reader, a list of the operators in Euclidean space R3 is included (see Table 2.1). Table 2.1 Differential operators in Euclidean space R3 . Symbol ∇ 𝐿 ∇· 𝐿· Δ

Differential Operator gradient curl gradient divergence curl Laplace operator

Of future interest are the following identities 𝐿 · ∇𝐹 = curl ∇𝐹 = 0, ∇ · (𝐿 · 𝑢) = div curl 𝑢 = 0, 𝐿 · (𝐿 · 𝑢) = curl curl 𝑢 = ∇ div 𝑢 − Δ𝑢 = ∇ (∇ · 𝑢) − Δ𝑢, 𝐿 · (∇𝑢) = curl ∇𝑢 = 0,   𝐿 · ∇𝑢𝑇 = curl (∇𝑢𝑇 ) = ∇curl 𝑢 = ∇ (𝐿 · 𝑢) ,   ∇ · (𝐿 · f) = div curl f = curl div f 𝑇 = 𝐿 · ∇ · f 𝑇 ,   (𝐿 · (𝐿 · f)) 𝑇 = (curl curl f) 𝑇 = curl curl f 𝑇 = 𝐿 · 𝐿 · f 𝑇 ,   ∇ · f 𝑇 𝑢 = div (f 𝑇 𝑢) = 𝑢 · div f + f · ∇𝑢 = 𝑢 · (∇ · f) + f · ∇𝑢,

(2.61) (2.62) (2.63) (2.64) (2.65) (2.66) (2.67) (2.68)

provided that 𝐹 is a scalar field, 𝑢 is a vector field, and f is a tensor field, sufficiently often differentiable on G. Furthermore, if ∇𝑢 = −∇𝑢𝑇 , it follows that ∇∇𝑢 = 0.

2.3 Euclidean Integral Calculus

We start with some basics of spherical vector analysis.

2.3 Euclidean Integral Calculus

41

Regular Regions. G ⊂ R3 is called a regular region in three-dimensional Euclidean space R3 , if it is an open and connected set G ⊂ R3 , for which (i) its boundary 𝜕G constitutes an orientable, piecewise smooth Lipschitzian surface, (ii) G uniquely divides R3 into the “inner space” G and the “outer space” G 𝑐 = R3 \G, G = G ∪ 𝜕G. Integral Theorems. The Gauss integral theorem (from 1813) and the related Green’s formulas (cf. G. Green (1838)) are among the basic tools of potential theory. They are also indispensable for a variety of problems in geosciences. Theorem 2.1 (Gauss’s Integral Theorem) Let G be a regular region. Let 𝐹 : G → R be a scalar field, 𝑓 : G → R3 a vector field, that is continuous on G and differentiable in G, respectively. Then ∫ ∫ 𝐹 (𝑦)𝜈(𝑦) 𝑑𝜔(𝑦) = ∇𝐹 (𝑦) 𝑑𝑦, (2.69) 𝜕G G ∫ ∫ 𝑓 (𝑦) · 𝜈(𝑦) 𝑑𝜔(𝑦) = ∇ · 𝑓 (𝑦) 𝑑𝑦, (2.70) 𝜕G G ∫ ∫ 𝜈(𝑦) ∧ 𝑓 (𝑦) 𝑑𝜔(𝑦) = ∇ ∧ 𝑓 (𝑦) 𝑑𝑦, (2.71) 𝜕G

G

provided that the integrand on the right hand side is Lebesgue-integrable on G. The vector field 𝜈 : 𝜕G → R3 is the (unit) normal field pointing into the exterior of G (𝑑𝜔 is the surface element). The identities (2.70) and (2.71) are valid for all vector fields, whatever their physical meaning. Of special interest is the case (2.70) in which 𝑓 may be understood to be the velocity vector of an incompressible fluid. Inside the surface 𝜕G there may be sources in which the fluid is generated or sinks in which the fluid is annihilated. The divergence ∇ · 𝑓 measures the strength of the sources and sinks. The volume integral ∫ ∇ · 𝑓 (𝑦) 𝑑𝑦 is the total amount of the fluid generated in unit time. The surface G ∫ integral 𝜕G 𝑓 (𝑦) · 𝜈(𝑦) 𝑑𝜔(𝑦) is the total amount of fluid flowing in unit time across the surface 𝜕G. Therefore, the Gauss formula expresses a balance equation, namely the evident fact that both integrals in (2.70) are equal.

Gravitational Interpretation. In the case where 𝑓 is the vector of the gravitational force, i.e., we especially choose instead of 𝑓 the field 𝑣 = ∇𝑉, the intuitive interpretation of the Gauss integral theorem is not so obvious, but the analogy to the balance equation of fluid flow is often helpful. In gravitation we can take advantage of the Poisson equation ∇ · 𝑣 = Δ𝑉 = −𝐶 𝜌. (2.72)

42

2 Basic Settings

This equation (cf. W.A. Heiskanen, H. Moritz (1967)) can be interpreted to mean that the masses are the sources of the gravitational field; the strength or the sources, ∇ · 𝑣, is proportional to the mass density 𝜌. The right-hand side of (2.72) is called the flux of force, in our case gravitational flux, also in analogy to the fluid flow. Next we come to the interior Green’s formulas for regular regions G ⊂ R3 . Suppose that 𝑓 = ∇𝐹, where 𝐹 ∈ C (1) (G) ∩ C (2) (G), i.e., 𝐹 : G → R is continuously differentiable in G and 𝐹 |G is twice continuously differentiable in G. Let Δ𝐹 be Lebesgue-integrable in G. Then we obtain from the Gauss theorem (2.70) ∫ ∫ 𝜕𝐹 (𝑦) 𝑑𝜔(𝑦) = Δ𝐹 (𝑦) 𝑑𝑦, (2.73) 𝜕G 𝜕𝜈 G where, as always, normal field 𝜈.

𝜕 𝜕𝜈

= 𝜈 · ∇ denotes the derivative in the direction of the outer (unit)

Under the special choice 𝑓 = 𝐹 ∇𝐺 the Gauss Theorem yields Theorem 2.2 (Interior First Green’s Theorem) Suppose that G ⊂ R3 is a regular region. For 𝐹 ∈ C (1) (G), 𝐺 ∈ C (1) (G) ∩ C (2) (G) with Δ𝐺 Lebesgue-integrable on G we have ∫ ∫ 𝜕𝐺 (𝐹 (𝑦)Δ𝐺 (𝑦) + ∇𝐹 (𝑦) · ∇𝐺 (𝑦)) 𝑑𝑦 = 𝐹 (𝑦) (𝑦) 𝑑𝜔(𝑦). (2.74) 𝜕𝜈 G 𝜕G Taking 𝑓 = 𝐹 ∇𝐺 − 𝐺 ∇𝐹 we finally obtain Theorem 2.3 (Interior Second Green’s Theorem) Suppose that G ∈ R3 is a regular region. For 𝐹, 𝐺 ∈ C (1) (G) ∩ C (2) (G) with Δ𝐹, Δ𝐺 Lebesgue-integrable on G we have ∫ (𝐺 (𝑦)Δ𝐹 (𝑦) − 𝐹 (𝑦)Δ𝐺 (𝑦)) 𝑑𝑦 (2.75) G  ∫  𝜕𝐹 𝜕𝐺 = 𝐺 (𝑦) (𝑦) − 𝐹 (𝑦) (𝑦) 𝑑𝜔(𝑦). 𝜕𝜈 𝜕𝜈 𝜕G

2.4 Spherical Nomenclature

43

2.4 Spherical Nomenclature

On the one hand, the sphere occurs as the boundary surface of a subdomain, i.e., a ball or its exterior, in R3 . On the other hand, intrinsically seen, the sphere is not representing a boundary surface, it is rather regarded as the underlying domain on which a certain problem in practice is formulated. Examples for this are the spherical Navier-Stokes equations and shallow water equations in meteorology and ocean modeling (see, e.g., R. Comblen et al. (2009), M.J. Fengler, W. Freeden (2005), M. Ganesh et al. (2009), A.A. Il’in (1991), J. Pedlovsky (1979)), but also other spherical differential equations occur in Geodesy and geomagnetism (see, e.g., G.E. Backus et al. (1996), M. Bayer et al. (2001), W. Freeden, M. Schreiner (2006b), T. Fehlinger et al. (2007, 2009), W. Freeden, C. Gerhards (2010), W. Freeden, H. Nutz (2018), W. Freeden, M. Schreiner (2020a), W. Freeden et al. (2020), C. Gerhards (2011, 2015, 2018), B. Hofmann-Wellenhof, H. Moritz (2005)) and vortex dynamics (see, e.g., R. Kidambi, K.P. Newton (2000, 1987), J. Pedlovsky (1979), L.M. Polvani, D.G. Dritschel (1993), C. Gerhards (2018)), more precisely, those based on the Beltrami operator (the spherical counterpart to the Laplace operator, which also is of interest on subdomains of the sphere, leading to potential theoretic concepts analogously to those of the Euclidean case. Subdomains appear naturally in geosciences, e.g., due to only regionally available data or coastal/continental boundaries. The sphere in R3 with radius 𝑅 around the origin will be denoted by Ω𝑅 :  Ω𝑅 = 𝑥 ∈ R3 | |𝑥| = 𝑅 .

(2.76)

ext We set Ωint 𝑅 for the “inner space” of Ω 𝑅 , while Ω 𝑅 denotes the “outer space” of Ω 𝑅 :

 3 Ωint 𝑅 = 𝑥 ∈ R | |𝑥| < 𝑅 ,

(2.77)

Ωext 𝑅

(2.78)

3

= {𝑥 ∈ R | |𝑥| > 𝑅}.

For the sphere, resp. the inner and outer spaces, centered at 𝑥 ∈ R3 , we write  Ω𝑅 (𝑥) = 𝑦 ∈ R3 | |𝑦 − 𝑥| = 𝑅 , (2.79)  int 3 Ω𝑅 (𝑥) = 𝑦 ∈ R | |𝑦 − 𝑥| < 𝑅 , (2.80)  ext 3 Ω𝑅 (𝑥) = 𝑦 ∈ R | |𝑦 − 𝑥| > 𝑅 . (2.81) It is well known that the total surface ∥Ω𝑅 ∥ of Ω𝑅 is equal to 4𝜋𝑅 2 : ∫ ∥Ω𝑅 ∥ = 𝑑𝜔(𝜉) = 4𝜋𝑅 2 Ω𝑅

(𝑑𝜔 denotes the surface element).

(2.82)

44

2 Basic Settings

Throughout this book, the unit sphere Ω1 in R3 is denoted by Ω. We set Ωint for the “inner space” of Ω, while Ωext denotes the “outer space” of Ω. We may represent the points 𝑥 ∈ R3 , 𝑥 = 𝑟𝜉, 𝜉 ∈ Ω, in polar coordinates as follows (see Fig. 2.1): 𝑥 = 𝑟𝜉, 𝑟 = |𝑥|, √ 𝜉 = 𝜉 (𝑡, 𝜑) = 𝑡𝜀 3 + 1 − 𝑡 2 (cos 𝜑𝜀 1 + sin 𝜑𝜀 2 ), −1 ≤ 𝑡 ≤ 1 , 0 ≤ 𝜑 < 2𝜋 , 𝑡 = cos 𝜗 ,

(2.83)

(𝜗 ∈ [0, 𝜋]: (co-)latitude, 𝜑: longitude, 𝑡: polar distance), i.e., 𝜉 = (sin 𝜗 cos 𝜑, sin 𝜗 sin 𝜑, cos 𝜗) 𝑇 .

(2.84)

𝜀3

𝑥

𝜉

𝜗

𝑥3 ·

𝜀2

𝜑 𝑥1 ·

· 𝑥2 𝜀1

Fig. 2.1: Polar coordinates in three-dimensional Euclidean space R3 .

More explicitly, 𝜉 = sin 𝜗 𝜉 cos 𝜑 𝜉 , sin 𝜗 𝜉 sin 𝜑 𝜉 , cos 𝜗 𝜉


𝑇

.

The scalar product between two unit vectors 𝜉 and 𝜂 reads as follows:

(2.85)

2.4 Spherical Nomenclature

45

𝜂 · 𝜉 = sin 𝜗 𝜂 cos 𝜑 𝜂 sin 𝜗 𝜉 cos 𝜑 𝜉

(2.86)

+ sin 𝜗 𝜂 sin 𝜑 𝜂 sin 𝜗 𝜉 sin 𝜑 𝜉 + cos 𝜗 𝜂 cos 𝜗 𝜉 = (cos 𝜑 𝜂 cos 𝜑 𝜉 + sin 𝜑 𝜂 sin 𝜑 𝜉 ) sin 𝜗 𝜂 sin 𝜗 𝜉 + cos 𝜗 𝜂 cos 𝜗 𝜉 = cos(𝜑 𝜂 − 𝜑 𝜉 ) sin 𝜗 𝜂 sin 𝜗 𝜉 + cos 𝜗 𝜂 cos 𝜗 𝜉 √︃ √︃ = cos(𝜑 𝜂 − 𝜑 𝜉 ) 1 − 𝑡 2𝜂 1 − 𝑡 2𝜉 + 𝑡 𝜂 𝑡 𝜉 .

Function Spaces. The set of scalar functions 𝐹 : Ω → R which are measurable and for which  𝑝1

∫ |𝐹 (𝜉)|

∥𝐹 ∥ L 𝑝 (Ω) =

𝑝

𝑑𝜔(𝜉)

< ∞,

1 ≤ 𝑝 < ∞,

(2.87)

Ω

is known as L 𝑝 (Ω). Clearly, L 𝑝 (Ω) ⊂ L𝑞 (Ω) for 1 ≤ 𝑞 < 𝑝. A function 𝐹 : Ω → R possessing 𝑘 continuous derivatives on the unit sphere Ω is said to be of class C (𝑘 ) (Ω), (0 ≤ 𝑘 ≤ ∞). C(Ω) (= C (0) (Ω)) is the class of continuous scalar-valued functions on Ω. C(Ω) is a complete normed space endowed with ∥𝐹 ∥ C(Ω) = sup |𝐹 (𝜉)|.

(2.88)

𝜉 ∈Ω

By 𝜇(𝐹; 𝛿), we denote the modulus of continuity of the function 𝐹 ∈ C(Ω) 𝜇(𝐹; 𝛿) =

max

𝜉 ,𝜁 ∈Ω;1− 𝜉 ·𝜁 ≤ 𝛿

|𝐹 (𝜉) − 𝐹 (𝜁)| ,

0 < 𝛿 < 2.

(2.89)

A function 𝐹 : Ω → R is said to be Lipschitz-continuous if there exists a (Lipschitz) constant 𝐶𝐹 > 0 such that the inequality √︁ √ |𝐹 (𝜉) − 𝐹 (𝜂)| ≤ 𝐶𝐹 |𝜉 − 𝜂| = 2𝐶𝐹 1 − 𝜉 · 𝜂 (2.90) holds for all 𝜉, 𝜂 ∈ Ω. The class of all Lipschitz-continuous functions on Ω is denoted by Lip(Ω). Clearly, C (1) (Ω) ⊂ Lip(Ω). L2 (Ω) is a Hilbert space with respect to the inner product (·, ·)L2 (Ω) defined by ∫ (𝐹, 𝐺)L2 (Ω) = 𝐹 (𝜉)𝐺 (𝜉) 𝑑𝜔(𝜉), 𝐹, 𝐺 ∈ L2 (Ω). (2.91) Ω

In connection with (·, ·)L2 (Ω) , C(Ω) is a pre-Hilbert space. For each 𝐹 ∈ C(Ω) we have the norm estimate √ ∥𝐹 ∥ L2 (Ω) ≤ 4𝜋 ∥𝐹 ∥ C(Ω) . (2.92) L2 (Ω) is the completion of C(Ω) with respect to the norm ∥ · ∥ L2 (Ω) , i.e.,

46

2 Basic Settings

L2 (Ω) = C(Ω)

∥ · ∥ L2 (Ω)

(2.93)

.

Any function of the form 𝐺 𝜉 : Ω → R, 𝜂 ↦→ 𝐺 𝜉 (𝜂) = 𝐺 (𝜉 · 𝜂),

𝜂 ∈ Ω,

(2.94)

is called a 𝜉-zonal function on Ω (or 𝜉-axial radial basis function). Zonal functions are constant on the sets {𝜂 ∈ Ω| 𝜉 · 𝜂 = ℎ}, ℎ ∈ [−1, 1]. The set of all 𝜉-zonal functions is isomorphic to the set of functions 𝐺 : [−1, 1] → R. This allows us to interpret C[−1, 1] and L 𝑝 [−1, 1] (with norms defined correspondingly) as subspaces of C(Ω) and L 𝑝 (Ω). Obviously,

and we define

∥𝐺 ∥ C[−1,1] = ∥𝐺 (𝜀 3 ·)∥ C(Ω) ,

(2.95)

∥𝐺 ∥ L 𝑝 [−1,1] = ∥𝐺 (𝜀 3 ·)∥ L 𝑝 (Ω) ∫  1/ 𝑝 = |𝐺 (𝜂 · 𝜀 3 )| 𝑝 𝑑𝜔(𝜂)  Ω∫ 1  1/ 𝑝 = 2𝜋 |𝐺 (𝑡)| 𝑝 𝑑𝑡 .

(2.96)

−1

Analogously, we define the inner product in L2 [−1, 1] by ∫ (𝐹, 𝐺) L2 [−1,1] = 2𝜋

1

𝐹 (𝑡)𝐺 (𝑡) 𝑑𝑡,

(2.97)

−1

𝐹, 𝐺 ∈ L2 [−1, 1]. Next, we give some preliminaries for the study of vector fields defined on the unit sphere Ω. Using the canonical orthonormal basis {𝜀 1 , 𝜀 2 , 𝜀 3 } of R3 , we may write any vector field 𝑓 : Ω → R3 in the form 𝑓 (𝜉) =

3 ∑︁

𝐹𝑖 (𝜉)𝜀 𝑖 ,

𝜉 ∈ Ω,

(2.98)

𝑖=1

where the component functions 𝐹𝑖 are given by 𝐹𝑖 (𝜉) = 𝑓 (𝜉) · 𝜀 𝑖 , 𝜉 ∈ Ω. l2 (Ω) denotes the space consisting of all square-integrable vector fields on Ω. In connection with the inner product ∫ ( 𝑓 , 𝑔)l2 (Ω) = 𝑓 (𝜉) · 𝑔(𝜉) 𝑑𝜔(𝜉), 𝑓 , 𝑔 ∈ l2 (Ω), (2.99) Ω

l2 (Ω) is a Hilbert space. The space c ( 𝑝) (Ω), 0 ≤ 𝑝 ≤ ∞, consists of all 𝑝-times continuously differentiable vector fields on Ω. For brevity, we usually write c(Ω) =

2.5 Spherical Differential Calculus

47

c (0) (Ω). The space c(Ω) is complete with respect to the norm ∥ 𝑓 ∥ c(Ω) = sup | 𝑓 (𝜉)|,

𝑓 ∈ c(Ω).

(2.100)

𝜉 ∈Ω

Furthermore, c(Ω)

∥ · ∥ l2 (Ω)

= l2 (Ω).

In analogy to (2.92), we have for all 𝑓 ∈ c(Ω) the norm estimate √ ∥ 𝑓 ∥ l2 (Ω) ≤ 4𝜋 ∥ 𝑓 ∥ c(Ω) .

(2.101)

(2.102)

The generalization of the preceding settings to tensor fields of rank two is straightforward. A tensor field is said to be of class c (𝑘 ) (Ω), 0 ≤ 𝑘 ≤ ∞, if its component functions with respect to the basis {𝜀 𝑖 ⊗ 𝜀 𝑘 }𝑖,𝑘=1,2,3 are in C (𝑘 ) (Ω). The space c(Ω)(= c (0) (Ω)) equipped with the norm ∥ · ∥ c(Ω) defined by ∥f ∥ c(Ω) = sup |f (𝜉)|,

f ∈ c(Ω),

(2.103)

𝜉 ∈Ω

is a Banach space. By l2 (Ω), we denote the Hilbert space of square-integrable tensor fields f : Ω → R3 ⊗ R3 with inner product ∫ (f, g)l2 (Ω) = f (𝜉) · g(𝜉) 𝑑𝜔(𝜉), f, g ∈ l2 (Ω), (2.104) Ω

and associated norm ∥ · ∥ l2 (Ω) . The space l2 (Ω) is the completion of c(Ω) with respect to the norm ∥ · ∥ l2 (Ω) .

2.5 Spherical Differential Calculus

In order to introduce a system of triads on spheres, we define the vector function

by

Φ : [0, ∞) × [0, 2𝜋) × [−1, 1] → R3

(2.105)

√ 2 © 𝑟 √1 − 𝑡 cos 𝜑 ª ­ Φ(𝑟, 𝜑, 𝑡) = ­ 𝑟 1 − 𝑡 2 sin 𝜑 ®® . 𝑟𝑡 « ¬

(2.106)

Setting 𝑟 = 1 we already know that a local coordinate system is obtainable on the unit sphere. In other words, instead of denoting any element of Ω by its vec-

48

2 Basic Settings

torial representation 𝜉, we may also use its coordinates (𝜑, 𝑡) in accordance with (2.83). Calculating the derivatives of Φ and setting 𝑟 = 1, the corresponding set of orthonormal unit vectors in the directions 𝑟, 𝜑, and 𝑡 is easily determined to be √ 2 © √1 − 𝑡 cos 𝜑 ª 𝑟 ­ 𝜀 (𝜑, 𝑡) = ­ 1 − 𝑡 2 sin 𝜑 ®® , 𝑡 « ¬ − sin 𝜑 © ª 𝜀 𝜑 (𝜑, 𝑡) = ­ cos 𝜑 ® , « 0 ¬ −𝑡 cos 𝜑 © sin 𝜑 ª 𝜀 𝑡 (𝜑, 𝑡) = ­ −𝑡 ®. √ 2 « 1−𝑡 ¬

(2.107)

(2.108)

(2.109)

Obviously, 𝜀 𝑡 (𝜑, 𝑡) = 𝜀𝑟 (𝜑, 𝑡) ∧ 𝜀 𝜑 (𝜑, 𝑡).

(2.110)

The vectors 𝜀 𝜑 and 𝜀 𝑡 mark the tangential directions. Since we associate 𝜉 with its representations using the local coordinates 𝜑 and 𝑡, we identify 𝜀𝑟 (𝜉) with 𝜀𝑟 (𝜑, 𝑡), etc. (cf. Fig. 2.2). From (2.107) to (2.109), we immediately obtain a representation of the Cartesian unit vectors in terms of the spherical ones: √︁ 𝜀 1 = 1 − 𝑡 2 cos 𝜑𝜀𝑟 (𝜑, 𝑡) − sin 𝜑𝜀 𝜑 (𝜑, 𝑡) − 𝑡 cos 𝜑𝜀 𝑡 (𝜑, 𝑡), (2.111) √︁ 2 𝑟 𝜑 𝑡 𝜀 = 1 − 𝑡 2 sin 𝜑𝜀 (𝜑, 𝑡) + cos 𝜑𝜀 (𝜑, 𝑡) − 𝑡 sin 𝜑𝜀 (𝜑, 𝑡), (2.112) √︁ 𝜀 3 = 𝑡𝜀𝑟 (𝜑, 𝑡) + 1 − 𝑡 2 𝜀 𝑡 (𝜑, 𝑡). (2.113) The system {𝜀 𝜑 , 𝜀 𝑡 } enables us to formulate a vector differential calculus. Gradient Fields. Gradient fields ∇𝐹 can be decomposed into a radial and a tangential component. More explicitly, the surface gradient ∇∗ contains the tangential derivatives of the gradient ∇ as follows: ∇ = 𝜀𝑟

𝜕 1 + ∇∗ . 𝜕𝑟 𝑟

(2.114)

Letting 𝑥 = 𝑟𝜉, 𝑟 = |𝑥|, 𝜉 ∈ Ω, we find with 𝜂 ∈ Ω ∇ 𝑥 (𝑥 · 𝜂) = 𝜂 = 𝜀𝑟 (𝜉 · 𝜂) + ∇∗𝜉 (𝜉 · 𝜂),

(2.115)

∇∗𝜉 (𝜉 · 𝜂) = 𝜂 − (𝜉 · 𝜂)𝜉.

(2.116)

such that The surface curl gradient 𝐿 ∗ is defined by

2.5 Spherical Differential Calculus

49

𝜀𝑡 ( 𝜉 )

𝜀𝑟 ( 𝜉 )

𝜀𝜑 ( 𝜉 )

𝜀 𝑡 ( 𝜂)

𝜀 𝑟 ( 𝜂) 𝜀 𝜑 ( 𝜂)

Fig. 2.2: The local triad 𝜀𝑟 , 𝜀 𝜑 , 𝜀 𝑡 with respect to two different points 𝜉 and 𝜂 on the unit sphere. 𝐿 ∗ 𝐹 (𝜉) = 𝜉 ∧ ∇∗ 𝐹 (𝜉),

𝜉 ∈ Ω,

(2.117)

𝐹 ∈ C (1) (Ω). According to its definition (2.117), 𝐿 ∗ 𝐹 is a tangential vector field perpendicular to ∇∗ 𝐹, i.e., ∇∗ 𝐹 (𝜉) · 𝐿 ∗ 𝐹 (𝜉) = 0,

𝜉 ∈ Ω.

(2.118)

∇∗ · = div ∗ and 𝐿 ∗ · = curl ∗ , respectively, denote the surface divergence and the surface curl given by 3 ∑︁ ∇∗ · 𝑓 (𝜉) = ∇∗ 𝐹𝑖 (𝜉) · 𝜀 𝑖 (2.119) 𝑖=1

and 𝐿 ∗ · 𝑓 (𝜉) =

3 ∑︁

𝐿 ∗ 𝐹𝑖 (𝜉) · 𝜀 𝑖 .

𝑖=1

Note that the surface curl as defined by (2.120), i.e.,

(2.120)

50

2 Basic Settings

𝜉 ↦→ 𝐿 ∗ · 𝑓 (𝜉) = curl ∗ 𝑓 (𝜉) = div ∗ ( 𝑓 (𝜉) ∧ 𝜉) = ∇∗ · ( 𝑓 (𝜉) ∧ 𝜉),

𝜉 ∈ Ω, (2.121)

represents a scalar-valued function on the unit sphere Ω in R3 . The aforementioned relations can be understood from the well-known role of the Beltrami operator Δ∗ in the representation of the Laplace operator Δ: 

𝜕 Δ= 𝜕𝑟

2 +

2 𝜕 1 + Δ∗ . 𝑟 𝜕𝑟 𝑟 2

(2.122)

In spherical coordinates, the operators Δ∗ , 𝐿 ∗ , ∇∗ , respectively, read as follows:  2  𝜕  1 𝜕 2 𝜕 Δ = 1−𝑡 + , 2 𝜕𝑡 𝜕𝑡 1 − 𝑡 𝜕𝜑 √︁ 1 𝜕 𝜕 ∇∗ = 𝜀 𝜑 √ + 𝜀𝑡 1 − 𝑡 2 , 2 𝜕𝜑 𝜕𝑡 1−𝑡 √︁ 𝜕 1 𝜕 𝐿 ∗ = −𝜀 𝜑 1 − 𝑡 2 + 𝜀 𝑡 √ . 2 𝜕𝑡 𝜕𝜑 1−𝑡 ∗

(2.123) (2.124) (2.125)

An easy calculation using (2.107)–(2.109) shows that 1

 𝜕 𝜕𝜑 1 − 𝑡2   𝜕 √︁ √︁ + 1 − 𝑡 2 −𝑡 cos 𝜑𝜀 1 − 𝑡 sin 𝜑𝜀 2 + 1 − 𝑡 2 𝜀 3 , 𝜕𝑡

(2.126)

  𝜕 √︁ 1 − 𝑡 2 sin 𝜑𝜀 1 − cos 𝜑𝜀 2 𝜕𝑡   𝜕 √︁ 1 +√ −𝑡 cos 𝜑𝜀 1 − 𝑡 sin 𝜑𝜀 2 + 1 − 𝑡 2 𝜀 3 . 𝜕𝜑 1 − 𝑡2

(2.127)

∇∗ = √

𝐿∗ =



− sin 𝜑𝜀 1 + cos 𝜑𝜀 2

It should be mentioned that the operators ∇∗ , 𝐿 ∗ , Δ∗ will be always used here in coordinate-free representation, thereby avoiding any singularity at the poles. Since the operators ∇∗ , 𝐿 ∗ and ∇∗ ·, 𝐿 ∗ · are of particular interest throughout this work, we list some of their properties: If 𝜉 ∈ Ω, then

2.5 Spherical Differential Calculus

51

∇∗ · ∇∗ 𝐹 (𝜉) = Δ∗ 𝐹 (𝜉),

(2.128)

𝐿 ∗ · 𝐿 ∗ 𝐹 (𝜉) = Δ∗ 𝐹 (𝜉),

(2.129)

∗

∗

(2.130)

∗

∗

(2.131)

∗

(2.132)

∇ · 𝐿 𝐹 (𝜉) = 0, 𝐿 · ∇ 𝐹 (𝜉) = 0, ∗

∇ 𝐹 (𝜉) · 𝐿 𝐹 (𝜉) = 0, ∇∗ · (𝐹 (𝜉) 𝑓 (𝜉)) = (∇∗ 𝐹 (𝜉)) · 𝑓 (𝜉) + 𝐹 (𝜉)(∇∗ · 𝑓 (𝜉)), ∗

∇ · 𝜉 = 2,

(2.133) (2.134)

𝜉 ∈ Ω. Moreover, we have ∇∗ ∧ (𝐹 (𝜉) 𝑓 (𝜉)) = ∇∗ 𝐹 (𝜉) ∧ 𝑓 (𝜉) + 𝐹 (𝜉)∇∗ ∧ 𝑓 (𝜉), 𝜉 ∈ Ω.

(2.135)

For a given function 𝐹 ∈ C (1) (Ω), the triple 𝐹 (𝜉)𝜉, ∇∗ 𝐹 (𝜉), 𝐿 ∗ 𝐹 (𝜉), 𝜉 ∈ Ω, supplies us with a system of three orthogonal vectors at each point 𝜉 ∈ Ω, provided that 𝐹 (𝜉) ≠ 0 and ∇∗ 𝐹 (𝜉) ≠ 0. Let 𝜂 ∈ Ω be fixed, then it is not difficult to see (cf. (2.115)) that, for 𝜉 ∈ Ω, ∇∗𝜉 (𝜉 · 𝜂) = 𝜂 − (𝜉 · 𝜂)𝜉,

(2.136)

𝐿 ∗𝜉 (𝜉

(2.137)

· 𝜂) = 𝜉 ∧

∇∗𝜉 (𝜉

· 𝜂) = 𝜉 ∧ 𝜂,

and Δ∗𝜉 (𝜉 · 𝜂) = −2(𝜉 · 𝜂).

(2.138)

More generally, if 𝐹 is of class C (1) [−1, 1] and 𝐹 ′ ∈ C[−1, 1] is its (onedimensional) derivative, then ∇∗𝜉 𝐹 (𝜉 · 𝜂) = 𝐹 ′ (𝜉 · 𝜂)(𝜂 − (𝜉 · 𝜂)𝜉),

(2.139)

L∗𝜉 𝐹 (𝜉

(2.140)

′

· 𝜂) = 𝐹 (𝜉 · 𝜂)(𝜉 ∧ 𝜂),

whereas, for 𝐹 ∈ C (2) [−1, 1], Δ∗𝜉 𝐹 (𝜉 · 𝜂) = −2(𝜉 · 𝜂)𝐹 ′ (𝜉 · 𝜂) + (1 − (𝜉 · 𝜂) 2 )𝐹 ′′ (𝜉 · 𝜂).

(2.141)

For the convenience of the reader, a list of spherical differential operators is included (see Table 2.2).

52

2 Basic Settings

Table 2.2 Important spherical differential operators on the sphere (symbols and their abbreviations). Symbol

Differential Operator

Abbreviation

∇∗

surface gradient surface curl gradient surface divergence surface curl divergence Beltrami operator

𝑜 (2) 𝑜 (3) 𝑜 (2) · 𝑜 (3) · 𝑜 (2) · 𝑜 (2) = 𝑜 (3) · 𝑜 (3)

𝐿∗ ∇∗ · 𝐿∗ · Δ∗

2.6 Spherical Integral Calculus

Spherical versions of the theorems of Gauss and Stokes are well-known. They can be found in any textbook on differential geometry (for example, K. Strubecker (1969)). Regular Region. Γ ⊂ Ω is called a regular region on Ω, if it is an open and connected set Γ ⊂ Ω, for which (i) its boundary 𝜕Γ constitutes an orientable, piecewise smooth Lipschitzian curve, (ii) Γ uniquely divides Ω into the “inner space” Γ and the “outer space” Γ 𝑐 = Ω\Γ, Γ = Γ ∪ 𝜕Γ. For more details about regular regions, the reader is referred, e.g., to W. Freeden, C. Gerhards (2013). Integral Theorems. The following Gauss integral theorem and the related Green’s formulas are among the basic tools of vector analysis on the sphere (see, e.g., W. Freeden, C. Gerhards (2013)). They are also indispensable for a variety of problems in geosciences.

2.6 Spherical Integral Calculus

53 𝜉

Γ 𝜉

𝜌 ·

𝜏(𝜉)

Γ𝜌 ( 𝜉 )

𝜕Γ 𝜈( 𝜉 )

Fig. 2.3: Examples for a general regular region Γ (left) and a spherical cap Γ𝜌 (𝜉) with center 𝜉 and radius 𝜌 (right) on the unit sphere Ω.

Theorem 2.4 (Surface Theorems of Gauss and Stokes) Let Γ ⊂ Ω be a regular region. If 𝑓 ∈ c (1) (Γ) is tangential, i.e., 𝜉 · 𝑓 (𝜉) = 0 for 𝜉 ∈ Γ, then ∫ ∫ ∇∗ · 𝑓 (𝜂) 𝑑𝜔(𝜂) = 𝜈(𝜂) · 𝑓 (𝜂) 𝑑𝜎(𝜂), (2.142) Γ 𝜕Γ ∫ ∫ 𝐿 ∗ · 𝑓 (𝜂) 𝑑𝜔(𝜂) = 𝜏(𝜂) · 𝑓 (𝜂) 𝑑𝜎(𝜂), (2.143) Γ

𝜕Γ

where 𝜈 : 𝜕Γ → Ω denotes the unit normal field, 𝜏 : 𝜕Γ → Ω the unit tangential field given by 𝜏(𝜉) = (|𝜈(𝜉) ∧ 𝜉 | −1 )(𝜈(𝜉) ∧ 𝜉) for 𝜉 ∈ Ω (note that we have 𝜕 ∗ ∗ 𝜕𝜈 𝐹 (𝜉) = 𝜈(𝜉) · ∇ 𝐹 (𝜉) = 𝜏(𝜉) · 𝐿 𝐹 (𝜉), 𝜉 ∈ 𝜕Γ); generally, “𝑑𝜔” denotes the surface element in Γ ⊂ Ω and “𝑑𝜎” the line element on 𝜕Γ. The identities of Theorem 2.4 are valid for all vector fields, whatever their physical meaning. Of special interest is the case (2.142) in which 𝑓 may be understood to be the velocity vector of an incompressible fluid on Ω. Inside the surface 𝜕Γ there may be sources in which the fluid is generated or sinks in which the fluid is annihilated. The divergence ∇ · 𝑓 measures the strength of the sources and sinks. The surface ∫ integral Γ ∇∗ · 𝑓 (𝜂) 𝑑𝜔(𝜂) is the total amount of the fluid generated in unit time. ∫ The line integral 𝜕Γ 𝜈(𝜂) · 𝑓 (𝜂) 𝑑𝜎(𝜂) is the total amount of fluid flowing in unit time across the surface 𝜕Γ. Therefore, the Surface Theorem of Gauss expresses a balance equation, namely the evident fact that both integrals in (2.142) are equal. An analogous interpretation holds true for the identity (2.143) involving the surface curl divergence 𝐿 ∗ · 𝑓 .

54

2 Basic Settings

It is important to point out the assumption of 𝑓 being tangential in Theorem 2.4. In comparison to the Euclidean case this causes an additional term in Green’s formulas involving ∇∗ , but it does not affect those for 𝐿 ∗ , which is due to the fact that ∇∗ · 𝜉 = 2 but 𝐿 ∗ · 𝜉 = 0, 𝜉 ∈ Ω. Whenever we refer to Green’s formulas (on the sphere) in this work, we mean the corresponding list of the following formulas. Lemma 2.5 (First Green’s Theorems) Let Γ ⊂ Ω be a regular region. Suppose that 𝐹, 𝐺 are of class C (1) (Γ). Then ∫ ∫ ∗ 𝐺 (𝜂)∇ 𝐹 (𝜂) 𝑑𝜔(𝜂) + 𝐹 (𝜂)∇∗ 𝐺 (𝜂) 𝑑𝜔(𝜂) (2.144) Γ Γ ∫ ∫ = 𝜈(𝜂) (𝐹 (𝜂)𝐺 (𝜂)) 𝑑𝜎(𝜂) + 2 𝜂 (𝐹 (𝜂)𝐺 (𝜂)) 𝑑𝜔(𝜂), Γ ∫ 𝜕Γ ∫ 𝐺 (𝜂)𝐿 ∗ 𝐹 (𝜂) 𝑑𝜔(𝜂) + 𝐹 (𝜂)𝐿 ∗ 𝐺 (𝜂) 𝑑𝜔(𝜂) (2.145) Γ Γ ∫ = 𝜏(𝜂) (𝐹 (𝜂)𝐺 (𝜂)) 𝑑𝜎(𝜂). 𝜕Γ

and

∫

∗

∫

𝐺 (𝜂)∇ 𝐹 (𝜂) 𝑑𝜔(𝜂) + Ω

∫

𝐹 (𝜂)∇∗ 𝐺 (𝜂) 𝑑𝜔(𝜂) = 0,

(2.146)

𝐹 (𝜂)𝐿 ∗ 𝐺 (𝜂) 𝑑𝜔(𝜂) = 0.

(2.147)

Ω

𝐺 (𝜂)𝐿 ∗ 𝐹 (𝜂) 𝑑𝜔(𝜂) +

Ω

∫ Ω

The proof can be found in W. Freeden, C. Gerhards (2013). In the same manner, we obtain the following versions. Lemma 2.6 (First Green’s Theorems) If 𝐹 is of class C (1) (Γ) and 𝑓 of class c (1) (Γ), then ∫ ∫ 𝑓 (𝜂) · ∇∗ 𝐹 (𝜂)𝑑𝜔(𝜂) + 𝐹 (𝜂)∇∗ · 𝑓 (𝜂)𝑑𝜔(𝜂) Γ Γ ∫ ∫ = 𝜈(𝜂) · (𝐹 (𝜂) 𝑓 (𝜂)) 𝑑𝜎(𝜂) + 2 𝜂 · (𝐹 (𝜂) 𝑓 (𝜂)) 𝑑𝜔(𝜂), Γ ∫ 𝜕Γ ∫ ∗ ∗ 𝑓 (𝜂) · 𝐿 𝐹 (𝜂) 𝑑𝜔(𝜂) + 𝐹 (𝜂)𝐿 · 𝑓 (𝜂) 𝑑𝜔(𝜂) Γ Γ ∫ = 𝜏(𝜂) · (𝐹 (𝜂) 𝑓 (𝜂)) 𝑑𝜎(𝜂). 𝜕Γ

For functions 𝑓 , 𝑔 of class c (1) (Γ), we have

(2.148)

(2.149)

2.6 Spherical Integral Calculus

∫

55

∫

(∇∗ ⊗ 𝑓 (𝜂)) 𝑇 𝑔(𝜂) 𝑑𝜔(𝜂) 𝑓 (𝜂) (∇∗ · 𝑔(𝜂)) 𝑑𝜔(𝜂) + (2.150) Γ ∫ ∫ ( 𝑓 (𝜂) ⊗ 𝜈(𝜂)) 𝑔(𝜂) 𝑑𝜎(𝜂) + 2 ( 𝑓 (𝜂) ⊗ 𝜂) 𝑔(𝜂) 𝑑𝜔(𝜂), = Γ ∫ 𝜕Γ ∫ (𝐿 ∗ ⊗ 𝑓 (𝜂)) 𝑇 𝑔(𝜂) 𝑑𝜔(𝜂) 𝑓 (𝜂) (𝐿 ∗ · 𝑔(𝜂)) 𝑑𝜔(𝜂) + (2.151) Γ Γ ∫ ( 𝑓 (𝜂) ⊗ 𝜏(𝜂)) 𝑔(𝜂)𝑑𝜎(𝜂). = Γ

𝜕Γ

Lemma 2.7 (First Green’s Theorems) If 𝐹 is of class C (1) (Γ) and 𝑓 of class c (1) (Γ), then ∫ ∫
∇∗ 𝐹 (𝜂) ∧ 𝑓 (𝜂)𝑑𝜔(𝜂) + 𝐹 (𝜂)∇∗ ∧ 𝑓 (𝜂)𝑑𝜔(𝜂) (2.152) Γ Γ ∫ ∫ = 𝜈(𝜂) ∧ (𝐹 (𝜂) 𝑓 (𝜂)) 𝑑𝜎(𝜂) + 2 𝜂 ∧ (𝐹 (𝜂) 𝑓 (𝜂)) 𝑑𝜔(𝜂), Γ ∫ 𝜕Γ ∫
∗ 𝐿 𝐹 (𝜂) ∧ 𝑓 (𝜂) 𝑑𝜔(𝜂) + 𝐹 (𝜂)𝐿 ∗ ∧ 𝑓 (𝜂) 𝑑𝜔(𝜂) (2.153) Γ Γ ∫ = 𝜏(𝜂) ∧ (𝐹 (𝜂) 𝑓 (𝜂)) 𝑑𝜎(𝜂). 𝜕Γ

Lemma 2.8 (Second Green’s Theorem) Suppose that 𝐹, 𝐻 are of class C (2) (Γ). Then ∫ ∫ ∗ 𝐹 (𝜂)Δ 𝐻 (𝜂) 𝑑𝜔(𝜂) − 𝐻 (𝜂)Δ∗ 𝐹 (𝜂) 𝑑𝜔(𝜂) Γ Γ ∫ ∫ 𝜕𝐻 𝜕𝐹 = 𝐹 (𝜂) (𝜂)𝑑𝜎(𝜂) − 𝐻 (𝜂) (𝜂)𝑑𝜎(𝜂). 𝜕𝜈 𝜕𝜈 𝜕Γ 𝜕Γ

(2.154)

The Green’s theorems hold true as well for the regular region Ω\Γ, however, with opposite orientation on the boundary 𝜕Γ. Hence, for the entire sphere Ω, we find that ∫ ∫ ∗ 𝑓 (𝜂) · ∇ 𝐹 (𝜂) 𝑑𝜔(𝜂) = − 𝐹 (𝜂)∇∗ · 𝑓 (𝜂) 𝑑𝜔(𝜂), (2.155) Ω Ω ∫ ∫ 𝑓 (𝜂) · 𝐿 ∗ 𝐹 (𝜂) 𝑑𝜔(𝜂) = − 𝐹 (𝜂)𝐿 ∗ · 𝑓 (𝜂) 𝑑𝜔(𝜂), (2.156) Ω

and

Ω

∫

∗

∫

∇ · 𝑓 (𝜂) 𝑑𝜔(𝜂) = Ω

Ω

𝐿 ∗ · 𝑓 (𝜂) 𝑑𝜔(𝜂) = 0.

(2.157)

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2 Basic Settings

are valid for functions 𝐹 of class C (1) (Ω) as well as tangential vector fields 𝑓 of (1) class ctan (Ω). There are immediate consequences of the above formulas due to the fact that the integral identities also hold true on Ω\Γ¯ (under suitable assumptions on the integrands). For the whole sphere Ω, this leads to ∫ ∫ ∗ 𝑓 (𝜉) · ∇ 𝐹 (𝜉) 𝑑𝜔(𝜉) = − 𝐹 (𝜉)∇∗ · 𝑓 (𝜉) 𝑑𝜔(𝜉), (2.158) Ω Ω ∫ ∫ 𝑓 (𝜉) · 𝐿 ∗ 𝐹 (𝜉) 𝑑𝜔(𝜉) = − 𝐹 (𝜉)𝐿 ∗ · 𝑓 (𝜉) 𝑑𝜔(𝜉), (2.159) Ω Ω ∫ ∫ ∇∗ 𝐹 (𝜉) · ∇∗ 𝐺 (𝜉) 𝑑𝜔(𝜉) = − 𝐹 (𝜉)Δ∗ 𝐺 (𝜉) 𝑑𝜔(𝜉) (2.160) Ω Ω ∫ = − 𝐺 (𝜉)Δ∗ 𝐹 (𝜉) 𝑑𝜔(𝜉). Ω

Furthermore,

∫

∇∗ · 𝑓 (𝜉) 𝑑𝜔(𝜉) = 0,

(2.161)

Ω

∫

𝐿 ∗ 𝐹 (𝜉) · 𝐿 ∗ 𝐺 (𝜉) 𝑑𝜔(𝜉) = −

Ω

∫

𝐹 (𝜉)Δ∗ 𝐺 (𝜉) 𝑑𝜔(𝜉),

(2.162)

Ω

∫

∇∗ · ( 𝑓 (𝜉) ∧ 𝜉) 𝑑𝜔(𝜉) = 0,

(2.163)

Ω

provided that 𝐹 : Ω → R (resp. 𝑓 : Ω → R3 ) are sufficiently often continuously differentiable. Decomposition. Let us consider a spherical vector field 𝑓 of class c(Ω). Of course, 𝑓 can be decomposed by using the three basis vectors 𝜀 1 , 𝜀 2 , 𝜀 3 : 𝑓 (𝜉) =

3 ∑︁ 𝑖=1

3 ∑︁
𝑓 (𝜉) · 𝜀 𝑖 𝜀 𝑖 = 𝐹𝑖 (𝜉) 𝜀 𝑖 ,

𝜉 ∈ Ω,

(2.164)

𝑖=1

where 𝐹𝑖 : Ω → R are differentiable functions with 𝐹𝑖 (𝜉) = 𝑓 (𝜉) · 𝜀 𝑖 , 𝜉 ∈ Ω, 𝑖 = 1, 2, 3. The representation (2.164) can be used to reduce vectorial differential or integral equations, but it has the drawback that essential properties (for example, surface divergence, surface curl, spherical symmetry, etc) of vector fields are ignored. This problem can be overcome by the Helmholtz decomposition formula (for more details the reader is referred to Section 5.2). To be more specific, the decomposition (2.164) of vector fields using the unit vectors 𝜀 𝑖 , 𝑖 ∈ {1, 2, 3}, is no longer adequate for a large class of problems, since none of them reflects either the tangential or the normal direction on the sphere. A first hint for a system of unit vectors that is more suitable to a physically motivated situation can be given by the representation:

2.6 Spherical Integral Calculus

57

𝑓 (𝜉) = 𝑓nor (𝜉) + 𝑓tan (𝜉),

(2.165)

𝑓nor (𝜉) = ( 𝑓 (𝜉) · 𝜉) 𝜉.

(2.166)

where The vector 𝜉 ∈ Ω points into the normal direction. Thus, we have to construct for 𝑓tan (𝜉) in each point 𝜉 ∈ Ω two unit vectors perpendicular to 𝜉 (that have to be of physical relevance). Clearly, for a continuous vector field 𝑓 : Ω → R3 , we call 𝜉 ↦→ 𝑓nor (𝜉) = ( 𝑓 (𝜉) · 𝜉) 𝜉,

𝜉 ∈ Ω,

(2.167)

the normal field of 𝑓 , while 𝜉 ↦→ 𝑓tan (𝜉) = 𝑓 (𝜉) − ( 𝑓 (𝜉) · 𝜉) 𝜉,

𝜉 ∈ Ω,

(2.168)

is called the tangential field of 𝑓 . Obviously, the identity (2.165) is valid and the normal field of 𝑓 is orthogonal to the tangential field of 𝑓 , i.e., for all 𝜉 ∈ Ω (( 𝑓 (𝜉) · 𝜉)𝜉) · (( 𝑓 (𝜉) − ( 𝑓 (𝜉) · 𝜉) 𝜉)) = ( 𝑓 (𝜉) · 𝜉) 2 − ( 𝑓 (𝜉) · 𝜉) 2 = 0.

(2.169)

Furthermore, for 𝑓 , 𝑔 ∈ c(Ω) and 𝜉 ∈ Ω, 𝑓 (𝜉) · 𝑔(𝜉) = 𝑓nor (𝜉) · 𝑔nor (𝜉) + 𝑓tan (𝜉) · 𝑔tan (𝜉).

(2.170)

Lemma 2.9 The tangential field of 𝑓 vanishes i.e., 𝑓tan (𝜉) = 0, 𝜉 ∈ Ω, if and only if 𝑓 (𝜉) · 𝜏(𝜉) ˆ = 0 for every unit vector 𝜏(𝜉) ˆ that is perpendicular to 𝜉, i.e., for which 𝜉 · 𝜏(𝜉) ˆ = 0, 𝜉 ∈ Ω. Proof First, assume 𝑓tan = 0. For all 𝜉 ∈ Ω we have 𝑓 (𝜉) · 𝜏(𝜉) ˆ = ( 𝑓 (𝜉) · 𝜉) (𝜉 · 𝜏(𝜉)) ˆ | {z } =0

+ ( 𝑓 (𝜉) − ( 𝑓 (𝜉) · 𝜉)𝜉) ·𝜏(𝜉) ˆ | {z }

(2.171)

=0

= 0. Conversely, assume that the tangential field is non-vanishing, i.e., 𝑓tan (𝜉) = 𝑓 (𝜉) − ( 𝑓 (𝜉) · 𝜉)𝜉 ≠ 0.

(2.172)

Then it follows that 𝑓tan (𝜉)| 𝑓tan (𝜉)| −1 is a unit vector field perpendicular to 𝜉. Hence, by our hypothesis, 𝑓tan (𝜉) = 0. (2.173) 𝑓tan (𝜉) · | 𝑓tan (𝜉)|

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2 Basic Settings

This implies | 𝑓tan (𝜉)| = 0, which is a contradiction. Thus it follows that 𝑓tan (𝜉) = 0, as required. Lemma 2.10 Suppose that 𝑓 is continuous on Ω. Moreover, let ∫ 𝜏𝜉 · 𝑓 (𝜉) 𝑑𝜎(𝜉) = 0

(2.174) □

(2.175)

𝜎

for every curve 𝜎 lying on Ω. Then 𝑓tan (𝜉) = 0

(2.176)

for all 𝜉 ∈ Ω, i.e., the tangential field of 𝑓 vanishes for all 𝜉 ∈ Ω. Proof Choose any point 𝜉0 ∈ Ω. Let 𝜏𝜉0 be any unit vector satisfying 𝜏𝜉0 · 𝜉0 = 0. Then, there is a curve 𝜎 on Ω passing through 𝜉0 whose unit tangent vector at 𝜉0 𝜉 is just 𝜏𝜉0 . Let 𝜎sub0 be any subset of 𝜎 containing 𝜉0 . Then, in accordance with our assumption, ∫ 𝜉

𝜏𝜉 · 𝑓 (𝜉) 𝑑𝜎(𝜉) = 0.

(2.177)

𝜎sub0 𝜉

Hence, letting the length of 𝜎sub0 tend to zero we find 𝜏𝜉0 · 𝑓 (𝜉0 ) = 0. Lemma 2.9 then yields 𝑓tan (𝜉0 ) = 𝑓 (𝜉0 ) − ( 𝑓 (𝜉0 ) · 𝜉0 )𝜉0 = 0. Since 𝜉0 can be any point on the sphere Ω, we have 𝑓tan (𝜉) = 𝑓 (𝜉) − ( 𝑓 (𝜉) · 𝜉)𝜉 = 0 for all 𝜉 ∈ Ω. □ The surface gradient acts like an ordinary gradient in R3 when we integrate it along lines on Ω. In more detail, suppose 𝐹 is continuously differentiable in an open set in R3 containing Ω, and 𝜎 is any curve lying on Ω, starting at 𝜉0 and ending at 𝜉1 (see Fig. 2.4). Suppose that 𝜏𝜉 is the unit tangent vector at 𝜉 on 𝜎 pointing from 𝜉0 to 𝜉1 . Then ∫ 𝐹 (𝜉1 ) − 𝐹 (𝜉0 ) = 𝜏𝜉 · ∇∗ 𝐹 (𝜉) 𝑑𝜎(𝜉) (2.178) 𝜎

(observe that 𝜏𝜉 · ∇ 𝜉 𝐹 (𝜉) = 𝜏𝜉 · ∇∗𝜉 𝐹 (𝜉), 𝜉 ∈ Ω, (cf. C. M¨uller (1969))). This result enables us to show the following lemma. Lemma 2.11 Let 𝐹 be of class C (1) (Ω). Assume that ∇∗ 𝐹 (𝜉) = 0 for all 𝜉 ∈ Ω, then 𝐹 is constant, and conversely. Proof If ∇∗ 𝐹 (𝜉) = 0, then we obtain, in connection with (2.178), 𝐹 (𝜉1 ) = 𝐹 (𝜉0 ) for any 𝜉0 , 𝜉1 on Ω. Conversely, if 𝐹 is constant, the identity (2.178) shows that 𝑓 = ∇∗ 𝐹 fulfills ∫ 𝜏𝜉 · 𝑓 (𝜉) 𝑑𝜎(𝜉) = 0 (2.179) 𝜎

2.6 Spherical Integral Calculus

59

for every curve 𝜎 lying on Ω. Consequently, following Lemma 2.10, 𝑓tan (𝜉) = 0

(2.180)

𝑓tan (𝜉) = 𝑓 (𝜉) − ( 𝑓 (𝜉) · 𝜉) · 𝜉 = 𝑓 (𝜉) = ∇∗ 𝐹 (𝜉) = 0

(2.181)

for all 𝜉 ∈ Ω. This shows that

for all 𝜉 ∈ Ω.

□

From Lemma 2.11 we are immediately able to deduce the following statement. Lemma 2.12 Let 𝐹 be of class C (1) (Ω). Assume that 𝐿 ∗ 𝐹 (𝜉) = 0 for all 𝜉 ∈ Ω, then 𝐹 is constant, and conversely. Proof If 𝐿 ∗ 𝐹 (𝜉) = 0, i.e., 𝜉 ∧ ∇∗ 𝐹 (𝜉) = 0 for all 𝜉 ∈ Ω. Then 𝜉 ∧ 𝜉 ∧ ∇∗ 𝐹 (𝜉) = (𝜉 · ∇∗ 𝐹 (𝜉))𝜉 − ∇∗ 𝐹 (𝜉)(𝜉 · 𝜉) = −∇∗ 𝐹 (𝜉) = 0 for all 𝜉 ∈ Ω. Thus, by virtue of Lemma 2.11, we find 𝐹 = const. Conversely, if 𝐹 is constant, then 𝐿 ∗ 𝐹 (𝜉) = 𝜉 ∧ ∇∗ 𝐹 (𝜉) = 𝜉 ∧ 0 = 0 for all 𝜉 ∈ Ω. This proves Lemma 2.12. □ Next, we prove the following well-known result of spherical vector analysis (see, e.g., G.E. Backus et al. (1996)). Lemma 2.13 Let 𝑓 ∈ c(Ω) be a tangent vector field, i.e., 𝑓 (𝜉) = 𝑓tan (𝜉) = 𝑓 (𝜉) − ( 𝑓 (𝜉) · 𝜉)𝜉, 𝜉 ∈ Ω. Furthermore, suppose that ∫ 𝜏𝜉 · 𝑓 (𝜉) 𝑑𝜎(𝜉) = 0 (2.182) 𝜎

for every closed curve 𝜎 on Ω. Then, there is a scalar field 𝑃 on Ω such that 𝑓 (𝜉) = ∇∗ 𝑃(𝜉),

𝜉 ∈ Ω.

(2.183)

The field 𝑃 is continuously differentiable and is unique up to a constant. Proof Take an arbitrary, but fixed 𝜉0 ∈ Ω. We let ∫

𝜉

𝜏𝜁 · 𝑓 (𝜁) 𝑑𝜎(𝜁),

𝑃(𝜉) =

(2.184)

𝜉0

the integral being along any curve 𝜎 that starts at 𝜉0 ∈ Ω and ends at 𝜉 ∈ Ω. Then, for any two points 𝜉0 , 𝜉 on Ω and any curve 𝜎 lying on Ω and starting at 𝜉0 and ending at 𝜉1 ,

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2 Basic Settings

∫

𝜉1

𝑃(𝜉1 ) − 𝑃(𝜉0 ) =

𝜏𝜁 · 𝑓 (𝜁) 𝑑𝜎(𝜁).

(2.185)

𝜏𝜁 · ∇∗ 𝑃(𝜁) 𝑑𝜎(𝜁).

(2.186)

𝜉0

Observing (2.178) we find ∫

𝜉1

𝑃(𝜉1 ) − 𝑃(𝜉0 ) = 𝜉0

Combining (2.185) and (2.186), we obtain ∫

𝜉1

𝜏𝜁 · ( 𝑓 (𝜁) − ∇∗ 𝑃(𝜁)) 𝑑𝜎(𝜁) = 0

(2.187)

𝜉0

for any curve 𝜎 on Ω. Lemma 2.10, therefore, tells us that 𝑓 (𝜉) − ∇∗ 𝑃(𝜉) = 0,

𝜉 ∈ Ω.

(2.188)

The proof that 𝑃 is continuously differentiable on Ω is omitted. The easiest way to construct such a proof is to take 𝑃 constant on each straight line passing through Ω in the normal direction (see, e.g., G.E. Backus et al. (1996)). In order to verify that 𝑃 is unique up to a constant, we observe that ∇∗ 𝑃1 (𝜉) = ∇∗ 𝑃2 (𝜉), 𝜉 ∈ Ω, implies ∇∗ (𝑃1 − 𝑃2 )(𝜉) = 0, 𝜉 ∈ Ω, i.e., by virtue of Lemma 2.11, 𝑃1 − 𝑃2 = const. □ Now we are able to formulate the following important theorem: Theorem 2.14 Let 𝑓 ∈ c (1) (Ω) be a tangential field, i.e., 𝑓 (𝜉) = 𝑓tan (𝜉) = 𝑓 (𝜉) − ( 𝑓 (𝜉) · 𝜉)𝜉

(2.189)

for all 𝜉 ∈ Ω. Then 𝐿 ∗ · 𝑓 (𝜉) = 0, 𝜉 ∈ Ω, if and only if there is a scalar field 𝑃 such that 𝑓 (𝜉) = ∇∗ 𝑃(𝜉), 𝜉 ∈ Ω, (2.190) and 𝑃 is unique up to an additive constant (𝑃 is called potential function for 𝑓 ). Similarly, ∇∗ · 𝑓 (𝜉) = 0, 𝜉 ∈ Ω, if and only if there is a scalar field 𝑆 such that 𝑓 (𝜉) = 𝐿 ∗ 𝑆(𝜉),

𝜉 ∈ Ω,

(2.191)

and 𝑆 is unique up to an additive constant (𝑆 is called stream function for 𝑓 ). Proof The condition 𝑓 = ∇∗ 𝑃 implies 𝐿 ∗ · 𝑓 = 0, and 𝑓 = 𝐿 ∗ 𝑆 implies ∇∗ · 𝑓 = 0. Conversely, assume that 𝐿 ∗ · 𝑓 (𝜉) = 0, 𝜉 ∈ Ω. Then the surface theorem of Stokes implies ∫ 𝜏𝜉 · 𝑓 (𝜉) 𝑑𝜎(𝜉) = 0 (2.192) 𝜎

2.6 Spherical Integral Calculus

61

for every closed curve 𝜎 on Ω. From Lemma 2.13, it follows that there exists a scalar field 𝑃 such that 𝑓 = ∇∗ 𝑃. Furthermore, 𝑃 is unique up to an additive constant. Finally, suppose ∇∗ · 𝑓 = 0. Then 𝐿 ∗ · (𝜉 ∧ 𝑓 (𝜉)) = 0, 𝜉 ∈ Ω. Hence, by the same arguments as above, there is a scalar field 𝑆, unique up to a constant, such that −𝜉 ∧ 𝑓 (𝜉) = ∇∗ 𝑆(𝜉),

𝜉 ∈ Ω.

(2.193)

This is equivalent to −𝜉 ∧ (𝜉 ∧ 𝑓 (𝜉)) = (𝜉 ∧ ∇∗ )𝑆(𝜉),

𝜉 ∈ Ω,

(2.194)

or 𝑓 = 𝐿∗ 𝑆

(2.195) □

on Ω. This proves Theorem 2.14.

For tangential fields, the validity of homogeneous “pre-Maxwell equations” implies that the field under consideration vanishes identically. This is the content of the next theorem. Theorem 2.15 Let 𝑓 be a continuously differentiable tangential vector field on Ω (i.e., 𝑓 (𝜉) = 𝑓tan (𝜉) = 𝑓 (𝜉) − ( 𝑓 (𝜉) · 𝜉)𝜉, 𝜉 ∈ Ω) such that ∇∗ · 𝑓 (𝜉) = 0,

𝜉 ∈ Ω,

𝐿 ∗ · 𝑓 (𝜉) = 0,

𝜉 ∈ Ω.

Then 𝑓 = 0 on Ω. Proof From 𝐿 ∗ · 𝑓 (𝜉) = 0 we obtain from Theorem 2.14 that there exists a scalar field 𝑃 such that 𝑓 (𝜉) = ∇∗ 𝑃(𝜉), 𝜉 ∈ Ω. (2.196) From ∇∗ · 𝑓 (𝜉) = 0 we can therefore deduce that ∇∗ · ∇∗ 𝑃(𝜉) = Δ∗ 𝑃(𝜉) = 0. Together with (2.160), this leads to ∫ (∇∗ 𝑃(𝜉)) 2 𝑑𝜔(𝜉) = 0.

(2.197)

Ω

Consequently, it follows that 𝑓 (𝜉) = ∇∗ 𝑃(𝜉) = 0. This is the required result.

□

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2 Basic Settings

2.7 Orthogonal Invariance

Systems of equations which maintain their form when the coordinate axes are subjected to an arbitrary rotation are said to be rotational-, or orthogonal-, invariant. The orthogonal invariance is, of course, closely related to the group 𝑂 (3) of all orthogonal transformations, i.e., the group of all t ∈ R3 ⊗ R3 such that tt𝑇 = t𝑇 t = i, i = (𝛿𝑖 𝑗 )𝑖, 𝑗=1,2,3 . The set of all rotations, i.e., 𝑆𝑂 (3) = {t ∈ 𝑂 (3) | det t = 1} is a subgroup called the special orthogonal group . We briefly recapitulate some properties of these groups (see, e.g., C. M¨uller (1998), N.J. Vilenkin (1968) and many others): (i) Let 𝜉, 𝜂 be members of Ω. Then, there exists an orthogonal transformation t ∈ 𝑂 (3) with 𝜂 = t𝜉 and an orthogonal transformation s ∈ 𝑆𝑂 (3) with 𝜂 = s𝜉. (ii) For every t ∈ 𝑂 (3) t𝜉 · t𝜂 = 𝜉 · 𝜂, 𝜉, 𝜂 ∈ Ω. (2.198) (iii) Suppose that 𝜉 ∈ Ω. The set 𝑂 𝜉 (3) = {t ∈ 𝑂 (3) | t𝜉 = 𝜉} is a subgroup of 𝑂 (3). Analogously, the set 𝑆𝑂 𝜉 (3) = {t ∈ 𝑆𝑂 (3) | t𝜉 = 𝜉} is a subgroup of 𝑆𝑂 (3). (iv) For every t ∈ 𝑂 (3), we have det t = ±1. If det t = 1, t is called a rotation, while for det t = −1, t is called a reflection . (v) Let t, t′ ∈ 𝑂 (3) with det t = 1, det t′ = −1. Then t𝜉 ∧ t𝜂 = t(𝜉 ∧ 𝜂), t′ 𝜉 ∧ t′ 𝜂 = −t′ (𝜉 ∧ 𝜂),

𝜉, 𝜂 ∈ Ω, 𝜉, 𝜂 ∈ Ω.

(2.199) (2.200)

(vi) Let t ∈ 𝑂 (3). Then, for the dyadic product, we obtain t(𝜉 ⊗ 𝜂)t𝑇 = t𝜉 ⊗ t𝜂,

𝜉, 𝜂 ∈ Ω.

(2.201)

The following definitions will prove useful for our later considerations. Definition 2.16 Let 𝐹 ∈ L2 (Ω), 𝑓 ∈ l2 (Ω), f ∈ l2 (Ω) and suppose that t ∈ 𝑂 (3). For scalar, vector, and tensor fields the operator 𝑅t is defined by 𝑅t : L2 (Ω) → L2 (Ω), 2

2

𝑅t 𝐹 (𝜉) = 𝐹 (t𝜉),

𝑅t : l (Ω) → l (Ω),

𝑅t 𝑓 (𝜉) = t𝑇 𝑓 (t𝜉),

𝑅t : l2 (Ω) → l2 (Ω),

𝑅t f (𝜉) = t𝑇 f (t𝜉)t,

respectively. 𝑅t 𝐹, 𝑅t 𝑓 , and 𝑅𝑡 f are called the t-transformed fields .

2.7 Orthogonal Invariance

63

𝜉

t𝜉

𝜉 ↦→ 𝐹 ( 𝜉 ) 𝜉 ↦→ 𝑅t 𝐹 ( 𝜉 )

Fig. 2.4: The operator 𝑅t acting on a function.

For examples illustrating how the operators 𝑅t act on functions and vector fields, see Figs. 2.4 and 2.5, respectively. Definition 2.17 Let F be a subspace of L2 (Ω) (l2 (Ω) or l2 (Ω)). F is called invariant with respect to orthogonal transformations or, equivalently, orthogonally invariant if, for all 𝐹 ∈ F and for all orthogonal transformations t ∈ 𝑂 (3), the function 𝑅𝑡 𝐹 is of class F . An orthogonally invariant F is called reducible if there exists a proper subspace F ′ ⊂ F which itself is invariant with respect to orthogonal transformations. 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 2.18 Let (F , (·, ·)) be an orthogonally invariant Hilbert subspace of L2 (Ω). Let F1 be an orthogonally invariant subspace of F . Then, the orthogonal complement F1⊥ of F1 is orthogonally invariant, as well. Proof For all 𝐹 ∈ F1 , 𝐹 ⊥ ∈ F1⊥ and for all orthogonal transformations t ∈ 𝑂 (3), we have ∫ ⊥ (𝐹, 𝑅t 𝐹 ) = 𝐹 (𝜉)𝑅t 𝐹 ⊥ (𝜉) 𝑑𝜔(𝜉) (2.202) Ω ∫ = (det t) 𝐹 (𝜉)𝑅t 𝐹 ⊥ (𝜉) 𝑑𝜔(𝜉) tΩ ∫ 2 = (det t) 𝑅t𝑇 𝐹 (𝜉)𝐹 ⊥ (𝜉) 𝑑𝜔(𝜉) Ω

= 0,

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2 Basic Settings

t𝜉 𝜉

𝜉 ↦→ 𝑓 ( 𝜉 ) 𝜉 ↦→ 𝑓 (t 𝜉 )

t𝜉 𝜉

𝜉 ↦→ 𝑓 ( 𝜉 ) 𝜉 ↦→ t𝑇 𝑓 (t 𝜉 )

Fig. 2.5: The definition of the operator 𝑅t for vector fields (note that it is necessary not only to substitute 𝜉 by t𝜉, but also to transform the directions of the vectors). since 𝑅t𝑇 𝐹 ∈ F1 . This implies that 𝑅t 𝐹 ⊥ ∈ F1⊥ and, therefore, F1⊥ is invariant with respect to orthogonal transformations. □ Analogous results can be formulated for Hilbert spaces of square-integrable vector and tensor fields. Lemma 2.18 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 definitely provide us with elements that are invariant with respect to certain orthogonal transformations. The following results help us to analyze the structure of such rotational-invariant functions. Lemma 2.19 Let 𝐹 be a function of class C(Ω) with 𝑅t 𝐹 (𝜉) = 𝐹 (𝜉) for all t ∈ 𝑆𝑂 (3) and all 𝜉 ∈ Ω. Then

2.7 Orthogonal Invariance

65

𝐹 = 𝐹 (𝜀 3 ) = 𝐶 = const. Proof For every 𝜉 ∈ Ω, there exists a rotation t ∈ 𝑆𝑂 (3) with t𝜉 = 𝜀 3 . Consequently, for every 𝜉 ∈ Ω, we have 𝐹 (𝜉) = 𝑅t 𝐹 (𝜉) = 𝐹 (t𝜉) = 𝐹 (𝜀 3 ) = 𝐶 = const. □ Lemma 2.20 Let 𝜂 ∈ Ω be fixed. Furthermore, let 𝐹 ∈ C(Ω) with 𝑅t 𝐹 (𝜉) = 𝐹 (𝜉) for all t ∈ 𝑆𝑂 𝜂 (3) and for all 𝜉 ∈ Ω. Then, 𝐹 can be represented in the form 𝐹 (𝜉) = Φ(𝜉 · 𝜂),

𝜉 ∈ Ω,

Φ being a function Φ : [−1, 1] → R. Proof Without loss of generality , let 𝜂 = 𝜀 3 (if this were not true, we could √ use the function 𝐺 (𝜉) = 𝑅t′ 𝐹 (𝜉), where t′ ∈ 𝑂 (3) with t′ 𝜀 3 = 𝜂). With 𝜉 = 𝑡𝜀 3 + 1 − 𝑡 2 𝜂 ′ we have, by assumption, that √︁ √︁ 𝐹 (𝑡𝜀 3 + 1 − 𝑡 2 𝜂 ′ ) = 𝐹 (𝑡𝜀 3 + 1 − 𝑡 2 𝜂 ′′ ), (2.203) for all points 𝜂 ′ , 𝜂 ′′ of the unit circle. Hence, 𝐹 depends only on 𝑡 = 𝜉 · 𝜀 3 and is, therefore, a function of 𝑡 alone, as desired. □ Lemma 2.21 Let 𝜂 ∈ Ω be fixed. Let 𝐹 ∈ C(Ω) with 𝑅t 𝐹 (𝜉) = (det t) F(𝜉) for all t ∈ 𝑂 𝜂 (3) and all 𝜉 ∈ Ω. Then 𝐹 = 0. Proof Suppose that 𝜉 is an element of Ω. There exists a reflection t ∈ 𝑂 𝜂 (3) with t𝜉 = 𝜉, but then – by assumption – we have 𝐹 (𝜉) = 𝑅t 𝐹 (𝜉) = −𝐹 (𝜉), hence, 𝐹 (𝜉) = 0. □ Note that in Lemma 2.19 and Lemma 2.20, 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 : Ω → R3 . Let 𝜂 be an element of Ω. In every point 𝜉 ≠ ±𝜂, we are able to introduce the so-called moving triad at the point 𝜉 𝜀 1𝜉 = 𝜉, 1 𝜀 2𝜉 = √︁ (𝜂 − (𝜉 · 𝜂)𝜉), 1 − (𝜉 · 𝜂) 2 1 𝜀 3𝜉 = √︁ 𝜂 ∧ 𝜉, 1 − (𝜉 · 𝜂) 2

(2.204) (2.205) (2.206)

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2 Basic Settings

such that there exist functions 𝐹1 , 𝐹2 , 𝐹3 : Ω → R with 𝑓 = 𝐹1 𝜀 1𝜉 + 𝐹2 𝜀 2𝜉 + 𝐹3 𝜀 3𝜉 .

(2.207)

For further investigations, the following lemma is helpful. Lemma 2.22 Let 𝜂 ∈ Ω be fixed, and let the moving triad 𝜀 𝑖𝜉 , 𝑖 = 1, 2, 3, be defined as in (2.204)–(2.206). Then, for all t ∈ 𝑂 (3), 𝑅t 𝜀 𝑖𝜉 = 𝜀 𝑖𝜉 ,

𝑖 = 1, 2

𝑅t 𝜀 3𝜉 = (det t) 𝜀 3𝜉 . Proof For t ∈ 𝑂 𝜂 (3), 𝑅t 𝜀 1𝜉 = t𝑇 𝜀 t1𝜉 = t𝑇 t 𝜉 = 𝜉 = 𝜀 1𝜉 .

(2.208)

For the tangential fields, we only show the case 𝑖 = 2 (the case 𝑖 = 3 follows similarly). We have 1 𝑅t 𝜀 2𝜉 = t𝑇 𝜀 t2𝜉 = √︁ t𝑇 (𝜂 − (t𝜉 · 𝜂)t𝜉) 1 − (t𝜉 · 𝜂) 2 1 = √︁ (t𝑇 𝜂 − (𝜉 · t𝑇 𝜂)t𝑇 t𝜉) 𝑇 2 1 − (𝜉 · t 𝜂) 1 = √︁ (𝜂 − (𝜉 · 𝜂)𝜉) 1 − (𝜉 · 𝜂) 2

(2.209)

= 𝜀 2𝜉 . We now extend our results for rotational-invariant functions to the vector case. Lemma 2.23 Let 𝑓 ∈ c(Ω) with 𝑅t 𝑓 (𝜉) = 𝑓 (𝜉) (or equivalently, 𝑓 (t𝜉) = t 𝑓 (𝜉)) for all t ∈ 𝑆𝑂 (3) and 𝜉 ∈ Ω. Then, there exists a constant 𝐶 ∈ R such that 𝑓 (𝜉) = 𝐶 𝜉 ,

𝜉 ∈ Ω.

Proof Consider the orthogonal matrix −1 0 © t = ­ 0 −1 « 0 0

0 ª 0®. 1¬

(2.210)

Then t𝜀 3 = 𝜀 3 and, by assumption, 𝑓 (𝜀 3 ) = t 𝑓 (𝜀 3 ). Hence, in connection with (2.210), we have 𝑓 (𝜀 3 ) = 𝐶𝜀 3 , 𝐶 ∈ R. For 𝜉 ∈ Ω, there exists a rotation t′ with t′ 𝜀 3 = 𝜉. Consequently, we have

2.7 Orthogonal Invariance

67

𝑓 (𝜉) = 𝑓 (t′ 𝜀 3 ) = t′ 𝑓 (𝜀 3 ) = 𝐶t′ 𝜀 3 = 𝐶𝜉.

(2.211)

Lemma 2.24 Let 𝜂 ∈ Ω. Let 𝑓 ∈ 𝑐(Ω) with 𝑅t 𝑓 (𝜉) = 𝑓 (𝜉) for all t ∈ 𝑆𝑂 𝜂 (3). Then, for 𝜉 ≠ ±𝜂, 𝑓 has the representation: 𝑓 (𝜉) = Φ1 (𝜉 · 𝜂)𝜀 1𝜉 + Φ2 (𝜉 · 𝜂)𝜀 2𝜉 + Φ3 (𝜉 · 𝜂)𝜀 3𝜉 , where Φ𝑖 , 𝑖 = 1, 2, 3, are functions Φ𝑖 : [−1, 1] → R. Proof From Lemma 2.22, it follows that the functions 𝐹𝑖 in (2.207) fulfill 𝑅t 𝐹𝑖 (𝜉) = 𝐹𝑖 (𝜉),

𝜉 ∈ Ω,

(2.212)

provided that t𝜂 = 𝜂. Therefore, via Lemma 2.20, we know that, for the functions 𝐹𝑖 , we have 𝐹𝑖 (𝜉) = Φ𝑖 (𝜉 · 𝜂). (2.213)

Lemma 2.25 Suppose that 𝜂 ∈ Ω. Let 𝑓 be of class c(Ω) with 𝑅t 𝑓 (𝜉) = 𝑓 (𝜉) for all t ∈ 𝑂 𝜂 (3). Then, for 𝜉 ≠ ±𝜂, 𝑓 has the representation, 𝑓 (𝜉) = Φ1 (𝜉 · 𝜂)𝜀 1𝜉 + Φ2 (𝜉 · 𝜂)𝜀 2𝜉 , Φ𝑖 , 𝑖 = 1, 2, being functions Φ𝑖 : [−1, 1] → R. Proof Starting from Lemma 2.24, we now have to consider reflections, as well. By our assumption and Lemma 2.22, we obtain Φ3 (𝜉 · 𝜂) = −Φ3 (𝜉 · 𝜂) and, therefore, Φ3 (𝜉 · 𝜂) = 0. □ Lemma 2.26 Suppose that 𝜂 ∈ Ω. Let 𝑓 be of class c(Ω) with 𝑅t 𝑓 (𝜉) = (det t) f (𝜉) for all t ∈ 𝑂 𝜂 (3). Then, for 𝜉 ≠ ±𝜂, the field 𝑓 can be represented as follows 𝑓 (𝜉) = Φ3 (𝜉 · 𝜂)𝜀 3𝜉 ,

(2.214)

with Φ3 being a function Φ3 : [−1, 1] → R. Proof Using the same reasoning as in the proof of Lemma 2.25, but now considering the change in sign under reflections, we end up with Φ1 (𝜉 · 𝜂) = −Φ1 (𝜉 · 𝜂), Φ2 (𝜉 · 𝜂) = −Φ2 (𝜉 · 𝜂), hence, Φ1 (𝜉 · 𝜂) = Φ2 (𝜉 · 𝜂) = 0.

(2.215) □

In order to extend our considerations to tensor fields of second rank , we assume f (𝜉) to be a matrix constituting a linear vector function for each 𝜉 ∈ Ω. Furthermore, let

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𝜂 be an element of Ω. Then, to the moving triad (2.204)–(2.206), there exist scalar spherical functions 𝐹𝑖 𝑗 : Ω → R such that f can be represented via dyadic products of the unit vectors, i.e., f (𝜉) =

3 ∑︁ 3 ∑︁

𝑗

𝐹𝑖 𝑗 (𝜉) 𝜀 𝑖𝜉 ⊗ 𝜀 𝜉 ,

𝜉 ≠ ±𝜂.

(2.216)

𝑖=1 𝑗=1

It should be remarked that, for 𝐹𝑖 𝑗 = 𝛿𝑖 𝑗 , (2.216) forms a partition of the unit matrix i. We now examine matrices with certain, rotational-invariant characteristics. Lemma 2.27 For every 𝜉 ∈ Ω and every t ∈ 𝑂 (3), let f be of class c(Ω) with 𝑅t f (𝜉) = f (𝜉) (i.e., f (t𝜉) = tf (𝜉)t𝑇 ),

(2.217)

𝜉 ∈ Ω. Then, there exist constants 𝐶1 , 𝐶2 ∈ R with f (𝜉) = 𝐶1 i + 𝐶2 𝜉 ⊗ 𝜉,

𝜉 ∈ Ω,

(2.218)

i being the unit matrix . Proof We start with the determination of the matrix f (𝜀 3 ) = (𝐹𝑖 𝑗 )𝑖, 𝑗=1,2,3 . Using the transformation −1 0 0 © ª t1 = ­ 0 1 0 ® (2.219) 0 0 1 « ¬ (2.217) leads to 𝐹12 = 𝐹13 = 𝐹21 = 𝐹31 = 0. Analogously, the application of the transformation 1 0 0 © ª t2 = ­ 0 −1 0 ® (2.220) «0 0 1¬ yields 𝐹23 = 𝐹32 = 0. Finally,

leads to 𝐹11

010 © ª t3 = ­ 1 0 0 ® (2.221) 0 0 1 « ¬ = 𝐹22 . Consequently, there exist constants 𝐶1 , 𝐶2 ∈ R such that f (𝜀 3 ) = 𝐶1 i + 𝐶2 𝜀 3 ⊗ 𝜀 3 .

(2.222)

If 𝜉 ∈ Ω, then there exits an orthogonal transformation t with t𝜀 3 = 𝜉. Thus, it follows that, for 𝜉 ∈ Ω,

2.7 Orthogonal Invariance

69

f (𝜉) = f (t𝜀 3 ) = tf (𝜀 3 )t𝑇

(2.223) 3

3

𝑇

= t(𝐶1 i + 𝐶2 𝜀 ⊗ 𝜀 )t = 𝐶1 i + 𝐶2 𝜉 ⊗ 𝜉.

Lemma 2.28 Let f be of class c(Ω) with 𝑅t f (𝜉) = t𝑇 𝑓 (t𝜉)t = (det t) f (𝜉) for all 𝜉 ∈ Ω and all t ∈ 𝑂 (3). Then there exists a constant 𝐶 ∈ R with f (𝜉) = 𝐶 i∗ (𝜉),

𝜉 ∈ Ω,

where 0 −𝜉3 𝜉2 © ª i∗ (𝜉) = ­ 𝜉3 0 −𝜉1 ® , « −𝜉2 𝜉1 0 ¬

𝜉 = (𝜉1 , 𝜉2 , 𝜉3 ) 𝑇 .

Proof In analogy to the proof of Lemma 2.27, we first determine f (𝜀 3 ) with the same transformations t1 , t2 and t3 as before. Now, our assumptions lead to 𝐹11 = 𝐹22 = 𝐹23 = 𝐹32 = 𝐹33 = 𝐹13 = 𝐹31 = 0, and 𝐹12 = −𝐹21 . Therefore, we can find a 𝐶 ∈ R such that f (𝜀 3 ) = C(𝜀 2 ⊗ 𝜀 1 − 𝜀 1 ⊗ 𝜀 2 ). (2.224) For every vector 𝑎 ∈ R3 , we obviously have f (𝜀 3 )𝑎 = 𝐶 𝜀 3 ∧ 𝑎. If 𝜉 ∈ Ω and if t ∈ 𝑂 (3) with t𝜀 3 = 𝜉, then f (𝜉)𝑎 = f (t𝜀 3 )𝑎 = (det t) tf (𝜀 3 )t𝑇 𝑎 = (det t) 𝐶 t(𝜀 3 ∧ (t𝑇 𝑎)) = 𝜉 ∧ 𝑎.

(2.225)

The vector product 𝜉 ∧ 𝑎 can easily be expressed by the antisymmetric matrix i∗ (𝜉), i.e., 𝜉 ∧ 𝑎 = i∗ (𝜉)𝑎. □ Lemma 2.29 Suppose that 𝜂 ∈ Ω. For every 𝜉 ∈ Ω and every t ∈ 𝑆𝑂 𝜂 (3), let f be of class c(Ω) with 𝑅t f (𝜉) = t𝑇 f (t𝜉)t = f (𝜉), 𝜉 ∈ Ω. Then, for 𝜉 ≠ ±𝜂, we have f (𝜉) =

3 ∑︁ 3 ∑︁

𝑗

Φ𝑖 𝑗 (𝜉 · 𝜂) 𝜀 𝑖𝜉 ⊗ 𝜀 𝜉 ,

(2.226)

𝑖=1 𝑗=1

with Φ𝑖 𝑗 being functions Φ𝑖 𝑗 : [−1, 1] → R. 𝑗

Proof We start from (2.216) and let 𝐹𝑖 𝑗 (𝜉) = 𝜀 𝑖𝜉 · (f (𝜉)𝜀 𝜉 ). By assumption, we have 𝐹𝑖 𝑗 (t𝜉) = 𝐹𝑖 𝑗 (𝜉), for every t ∈ 𝑆𝑂 𝜂 (3). Due to Lemma 2.20, we have 𝐹𝑖 𝑗 = Φ𝑖 𝑗 (𝜉 · 𝜂). This is the wanted result. □ Lemma 2.30 Suppose that 𝜂 is a point of Ω. For all 𝜉 ∈ Ω and for all t ∈ 𝑂 𝜂 (3), let f be a of class c(Ω) with 𝑅t f (𝜉) = f (𝜉). Then, for 𝜉 ≠ ±𝜂, f (𝜉) can be written as follows

70

2 Basic Settings

f (𝜉) = Φ11 (𝜉 · 𝜂) 𝜀 1𝜉 ⊗ 𝜀 1𝜉 + Φ12 (𝜉 · 𝜂) 𝜀 1𝜉 ⊗ 𝜀 2𝜉 + Φ21 (𝜉 · 𝜂) 𝜀 2𝜉 ⊗ 𝜀 1𝜉 + Φ22 (𝜉 · 𝜂) 𝜀 2𝜉 ⊗ 𝜀 2𝜉 + Φ33 (𝜉 · 𝜂) 𝜀 3𝜉 ⊗ 𝜀 3𝜉 , with Φ𝑖 𝑗 being functions Φ𝑖 𝑗 : [−1, 1] → R. Proof In contrast to Lemma 2.29, we also have to consider the use of reflections, i.e., we have to take into account that the transformation of cross-products by reflections leads to a change in sign. Consequently, Φ13 (𝜉 · 𝜂) = Φ23 (𝜉 · 𝜂) = Φ31 (𝜉 · 𝜂) = Φ32 (𝜉 · 𝜂) = 0. This proves Lemma 2.30. □ Lemma 2.31 Let 𝜂 ∈ Ω. For all 𝜉 ∈ Ω and for all t ∈ 𝑂 𝜂 (3) let f be of class c(Ω) with 𝑅t f (𝜉) = (dett) f (𝜉). Then, for 𝜉 ≠ ±𝜂, f (𝜉) can be written in the form f (𝜉) = Φ13 (𝜉 · 𝜂) 𝜀 1𝜉 ⊗ 𝜀 3𝜉 + Φ31 (𝜉 · 𝜂) 𝜀 3𝜉 ⊗ 𝜀 1𝜉 +Φ23 (𝜉 · 𝜂) 𝜀 2𝜉 ⊗ 𝜀 3𝜉 + Φ32 (𝜉 · 𝜂) 𝜀 3𝜉 ⊗ 𝜀 2𝜉 , with Φ𝑖 𝑗 being functions Φ𝑖 𝑗 : [−1, 1] → R. Proof Considering that the transformation of f (𝜉) using reflections leads to a minus sign, we find that in (2.226) the terms with Φ11 , Φ12 , Φ21 , Φ22 and Φ33 vanish. □ Remark 2.32 For 𝜉 ≠ ±𝜂, another basis system is given by 𝜀 1𝜂 = 𝜂, 𝜀 2𝜂 = √︁ 𝜀 3𝜂 = √︁

(2.227) 1 1 − (𝜉 · 𝜂) 2 1

(𝜉 − (𝜉 · 𝜂)𝜂),

(2.228)

𝜂 ∧ 𝜉.

(2.229)

1 − (𝜉 · 𝜂) 2

This shows us that analogous results to Lemma 2.31 can be based on (2.227), (2.228), (2.229). For example, under the assumptions of Lemma 2.30, we obtain 𝑓 (𝜉) = Φ11 (𝜉 · 𝜂)𝜀 1𝜉 ⊗ 𝜀 1𝜂 + Φ12 (𝜉 · 𝜂)𝜀 1𝜉 ⊗ 𝜀 2𝜂 2

+ Φ21 (𝜉 · 𝜂)𝜀 ⊗ 𝜉 ∈ Ω.

𝜀 1𝜂

+ Φ22 (𝜉 ·

𝜂)𝜀 2𝜉

⊗

𝜀 2𝜂

(2.230) + Φ33 (𝜉 ·

𝜂)𝜀 3𝜂

⊗

𝜀 3𝜂 ,

Chapter 3

Circular Harmonics

Our considerations in the following chapters will be exclusively undertaken with respect to two-dimensional spheres embedded in the Euclidean space R3 . Of course, most of the material can be generalized to (𝑞 − 1)-dimensional spheres Ω𝑞−1 embedded in the Euclidean space R𝑞 (for 𝑞 ≥ 3 see, e.g., C. M¨uller (1966, 1998), W. Freeden, M. Gutting (2013)). Nonetheless, as a preparatory context and for the sake of completeness, we give a brief overview on the case involving Ω1𝑅 = {𝑥 ∈ R2 | |𝑥| = 𝑅},

(3.1)

i.e., the circle of radius 𝑅 > 0 around the origin in R2 . As before, Ω𝑅 = Ω2𝑅 still denotes the sphere of radius 𝑅 > 0 in R3 ; Ω and Ω1 denote the unit sphere in R3 and the unit circle in R2 , respectively. The disc with radius 𝑅 > 0 and center 𝑥 in R2 is denoted by 2 Ω1;int (3.2) 𝑅 (𝑥) = {𝑦 ∈ R | |𝑥 − 𝑦| < 𝑅}, while the ball with radius 𝑅 > 0 and center 𝑥 in R3 is still denoted by Ωint 𝑅 , so that int Ω𝑅 = 𝜕Ω𝑅 . If the radius 𝑅 = 1, we also write Ω1 = {𝑥 ∈ R2 | |𝑥| = 1}, 1;int

Ω

2

(𝑥) = {𝑦 ∈ R | |𝑥 − 𝑦| < 1}.

© The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2022 W. Freeden and M. Schreiner, Spherical Functions of Mathematical Geosciences, Geosystems Mathematics, https://doi.org/10.1007/978-3-662-65692-1_3

(3.3) (3.4)

71

72

3 Circular Harmonics

3.1 Homogeneous Harmonic Polynomials and Circular Harmonics

Actually, the bivariate case mainly is a more abstract reformulation of well-known results for the trigonometric sine and cosine functions. Definition 3.1 Let 𝐻𝑛 : R2 → R be a homogeneous and harmonic polynomial of degree 𝑛 ∈ N0 , i.e., 𝐻𝑛 (𝜆𝑥) = 𝜆 𝑛 𝐻𝑛 (𝑥), 𝜆 ∈ R, and Δ𝐻𝑛 (𝑥) = 0, 𝑥 ∈ R2 . Then, the restriction 𝑌𝑛 (2; ·) = 𝐻𝑛 |Ω1 (3.5) is called a (scalar) circular harmonic (of degree 𝑛). The space of all circular harmonics of degree 𝑛 is denoted by Harm𝑛 (Ω1 ) (note that the notation 𝑌𝑛 (2; ·) is
used to distinguish the case 𝑞 = 2 from the case 𝑞 = 3, where we use 𝑌𝑛 = 𝑌𝑛 (3; ·) ). The circular harmonics of degree 𝑛 form a linear space of dimension  1, 𝑛 = 0, dim(Harm𝑛 (Ω1 )) = 2, 𝑛 ≥ 1.

(3.6)

3.2 Eigenfunctions of the Beltrami Operator Similar to the spherical harmonics of higher dimensions, any 𝑌𝑛 (2; ·) ∈ Harm𝑛 (Ω1 ) is an infinitely often differentiable eigenfunction of the Beltrami operator Δ∗ . More precisely, Δ∗𝑌𝑛 (2; 𝜉) = −𝑛2 𝑌𝑛 (2; 𝜉), 𝜉 ∈ Ω1 . (3.7) Note that, for the two-dimensional case, the Beltrami operator Δ∗ reads Δ=

1 𝜕 𝜕 1 𝑟 + Δ∗ , 𝑟 𝜕𝑟 𝜕𝑟 𝑟 2

(3.8)

where Δ denotes the Laplace operator in R2 . Using a representation in polar coordinates (i.e., 𝑥 = 𝑟𝜉 = (𝑟 cos(𝜏), 𝑟 sin(𝜏)) 𝑇 , 𝑟 > 0, 𝜏 ∈ [0, 2𝜋)), for any vector 𝑥 ∈ R2 \ {0}) the Beltrami operator is given by Δ∗ =



𝜕 𝜕𝜏

2 .

(3.9)

Circular harmonics of different degrees are orthogonal with respect to the L2 (Ω1 )inner product, i.e.,

3.3 Addition Theorem and Chebyshev Polynomial

73

∫ (𝑌𝑛 (2; ·), 𝑌𝑚 (2; ·)) L2 (Ω1 ) =

𝑌𝑛 (2; 𝜂)𝑌𝑚 (2; 𝜂) 𝑑𝜎(𝜂) = 0,

𝑛 ≠ 𝑚,

(3.10)

Ω1

where 𝑑𝜎 is the line element.

3.3 Addition Theorem and Chebyshev Polynomial A set {𝑌𝑛,𝑘 (2; ·)} 𝑘=1,2 ⊂ Harm𝑛 (Ω1 ) always denotes an orthonormal basis of Harm𝑛 (Ω1 ), 𝑛 ∈ N, with respect to the L2 (Ω1 )-inner product. For 𝑛 = 0, we 1 set 𝑌0,1 (2; 𝜉) = (2𝜋) − 2 , 𝜉 ∈ Ω1 . It can be easily seen that a possible orthonormal basis of Harm𝑛 (Ω1 ) is given by 1 𝑌𝑛,1 (2; 𝜉) = √ cos(𝑛𝜏), 𝜋

1 𝑌𝑛,2 (2; 𝜉) = √ sin(𝑛𝜏), 𝜋

(3.11)

for 𝑛 ∈ N, 𝜉 = (cos(𝜏), sin(𝜏)) 𝑇 and 𝜏 ∈ [0, 2𝜋).

For the case 𝑞 = 2, the Chebyshev polynomials take over the role of the Legendre polynomials 𝑃𝑛 (= 𝑃𝑛 (3; ·)) (for more details about Chebyshev polynomials, see, e.g., N.N. Lebedev (1973), and G. Szeg¨o (1967)). Definition 3.2 A polynomial 𝑃𝑛 (2; ·) : [−1, 1] → R of degree 𝑛 ∈ N0 is called Chebyshev polynomial (of degree 𝑛) if it satisfies ∫

1

𝑃𝑛 (2; 𝑡)𝑃𝑚 (2; 𝑡) √

(i) −1

1 1 − 𝑡2

𝑑𝑡 = 0,

𝑛 ≠ 𝑚,

(ii) 𝑃𝑛 (2; 1) = 1. The Chebyshev polynomials are uniquely determined by Definition 3.2 and have the explicit representation 𝑛

⌊2⌋ 𝑛 ∑︁ (𝑛 − 1 − 𝑘)! 𝑃𝑛 (2; 𝑡) = (−1) 𝑘 2𝑘−𝑛 𝑡 𝑛−2𝑘 = cos(𝑛 arccos(𝑡)), 2 𝑘=0 2 (𝑛 − 2𝑘)!𝑘!

for 𝑡 ∈ [−1, 1]. Furthermore, they satisfy the differential equation √︁  𝑑 𝑑 1 − 𝑡 2 (1 − 𝑡 2 ) 𝑃𝑛 (2; 𝑡) = −𝑛2 𝑃𝑛 (2; 𝑡), 𝑡 ∈ [−1, 1]. 𝑑𝑡 𝑑𝑡

(3.12)

(3.13)

74

3 Circular Harmonics

Similarly as for the Legendre polynomials, we have the following addition theorem connecting the Chebyshev polynomials to the circular harmonics. Theorem 3.3 (Addition Theorem) For 𝑛 ∈ N, we have 2 ∑︁

𝑌𝑛,𝑘 (2; 𝜉)𝑌𝑛,𝑘 (2; 𝜂) =

1 𝑃𝑛 (2; 𝜉 · 𝜂), 𝜋

𝜉, 𝜂 ∈ Ω1 .

(3.14)

𝑘=1

Theorem 3.3 is just an abstract version of well-known trigonometric addition theorems like cos(𝑛𝜏) cos(𝑛𝜗) + sin(𝑛𝜏) sin(𝑛𝜗) = cos(𝑛(𝜏 − 𝜗)). (3.15) Corollary 3.4 For 𝑛 ∈ N0 , we have |𝑃𝑛 (2; 𝑡)| ≤ 1,

𝑡 ∈ [−1, 1].

(3.16)

For 𝑛 ∈ N, 𝑘 = 1, 2, we obtain 1 |𝑌𝑛,𝑘 (2; 𝜉)| ≤ √ , 𝜋

𝜉 ∈ Ω1 .

(3.17)

It is possible to derive closed representations for different generating series expansion of the Chebyshev polynomials. Lemma 3.5 For 𝑡 ∈ [−1, 1] and ℎ ∈ (−1, 1), we have ∞ ∑︁ 𝑛=0 ∞ ∑︁ 𝑛=1

𝑃𝑛 (2; 𝑡)ℎ 𝑛 =

1 − ℎ𝑡 , 1 + ℎ2 − 2ℎ𝑡

1 1 𝑃𝑛 (2; 𝑡)ℎ 𝑛 = − ln(1 + ℎ2 − 2ℎ𝑡). 𝑛 2

(3.18) (3.19)

While the identity (3.18) is the canonical counterpart to that formulated later in Lemma 4.37, Equation (3.19) states the relevant expression to achieve a multipole expansion of the fundamental solution for the Laplace operator in R2 . More precisely, we obtain

3.4 Stereographic Projection

75

2 |𝑥| |𝑥| ln(|𝑥 − 𝑦|) = ln(|𝑦|) + ln 1 + − 2 (𝜉 · 𝜂) |𝑦| |𝑦|  𝑛 ∞ ∑︁ 2 |𝑥| = ln(|𝑦|) − 𝑃𝑛 (2; 𝜉 · 𝜂), 𝑛 |𝑦| 𝑛=1 

for 𝑥, 𝑦 ∈ R2 , |𝑥| < |𝑦|, and 𝜉 =

𝑥 |𝑥| ,

𝜂=

! (3.20)

𝑦 |𝑦| .

3.4 Stereographic Projection

The stereographic projection is a frequently used mapping to transform problems on the sphere to counterparts in the two-dimensional plane, and vice versa. For more information on such projections, the reader is, e.g., referred to P.W. MacDonell (1979). Definition 3.6 The mapping 𝑝 stereo : Ω \ {−𝜀 3 } → R2 defined by 

2𝜉1 2𝜉2 𝑝 stereo (𝜉) = , 1 + 𝜉3 1 + 𝜉3

𝑇 ,

𝜉 ∈ Ω \ {−𝜀 3 },

(3.21)

where 𝜉 = (𝜉1 , 𝜉2 , 𝜉3 ) 𝑇 , is called stereographic projection (with respect to 𝜀 3 ). Its −1 inverse 𝑝 stereo : R2 → Ω \ {−𝜀 3 } is given by −1 𝑝 stereo (𝑥)

4𝑥 1 4𝑥 2 4 − |𝑥| 2 = , , 4 + |𝑥| 2 4 + |𝑥| 2 4 + |𝑥| 2 

𝑇 ,

𝑥 ∈ R2 ,

(3.22)

where 𝑥 = (𝑥1 , 𝑥2 ) 𝑇 . Let now Γ ⊂ Ω \ {−𝜖 3 } be a region and 𝐵 = 𝑝 stereo (Γ) ⊂ R2 its image in the plane, then we define the mapping 𝑃stereo : C (0) (Γ) → C (0) (𝐵) by  𝑃stereo [𝐹] (𝑥) = 𝐹

4𝑥 1 4𝑥2 4 − |𝑥| 2 , , 2 2 4 + |𝑥| 4 + |𝑥| 4 + |𝑥| 2

𝑇 

−1 = 𝐹 ( 𝑝 stereo (𝑥)),

𝑥 ∈ 𝐵, (3.23)

−1 for 𝐹 of class C (0) (Γ), and its inverse Pstereo : C (0) (𝐵) → C (0) (Γ) by −1 𝑃stereo [𝐹] (𝜉) = 𝐹

for 𝐹 of class C (0) (𝐵).



2𝜉1 2𝜉2 , 1 + 𝜉3 1 + 𝜉3

𝑇  = 𝐹 ( 𝑝 stereo (𝜉)) ,

𝜉 ∈ Γ,

(3.24)

76

3 Circular Harmonics 𝑝stereo ( 𝜁 )

𝑝stereo ( 𝜂)

𝑝stereo ( 𝜀 3 )

𝑝stereo ( 𝜉 )

𝜂

R2

𝜉

𝜁 Ω

− 𝜀3

Fig. 3.1: Two-dimensional illustration of the stereographic projection.

For brevity, we introduced the stereographic projection only with respect to 𝜀 3 , and we mostly use this projection for explicit calculations. However, if t ∈ R3×3 is the rotation matrix with t𝜀 3 = 𝜉˜ for a 𝜉˜ ∈ Ω, then the stereographic projection (with ˜ is defined by (cf. Fig. 3.1) respect to 𝜉) 2𝜉 · (t𝜖 1 ) 2𝜉 · (t𝜀 2 ) 𝑝 stereo (𝜉) = , 1 + 𝜉 · 𝜉˜ 1 + 𝜉 · 𝜉˜ 

𝑇 ,

˜ 𝜉 ∈ \{−𝜉}.

(3.25)

𝜉˜ An index 𝜉˜ indicating the dependence 𝑝 stereo = 𝑝 stereo is generally omitted. The −1 , 𝑃 −1 mappings 𝑝 stereo stereo , and 𝑃stereo can be generalized analogously.

˜ be a regular Properties Involving the Stereographic Projection. Let Γ ⊂ Ω \ {−𝜉} 2 region and G = 𝑝 stereo (Γ) ⊂ R its image in the plane. Then the stereographic projection 𝑝 stereo [𝐹] : Γ → R of a function 𝐹 : G → R is defined by 𝑝 stereo [𝐹] (𝜉) = 𝐹 ( 𝑝 stereo (𝜉)),

𝜉 ∈ Γ.

(3.26)

An interesting connection between the Laplace operator and the Beltrami operator is given by the following lemma (see, e.g., W. Freeden, C. Gerhards (2013)). Lemma 3.7 Let Γ ⊂ Ω \ {−𝜂}, with 𝜉˜ ∈ Ω fixed, be a regular region, and G = 𝑝 stereo (Γ) ⊂ R2 its image in the plane. If 𝐹 is of class C (2) (G), then (1 + 𝜉 · 𝜂) 2 ∗ Δ 𝑝 stereo [𝐹] (𝜉) = Δ𝐹 (𝑥)| 𝑥= 𝑝stereo ( 𝜉 ) , 4

𝜉 ∈ Γ.

(3.27)

3.5 Closure and Completeness of Circular Harmonics

77

3.5 Closure and Completeness of Circular Harmonics

In what follows we deal with the closure and completeness of the circular harmonics in C (0) (Ω1 ) as well as L2 (Ω1 ). Theorem 3.8 The system of circular harmonics {𝑌0,1 (2; ·)} ⊕ {𝑌𝑛,𝑘 (2; ·)} 𝑛=1,2,...,𝑘=1,2

(3.28)

is closed in C (0) (Ω1 ) (with respect to ∥ · ∥ C (0) (Ω1 ) and ∥ · ∥ L2 (Ω1 ) ) and in L2 (Ω1 ) (with respect to ∥ · ∥ L2 (Ω1 ) )). Lemma 3.7 implies that the stereographic projection of a function that is harmonic with respect to the Laplace operator is harmonic with respect to the Beltrami operator. This observation allows us to transfer some results for harmonic functions from the Euclidean setting in the plane R2 to the sphere Ω. Definition 3.9 A circular harmonic (of degree 𝑛 and order 𝑘) on the boundary of a ˜ ⊂ Ω with radius 𝜌 ∈ (0, 2) and center 𝜉 ∈ Ω is defined as spherical cap Γ𝜌 ( 𝜉𝜂)  1  1 𝜌,stereo 𝜌 4 (2−𝜌) 4 𝑌𝑛,𝑘 (𝜉) = 𝑝 stereo 𝑌𝑛,𝑘 (2; ·) (𝜉), 𝜉 ∈ 𝜕Γ𝜌 (𝜂). (3.29) 𝑥 𝑅 (2; 𝑥) = 𝑅 −1𝑌 1 By 𝑌𝑛,𝑘 𝑛,𝑘 (2; | 𝑥 | ), 𝑥 ∈ Ω 𝑅 , we denote a set of orthonormalized circular harmonics on the circle of radius 𝑅 > 0.

𝜌,stereo

Remark 3.10 The functions 𝑌𝑛,𝑘 form an orthonormal system with respect to the L2 (𝜕Γ𝜌 (𝜂))-inner product. Furthermore, observing that the stereographic projection does not affect the longitude of a point on the sphere Ω (at least for the choice 𝜂 = 𝜀 3 ), 1

𝜌,stereo

1

𝜌 4 (2−𝜌) 4

(2; ·) have the same representation when it can be seen that 𝑌𝑛,𝑘 and 𝑌𝑛,𝑘 using spherical coordinates and polar coordinates, respectively. More precisely, for 𝑛 ∈ N, 𝜌,stereo

𝑌𝑛,1

(𝜉) =

cos(𝑛𝜏) 1 2

1 4

𝜋 𝜌 (2 − 𝜌) 1

1

𝜌,stereo 1 4 1

,

𝑌𝑛,2 1

sin(𝑛𝜏)

(𝜉) =

1 2

1

1

,

(3.30)

𝜋 𝜌 4 (2 − 𝜌) 4

where 𝜉 = 𝜌 2 (2 − 𝜌) 2 cos(𝜏), 𝜌 2 (2 − 𝜌) 2 sin(𝜏), 1 − 𝜌


𝑇

∈ 𝜕Γ𝜌 (𝜀 3 ), 𝜏 ∈ [0, 2𝜋).

78

3 Circular Harmonics

3.6 Inner/Outer Harmonics Next we deal with the bivariate case, i.e., the Euclidean space R2 (see, e.g., W. Freeden, C. Gerhards (2013) for more details). Here, inner and outer harmonics can be defined in accordance with Definition 3.11. Only for the outer harmonics there occurs a minor adaptation due to the reduced dimension. Definition 3.11 (Inner/Outer Harmonics) (i) The functions  𝑅 𝐻𝑛,𝑘 (2; 𝑥) =

|𝑥| 𝑅

𝑛 𝑅 (2; 𝑥) , 𝑌𝑛,𝑘

𝑥 ∈ R2 ,

(3.31)

for 𝑛 ∈ N, 𝑘 = 1, 2, or 𝑛 = 0, 𝑘 = 1, are called circular inner harmonics (of degree 𝑛 and order 𝑘). (ii) The functions 

𝑅 𝐻 −𝑛,𝑘 (2; 𝑥)

𝑅 = |𝑥|

𝑛 𝑅 (2; 𝑥) , 𝑌𝑛,𝑘

𝑥 ∈ R2 \{0},

(3.32)

for 𝑛 ∈ N, 𝑘 = 1, 2, or 𝑛 = 0, 𝑘 = 1, are called circular outer harmonics (of degree 𝑛 and order 𝑘). Remark 3.12 Similar as for the already known versions of inner harmonics, it can easily be seen that the following properties hold true: (i) (ii) (iii) (iv)

𝑅 (2; ·) is of class C (∞) (R2 ), 𝐻𝑛,𝑘 𝑅 (2; ·) satisfies Δ𝐻 𝑅 (2; 𝑥) = 0, 𝑥 ∈ R2 , 𝐻𝑛,𝑘 𝑛,𝑘 𝑅 (2; ·) Ω1 = 𝑌 𝑅 (2; ·), 𝐻𝑛,𝑘 𝑅 𝑛,𝑘 𝑅 (2; ·), 𝐻 𝑅 (2; ·)) (𝐻𝑛,𝑘 = 𝛿 𝑛 𝑝 𝛿 𝑘𝑞 . 𝑝,𝑞 L2 (Ω1 ) 𝑅

For the outer circular harmonics, we obtain the following properties: (i) (ii) (iii) (iv) (v)

𝑅 𝐻 −𝑛,𝑘 (2; ·) is of class C (∞) (R2 \{0}), 𝑅 𝑅 𝐻 −𝑛,𝑘 (2; ·) satisfies Δ𝐻 −𝑛,𝑘 (2; 𝑥) = 0, 𝑥 ∈ R2 \{0}, 𝑅 𝑅 𝐻 −𝑛,𝑘 (2; ·) is regular at infinity, i.e., 𝐻 −𝑛−1 (𝑥) = 𝑂 (ln |𝑥|), |𝑥| → ∞, 𝑅 1 𝑅 𝐻 −𝑛,𝑘 (2; ·) Ω𝑅 = 𝑌𝑛,𝑘 (2; ·), 𝑅 (𝐻 −𝑛,𝑘 (2; ·), 𝐻 −𝑅𝑝,𝑞 (2; ·)) L2 (Ω1 ) = 𝛿 𝑛 𝑝 𝛿 𝑘𝑞 . 𝑅

3.6 Inner/Outer Harmonics

79

For later considerations we are not interested in the circular inner and outer harmonics in the plane R2 but their spherical counterparts on the sphere Ω𝑅 . Of course, this can be achieved by stereographic projection. Definition 3.13 (Inner/Outer Harmonics on the Sphere) The following settings are related to the spherical cap Γ𝜌 (𝜂) with radius 𝜌 ∈ (0, 2) and center 𝜂 ∈ Ω. (i) The functions 𝜌,stereo 𝐻𝑛,𝑘 (𝜉)



=

1

1

𝜌 4 (2−𝜌) 4 𝑝 stereo 𝐻𝑛,𝑘

 (2; ·) (𝜉) ,

𝜉 ∈ Ω \ {−𝜂},

(3.33)

for 𝑛 ∈ N, 𝑘 = 1, 2, or 𝑛 = 0, 𝑘 = 1, are called inner harmonics (of degree 𝑛 and order 𝑘) with respect to the spherical cap Γ𝜌 (𝜂) . (ii) The functions  1  1 𝜌,stereo 𝜌 4 (2−𝜌) 4 𝐻 −𝑛,𝑘 (𝜉) = 𝑝 stereo 𝐻 −𝑛,𝑘 (2; ·) (𝜉) ,

𝜉 ∈ Ω \ {𝜂},

(3.34)

for 𝑛 ∈ N, 𝑘 = 1, 2, or 𝑛 = 0, 𝑘 = 1, are called outer harmonics (of degree 𝑛 and order 𝑘) with respect to the spherical cap Γ𝜌 (𝜂) . Since the stereographic projection depends on the center 𝜂 of a spherical cap, the inner and outer harmonics depend on the choice of 𝜂, too. For brevity, however, we usually do not indicate this in our notation. As expected, the inner harmonics with respect to the cap Γ𝜌 (𝜂) satisfy the properties: (i) (ii) (iii) (iv)

𝐻𝑛,𝑘 is of class C (∞) (Ω \ {−𝜂}), 𝜌,stereo 𝜌,stereo 𝐻𝑛,𝑘 satisfies Δ∗ 𝐻𝑛,𝑘 (𝜉) = 0, 𝜉 ∈ Ω \ {−𝜂}, 𝜌,stereo 𝜌,stereo 𝐻𝑛,𝑘 𝜕Γ𝜌 (𝜂) = 𝑌𝑛,𝑘 , 𝜌,stereo 𝜌,stereo (𝐻𝑛,𝑘 , 𝐻 𝑝,𝑞 ) 2 = 𝛿 𝑛 𝑝 𝛿 𝑘𝑞 . 𝜌,stereo

L (𝜕Γ𝜌 ( 𝜂) )

The properties for the outer harmonics with respect to the cap Γ𝜌 (𝜂), however, show slightly modified properties in comparison to the Euclidean cases (cf. W. Freeden, C. Gerhards (2013)):

80

(i) (ii) (iii) (iv)

3 Circular Harmonics

𝐻 −𝑛,𝑘 is of class C (∞) (Ω \ {𝜂}), 𝜌,stereo 𝜌,stereo 𝐻 −𝑛,𝑘 satisfies Δ∗ 𝐻 −𝑛,𝑘 (𝜉) = 0, 𝜉 ∈ Ω \ {𝜂}, 𝜌,stereo 𝜌,stereo 𝐻 −𝑛,𝑘 𝜕Γ𝜌 (𝜂) = 𝑌𝑛,𝑘 , 𝜌,stereo 𝜌,stereo (𝐻 −𝑛,𝑘 , 𝐻 − 𝑝,𝑞 ) 2 = 𝛿 𝑛 𝑝 𝛿 𝑘𝑞 . 𝜌,stereo

L (𝜕Γ𝜌 ( 𝜂) )

While there is an actual difference between inner and outer harmonics in the Euclidean setting of R2 and R3 , due to the fact that the exterior of a regular region is unbounded, this is not true in the spherical framework. As a consequence, the 𝜌,stereo inner harmonics 𝐻𝑛,𝑘 for the spherical cap Γ𝜌 (𝜂) coincide with outer harmonics 2−𝜌,stereo

𝐻 −𝑛,𝑘

for the spherical cap Γ2−𝜌 (−𝜂).

Of special interest is the multipole representation (3.20). The addition theorem implies ln(|𝑥 − 𝑦|) = ln(|𝑦|) − 𝜋𝑅 2

∞ ∑︁ 2 ∑︁ 2

𝑛

𝑅 𝑅 𝐻𝑛,𝑘 (2; 𝑥)𝐻 −𝑛,𝑘 (2; 𝑦),

(3.35)

𝑛=1 𝑘=1

for 𝑥, 𝑦 ∈ R2 , |𝑥| < |𝑦|, and some fixed 𝑅 > 0. Applying the stereographic projection to (3.35), we obtain the following spherical version. Lemma 3.14 Suppose that 𝜂 ∈ Ω. Then, for 𝜌 ∈ (0, 2), we have ln(1 − 𝜉 · 𝜁) = − ln(2) + ln(1 + 𝜉 · 𝜂) + ln(1 − 𝜁 · 𝜂) −

√︁

𝜌(2 − 𝜌)𝜋

∞ ∑︁ 2 ∑︁ 2

𝑛

𝜌,stereo

𝐻𝑛,𝑘

(3.36) 𝜌,stereo

(𝜉)𝐻 −𝑛,𝑘 (𝜁),

𝑛=1 𝑘=1

for 𝜉 ∈ Ω \ {−𝜂}, 𝜂 ∈ Ω \ {𝜂}, and | 𝑝 stereo (𝜉)| < | 𝑝 stereo (𝜁)|.

Part III

Spherical Harmonics

Chapter 4

Scalar Spherical Harmonics

In this chapter we develop polynomial function systems for scalar-valued functions on spheres in the three-dimensional Euclidean space, namely the scalar spherical harmonics. Scalar spherical harmonics are essential for any analysis of spherical functions. The main features are the addition theorem, the Funk-Hecke formula, and the orthogonal invariance leading to expressions in the terms of Legendre polynomials. The scalar spherical harmonics also provide the foundation for vector spherical harmonics (see Chapter 6) as well as tensor spherical harmonics (see Chapter 7). Our approach is essentially based on the work due to C. M¨uller (1952, 1966, 1998) and W. Freeden (1979, 1981b), W. Freeden, M. Gutting (2013). In fact, it is led by the observation (see H. Weyl (1934, 1946, 1965)) that spherical harmonics must be more than a fortunate guess in Fourier (orthogonal) expansions for providing tables of potential coefficients for geophysical quantities. This opinion arose from the occupation with theoretical physics (in particular, gravitational and geomagnetic theory, quantum mechanics, and general relativity) and was supported by many physicists during the last century. Today, even problems in medicine, e.g., the electroencephalographic description of scalp potential fields, can be tackled appropriately in terms of spherical harmonics. Our strategy to handle scalar spherical harmonics is as follows: The scalar spherical harmonics are introduced as the restrictions of the homogeneous harmonic polynomials to the unit sphere. In consequence, the addition theorem of homogeneous harmonic polynomials canonically goes over to the theory of scalar spherical harmonics. Maxwell’s representation formula shows that the (one-dimensional) Legendre polynomials may be obtained by repeated differentiation of the fundamental solutions of the Laplace operator. The closure and completeness of orthonormal systems in the space L2 (Ω) is fundamental for approximating square-integrable functions on the sphere by Fourier (spherical harmonic) expansions. The closure in L2 (Ω) can be derived from Bernstein or Abel-Poisson summability. The Funk-Hecke © The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2022 W. Freeden and M. Schreiner, Spherical Functions of Mathematical Geosciences, Geosystems Mathematics, https://doi.org/10.1007/978-3-662-65692-1_4

83

84

4 Scalar Spherical Harmonics

formula establishes the close connection between the orthogonal invariance of the sphere and the addition theorem. It turns out that any spherical harmonic may be intrinsically defined on the sphere as an eigenfunction of the Beltrami operator. The angular derivatives, i.e., the operators of the longitude and latitude, are shown to act as anisotropic operators within the framework of scalar spherical harmonics. Finally, the usually (in geosciences) used L2 (Ω)-orthonormal system of scalar spherical harmonics involving associated Legendre functions is introduced; its (real as well as complex) representation in terms of trigonometric functions is worth to be discussed in more detail. Associated Legendre harmonics are generated exactly entirely by integer operations.

4.1 Homogeneous Harmonic Polynomials Let Hom𝑛 (more accurately: Hom𝑛 (R3 )) consist of all polynomials 𝐻𝑛 in three variables which are homogeneous of degree 𝑛 (i.e., 𝐻𝑛 (𝜆𝑥) = 𝜆 𝑛 𝐻𝑛 (𝑥) for all 𝜆 ∈ R and all 𝑥 ∈ R3 ). Thus, if 𝐻𝑛 ∈ Hom𝑛 , then there exist real numbers 𝐶 𝛼 = 𝐶 𝛼1 𝛼2 𝛼3 such that ∑︁ 𝐻𝑛 (𝑥) = 𝐶𝛼 𝑥 𝛼. (4.1) [ 𝛼]=𝑛

In Cartesian coordinates, ∑︁

𝐻𝑛 (𝑥1 , 𝑥2 , 𝑥3 ) =

𝐶 𝛼1 𝛼2 𝛼3 𝑥1𝛼1 𝑥2𝛼2 𝑥3𝛼3 .

(4.2)

𝛼1 +𝛼2 +𝛼3 =𝑛

It is obvious that the set of monomials 𝑥 ↦→ 𝑥 𝛼 , [𝛼] = 𝛼1 + 𝛼2 + 𝛼3 = 𝑛, is a basis for the space Hom𝑛 . The number of such monomials is precisely the number of ways a triple can be chosen so that we have [𝛼] = 𝑛, i.e., the number of ways of selecting 2 elements out of a collection of 𝑛 + 2. This means that the dimension 𝑑 (Hom𝑛 ) of Hom𝑛 is equal to   (𝑛 + 1)(𝑛 + 2) 𝑛+2 𝑑 (Hom𝑛 ) = = . (4.3) 2 2 Let 𝐻𝑛 (∇ 𝑥 ) be the differential operator associated to 𝐻𝑛 (𝑥) (i.e., replace 𝑥 𝛼 formally by (∇ 𝑥 ) 𝛼 in the expression of 𝐻𝑛 (𝑥)): 𝐻𝑛 (∇ 𝑥 ) =

∑︁ 𝛼1 +𝛼2 +𝛼3 =𝑛

𝐶 𝛼1 𝛼2 𝛼3

∑︁ 𝜕 [ 𝛼] = 𝐶 𝛼 (∇ 𝑥 ) 𝛼 . 𝜕𝑥 1𝛼1 𝜕𝑥2𝛼2 𝜕𝑥3𝛼3 [ 𝛼]=𝑛

(4.4)

4.1 Homogeneous Harmonic Polynomials

85

If such an operator is applied to a homogeneous polynomial 𝑈𝑛 of the same degree ∑︁ 𝑈𝑛 (𝑥) = 𝐷𝛽 𝑥𝛽, (4.5) [𝛽 ]=𝑛

we obtain as result a real number (𝐻𝑛 (∇ 𝑥 )) 𝑈𝑛 (𝑥)   𝛼1   𝛼2   𝛼3 ∑︁ ∑︁ 𝜕 𝜕 𝜕 𝛽1 𝛽2 𝛽 = 𝐶𝛼 𝐷 𝛽 𝑥1 𝑥2 𝑥3 3 𝜕𝑥1 𝜕𝑥2 𝜕𝑥 3 [ 𝛼]=𝑛 [𝛽 ]=𝑛 ∑︁ = 𝐶 𝛼 𝐷 𝛼 𝛼! ,

(4.6)

[ 𝛼]=𝑛

where the factorial of a multi-index is defined as 𝛼! = 𝛼1 !𝛼2 !𝛼3 !. Clearly, we find (𝐻𝑛 (∇ 𝑥 )) 𝑈𝑛 (𝑥) = (𝑈𝑛 (∇ 𝑥 )) 𝐻𝑛 (𝑥),

(4.7)

(𝐻𝑛 (∇ 𝑥 )) 𝐻𝑛 (𝑥) ≥ 0.

(4.8)

This enables us to introduce an inner product (·, ·)Hom𝑛 on the space Hom𝑛 by letting (𝐻𝑛 , 𝑈𝑛 )Hom𝑛 = (𝐻𝑛 (∇ 𝑥 )) 𝑈𝑛 (𝑥).

(4.9)

The space Hom𝑛 equipped with the inner product (·, ·)Hom𝑛 is a finite-dimensional Hilbert space. The set of monomials {𝑥 ↦→ (𝛼!) −1/2 𝑥 𝛼 | [𝛼] = 𝑛} forms an orthonormal system in the space Hom𝑛 . For each 𝐻𝑛 ∈ Hom𝑛 , we have in connection with (4.4) ∑︁ 1 (𝐻𝑛 (∇ 𝑦 )) 𝑦 𝛼 𝑥 𝛼 𝛼! [ 𝛼]=𝑛 1 ∑︁ 𝑛! 𝛼 𝛼 = (𝐻𝑛 (∇ 𝑦 )) 𝑥 𝑦 𝑛! 𝛼!

𝐻𝑛 (𝑥) =

(4.10)

[ 𝛼]=𝑛

(𝑥 · 𝑦) 𝑛 = (𝐻𝑛 (∇ 𝑦 )) 𝑛! 1 𝑛 = (𝑥 · ∇ 𝑦 ) 𝐻𝑛 (𝑦). 𝑛! In other words, 𝐻𝑛 (𝑥) = (

(𝑥· ) 𝑛 , 𝐻𝑛 )Hom𝑛 . 𝑛!

(4.11)

Theorem 4.1 Hom𝑛 equipped with the inner product (·, ·)Hom𝑛 is a finite-dimensional Hilbert space of dimension (𝑛+1)2(𝑛+2) with the reproducing kernel

86

4 Scalar Spherical Harmonics

𝐾Hom𝑛 (𝑥, 𝑦) =

(𝑥 · 𝑦) 𝑛 , 𝑛!

𝑥, 𝑦 ∈ R3 ,

(4.12)

i.e., (i) for every fixed y, the function 𝐾Hom𝑛 (·, 𝑦) belongs to Hom𝑛 , (ii) for any 𝐻𝑛 ∈ Hom𝑛 and any point 𝑥 the reproducing property 𝐻𝑛 (𝑥) = (𝐾Hom𝑛 (𝑥, ·), 𝐻𝑛 )Hom𝑛 is valid. Let {𝐻𝑛, 𝑗 } 𝑗=1,...,𝑑 (Hom𝑛 ) , {𝑈𝑛, 𝑗 } 𝑗=1,...,𝑑 (Hom𝑛 ) be two orthonormal systems in the space Hom𝑛 : (𝐻𝑛, 𝑗 , 𝐻𝑛,𝑘 )Hom𝑛 = 𝛿 𝑗 𝑘 ,

(4.13)

(𝑈𝑛, 𝑗 , 𝑈𝑛,𝑘 )Hom𝑛 = 𝛿 𝑗 𝑘 ,

(4.14)

where 𝛿 𝑗 𝑘 is the usual Kronecker symbol. Then, for 𝑗 = 1, ..., 𝑑 (Hom𝑛 ), we have 𝐻𝑛, 𝑗 =

𝑑 (Hom ∑︁ 𝑛 )

(𝐻𝑛, 𝑗 , 𝑈𝑛,𝑘 )Hom𝑛 𝑈𝑛,𝑘 ,

(4.15)

(𝑈𝑛, 𝑗 , 𝐻𝑛,𝑘 )Hom𝑛 𝐻𝑛,𝑘 .

(4.16)

𝑘=1

𝑈𝑛, 𝑗 =

𝑑 (Hom ∑︁ 𝑛 ) 𝑘=1

Therefore, it follows that 𝑑 (Hom ∑︁ 𝑛 )

𝐻𝑛, 𝑗 (𝑥) 𝐻𝑛, 𝑗 (𝑦) =

𝑗=1

𝑑 (Hom ∑︁ 𝑛 )

𝑈𝑛, 𝑗 (𝑥)𝑈𝑛, 𝑗 (𝑦).

(4.17)

𝑗=1

Hence, in particular, for the orthonormal system of monomials, we obtain the following result. Theorem 4.2 Let {𝐻𝑛, 𝑗 } 𝑗=1,...,𝑑 (Hom𝑛 ) be an orthonormal system in Hom𝑛 . Then 𝐾Hom𝑛 (𝑥, 𝑦) =

(𝑥 · 𝑦) 𝑛 = 𝑛!

𝑑 (Hom ∑︁ 𝑛 )

𝐻𝑛, 𝑗 (𝑥)𝐻𝑛, 𝑗 (𝑦),

𝑥, 𝑦 ∈ R3 .

(4.18)

𝑗=1

𝐾Hom𝑛 (·, ·) is the only reproducing kernel in Hom𝑛 . Suppose that there are given 𝑑 (Hom𝑛 ) points 𝑥 1 , ..., 𝑥 𝑑 (Hom𝑛 ) ∈ R3 and 𝑑 (Hom𝑛 )values 𝑑1 , ..., 𝑑 𝑑 (Hom𝑛 ) ∈ R. We are able to solve the Hom𝑛 - interpolation problem

4.1 Homogeneous Harmonic Polynomials 𝑑 (Hom ∑︁ 𝑛 )

87

𝑏 𝑗 𝐻𝑛, 𝑗 (𝑥 𝑘 ) = 𝑑 𝑘 ,

𝑘 = 1, ..., 𝑑 (Hom𝑛 ),

(4.19)

𝑗=1

if and only if the matrix matr { 𝑥1 ,..., 𝑥𝑑 (Hom𝑛 ) } (𝐻𝑛,1 , . . . , 𝐻𝑛,𝑑 (Hom𝑛 ) ) 𝐻𝑛,1 (𝑥1 ) © .. ­ =­ . « 𝐻𝑛,𝑑 (Hom𝑛 ) (𝑥 1 )

(4.20)

... 𝐻𝑛,1 (𝑥 𝑑 (Hom𝑛 ) ) ª .. .. ® . . ® . . . 𝐻𝑛,𝑑 (Hom𝑛 ) (𝑥 𝑑 (Hom𝑛 ) ) ¬

is non-singular. A system of 𝑑 (Hom𝑛 ) points 𝑥 1 , ..., 𝑥 𝑑 (Hom𝑛 ) is called a fundamental system relative to Hom𝑛 if the matrix (4.20) is non-singular. In what follows, we guarantee the existence of a fundamental system relative to Hom𝑛 (cf. C. M¨uller (1966)). Lemma 4.3 There exists a system {𝑥1 , . . . , 𝑥 𝑑 (Hom𝑛 ) } ⊂ R3 such that (4.20) is nonsingular. Proof As orthonormal system, the functions 𝐻𝑛,1 , ..., 𝐻𝑛,𝑑 (Hom𝑛 ) are linearly independent. Hence, there exists a point 𝑥1 for which 𝐻𝑛,1 (𝑥1 ) ≠ 0. Now, there must also be a point 𝑥2 such that 𝐻𝑛,1 (𝑥1 ) 𝐻𝑛,1 (𝑥2 ) 𝐻𝑛,2 (𝑥1 ) 𝐻𝑛,2 (𝑥 2 ) ≠ 0,

(4.21)

(4.22)

for else, we would have a contradiction to the linear independence of 𝐻𝑛,1 , 𝐻𝑛,2 . In the same way, the existence of a point 𝑥3 can be deduced by the requirement 𝐻𝑛,1 (𝑥 1 ) 𝐻𝑛,1 (𝑥2 ) 𝐻𝑛,1 (𝑥3 ) 𝐻𝑛,2 (𝑥 1 ) 𝐻𝑛,2 (𝑥2 ) 𝐻𝑛,2 (𝑥3 ) ≠ 0. (4.23) 𝐻𝑛,3 (𝑥 1 ) 𝐻𝑛,3 (𝑥2 ) 𝐻𝑛,3 (𝑥3 ) Finally, by induction, we obtain a system of points 𝑥1 , ..., 𝑥 𝑑 (Hom𝑛 ) such that 𝐻𝑛,1 (𝑥1 ) ... 𝐻𝑛,1 (𝑥 𝑑 (Hom𝑛 ) ) .. . . . . (4.24) ≠ 0, . . . 𝐻𝑛,𝑑 (Hom ) (𝑥1 ) . . . 𝐻𝑛,𝑑 (Hom ) (𝑥 𝑑 (Hom ) ) 𝑛 𝑛 𝑛 i.e., {𝑥1 , . . . , 𝑥 𝑑 (Hom𝑛 ) } constitutes a fundamental system relative to Hom𝑛 .

□

88

4 Scalar Spherical Harmonics

To every 𝐻𝑛 ∈ Hom𝑛 , there exist real numbers 𝑏 1 , ..., 𝑏 𝑑 (Hom𝑛 ) such that 𝐻𝑛 =

𝑑 (Hom ∑︁ 𝑛 )

(4.25)

𝑏 𝑘 𝐻𝑛,𝑘 .

𝑘=1

Under the assumption that {𝑥1 , ..., 𝑥 𝑑 (Hom𝑛 ) } is a fundamental system relative to Hom𝑛 , the linear equations 𝑑 (Hom ∑︁ 𝑛 )

𝑎 𝑗 𝐻𝑛,𝑘 (𝑥 𝑗 ) = 𝑏 𝑘 ,

𝑘 = 1, ..., 𝑑 (Hom𝑛 ) ,

(4.26)

𝑗=1

are uniquely solvable in the unknowns 𝑎 1 , ..., 𝑎 𝑑 (Hom𝑛 ) . Thus, we obtain 𝐻𝑛 =

𝑑 (Hom ∑︁ 𝑛 ) 𝑑 (Hom ∑︁ 𝑛 ) 𝑘=1

𝑎 𝑗 𝐻𝑛,𝑘 (𝑥 𝑗 ) 𝐻𝑛,𝑘 .

(4.27)

𝑗=1

Theorem 4.4 Let {𝐻𝑛, 𝑗 } 𝑗=1,...,𝑑 (Hom𝑛 ) be an orthonormal system in Hom𝑛 . Assume that {𝑥 𝑘 } 𝑘=1,...,𝑑 (Hom𝑛 ) is a fundamental system relative to Hom𝑛 . Then, each member 𝐻𝑛 ∈ Hom𝑛 is uniquely representable in the form 𝐻𝑛 (𝑥) =

𝑑 (Hom ∑︁ 𝑛 )

𝑎 𝑗 𝐾Hom𝑛 (𝑥 𝑗 , 𝑥) =

𝑑 (Hom ∑︁ 𝑛 )

𝑗=1

𝑎𝑗

(𝑥 𝑗 · 𝑥) 𝑛 . 𝑛!

(4.28)

𝑗=1

Let Harm𝑛 (more accurately: Harm𝑛 (R3 )) be the class of all polynomials in Hom𝑛 that are harmonic: Harm𝑛 = {𝐻𝑛 ∈ Hom𝑛 | Δ 𝑥 𝐻𝑛 (𝑥) = 0, 𝑥 ∈ R3 }.

(4.29)

For 𝑛 < 2, of course, all homogeneous polynomials are harmonic. Any homogeneous harmonic polynomial of degree 𝑛 can be represented in the form 𝐻𝑛 (𝑥) = 𝐻𝑛 (𝑥1 , 𝑥2 , 𝑥3 ) =

𝑛 ∑︁

𝑗

𝑥3 𝐴𝑛− 𝑗 (𝑥 1 , 𝑥2 ),

(4.30)

𝑗=0

where 𝐴𝑛− 𝑗 is a homogeneous polynomial of degree 𝑛 − 𝑗 in the variables 𝑥 1 , 𝑥2 . Application of the Laplace operator gives

4.1 Homogeneous Harmonic Polynomials

89

2!

 2  2  𝑛 ∑︁ 𝜕 𝜕 𝜕 𝑗 0 = Δ𝐻𝑛 (𝑥) = + + 𝑥3 𝐴𝑛− 𝑗 (𝑥 1 , 𝑥2 ) 𝜕𝑥 𝜕𝑥 𝜕𝑥 1 2 3 𝑗=0 !     𝑛−2 2 2 ∑︁ 𝜕 𝜕 𝑗 = 𝑥3 + 𝐴𝑛− 𝑗 (𝑥1 , 𝑥2 ) 𝜕𝑥1 𝜕𝑥 2 𝑗=0 +

𝑛−2 ∑︁

𝑗

𝑥3 ( 𝑗 + 2)( 𝑗 + 1) 𝐴𝑛− 𝑗 −2 (𝑥1 , 𝑥2 ),

(4.31)

𝑗=0

where we have used the facts that  2 𝜕 + 𝜕𝑥1  2 𝜕 + 𝜕𝑥1





𝜕 𝜕𝑥2 𝜕 𝜕𝑥2

2! 𝐴0 (𝑥1 , 𝑥2 ) = 0,

(4.32)

𝐴1 (𝑥1 , 𝑥2 ) = 0.

(4.33)

2!

Thus, the functions 𝐴𝑛− 𝑗 : R2 → R satisfy the recursion relation  2  2! 𝜕 𝜕 + 𝐴𝑛− 𝑗 (𝑥 1 , 𝑥2 ) + ( 𝑗 + 2)( 𝑗 + 1) 𝐴𝑛− 𝑗 −2 (𝑥 1 , 𝑥2 ) = 0 𝜕𝑥1 𝜕𝑥2

(4.34)

for 𝑗 = 0, 1, ..., 𝑛 − 2. Therefore, all polynomials 𝐴𝑛− 𝑗 are determined if we know 𝐴𝑛 and 𝐴𝑛−1 . Theorem 4.5 Let 𝐴𝑛 and 𝐴𝑛−1 be homogeneous polynomials of degree 𝑛 and 𝑛 − 1 in R2 , respectively. For 𝑗 = 0, ..., 𝑛 − 2 we set recursively  2  2! 1 𝜕 𝜕 𝐴𝑛− 𝑗 −2 (𝑥1 , 𝑥2 ) = − + 𝐴𝑛− 𝑗 (𝑥1 , 𝑥2 ). (4.35) ( 𝑗 + 1)( 𝑗 + 2) 𝜕𝑥1 𝜕𝑥2 Then 𝐻𝑛 : R3 → R given by 𝐻𝑛 (𝑥1 , 𝑥2 , 𝑥3 ) =

𝑛 ∑︁

𝑗

𝑥3 𝐴𝑛− 𝑗 (𝑥1 , 𝑥2 )

(4.36)

𝑗=0

is a homogeneous harmonic polynomial of degree 𝑛 in R3 , i.e., 𝐻𝑛 ∈ Harm𝑛 . The number of linearly independent homogeneous harmonic polynomials is equal to the number of coefficients of 𝐴𝑛 and 𝐴𝑛−1 , i.e., 𝑑 (Harm𝑛 ) = 𝑛 + 𝑛 + 1 = 2𝑛 + 1.

90

4 Scalar Spherical Harmonics

Assume that 𝑛 is an integer with 𝑛 ≥ 2. Let 𝐻𝑛−2 be a homogeneous polynomial of degree 𝑛 − 2, i.e., 𝐻𝑛−2 ∈ Hom𝑛−2 . Then, for each homogeneous harmonic polynomial 𝐾𝑛 , we have (| · | 2 𝐻𝑛−2 , 𝐾𝑛 )Hom𝑛 = (𝐻𝑛−2 (∇))Δ𝐾𝑛 (𝑥) = 0.

(4.37)

This means | · | 2 𝐻𝑛−2 is orthogonal to 𝐾𝑛 in the sense of the inner product (·, ·)Hom𝑛 . Conversely, suppose that 𝐾𝑛 ∈ Hom𝑛 is orthogonal to all elements 𝐿 𝑛 of the form 𝐿 𝑛 (𝑥) = |𝑥| 2 𝐻𝑛−2 (𝑥) ,

𝐻𝑛−2 ∈ Hom𝑛−2 .

(4.38)

Then it follows that 0 = (| · | 2 𝐻𝑛−2 , 𝐾𝑛 )Hom𝑛 = (𝐻𝑛−2 (∇))Δ𝐾𝑛 (𝑥) = (𝐻𝑛−2 , Δ𝐾𝑛 )Hom𝑛−2

(4.39)

for all 𝐻𝑛−2 ∈ Hom𝑛−2 . This is true only if Δ𝐾𝑛 = 0, i.e., 𝐾𝑛 is a homogeneous harmonic polynomial. Theorem 4.6 (Decomposition Theorem of Hom𝑛 ) Hom𝑛 , 𝑛 ≥ 2, is the orthogonal ⊥ 2 direct sum of Harm𝑛 and Harm⊥ 𝑛 , where Harm𝑛 = | · | Hom𝑛−2 is the space of all 2 𝐿 𝑛 with 𝐿 𝑛 (𝑥) = |𝑥| 𝐻𝑛−2 (𝑥), 𝐻𝑛−2 ∈ Hom𝑛−2 . Consequently, each homogeneous polynomial 𝐻𝑛 of degree 𝑛 can be uniquely decomposed in the form 𝐻𝑛 (𝑥) = 𝐾𝑛 (𝑥) + |𝑥| 2 𝐻𝑛−2 (𝑥),

(4.40)

where 𝐾𝑛 is a homogeneous harmonic polynomial of degree 𝑛 and 𝐻𝑛−2 is a homogeneous polynomial of degree 𝑛 − 2. Denote by ProjHarm𝑛 and ProjHarm⊥𝑛 the projection operators in Hom𝑛 onto Harm𝑛 and Harm⊥ 𝑛 , respectively. Then 𝐻𝑛 = ProjHarm𝑛 𝐻𝑛 + ProjHarm⊥𝑛 𝐻𝑛 .

(4.41)

In other words, 𝐾𝑛 (𝑥) = ProjHarm𝑛 𝐻𝑛 (𝑥),

(4.42)

|𝑥| 𝐻𝑛−2 (𝑥) = ProjHarm⊥𝑛 𝐻𝑛 (𝑥).

(4.43)

(ProjHarm𝑛 𝐻𝑛 , 𝑈𝑛 )Hom𝑛 = (𝐻𝑛 , ProjHarm𝑛 𝑈𝑛 )Hom𝑛 .

(4.44)

2

For all 𝐻𝑛 , 𝑈𝑛 ∈ Hom𝑛 ,

Moreover, we have ProjHarm𝑛 𝐻𝑛 = ProjHarm𝑛 𝐾𝑛 = 𝐾𝑛 . Observe that

4.1 Homogeneous Harmonic Polynomials

91

𝑑 (Harm𝑛 ) = 𝑑 (Hom𝑛 ) − 𝑑 (Harm⊥ 𝑛)

(4.45)

= 𝑑 (Hom𝑛 ) − 𝑑 (Hom𝑛−2 )     𝑛+2 𝑛 = − = 2𝑛 + 1. 2 2 If we apply Theorem 4.6 recursively to 𝐻𝑛−2 , 𝐻𝑛−4 , ..., we obtain the following result. Theorem 4.7 Each homogeneous polynomial of degree 𝑛 can be uniquely decomposed in the form 𝑛

𝐻𝑛 (𝑥) =

⌊2⌋ ∑︁

|𝑥| 2𝑖 𝐾𝑛−2𝑖 (𝑥),

𝐾𝑛−2𝑖 ∈ Harm𝑛−2𝑖 ,

𝑥 ∈ R3 ,

(4.46)

𝑖=0

where ⌊𝑛/2⌋ is the largest integer which is less than or equal to 𝑛/2. In other words, Hom𝑛 admits the direct sum decomposition 𝑛

3

Hom𝑛 (R ) =

⌊2⌋ Ê

| · | 2𝑖 Harm𝑛−2𝑖 (R3 ).

(4.47)

𝑖=0

This result gives rise to the following corollary. Corollary 4.8 For 𝑛 = 0, 1, . . . 𝑛

3

Hom𝑛 (R )|Ω = Hom𝑛 (Ω) =

⌊2⌋ Ê

Harm𝑛−2𝑖 (R3 )|Ω.

𝑖=0

Since the space Pol0,...,𝑛 (R3 ) of polynomials in three variables of degree ≤ 𝑛 can be written as direct sum decomposition of Hom𝑛 (R3 ) and Hom𝑛−1 (R3 ), when restricted to Ω, i.e., Pol0,...,𝑛 (R3 )|Ω = (Hom𝑛 (R3 )|Ω) ⊕ (Hom𝑛−1 (R3 )|Ω) we finally obtain the following corollary. Corollary 4.9 For 𝑛 = 0, 1, . . . Pol0,...,𝑛 (R3 )|Ω =

𝑛 Ê 𝑖=0

Harm𝑖 (R3 )|Ω.

(4.48)

92

4 Scalar Spherical Harmonics

In other words, the restriction to the unit sphere Ω of any polynomial of three variables is a sum of restrictions to Ω of homogeneous harmonic polynomials.

4.2 Addition Theorem

We are now interested in giving the explicit representation of the orthogonal projection ProjHarm𝑛 𝐻𝑛 of a given homogeneous polynomial 𝐻𝑛 . For that purpose, we need some preliminaries. By induction, we are able to prove that for 𝑖 = 1, 2, 3 and |𝑥| ≠ 0 (cf. E.W. Hobson (1955))  𝑛 𝜕 1 (4.49) 𝜕𝑥𝑖 |𝑥| 𝑛

⌊2⌋ (2𝑛)! 1 ©∑︁ 𝑛!(2𝑛 − 2𝑠)! ª = (−1) (−1) 𝑠 |𝑥| 2𝑠 Δ𝑠 ® 𝑥𝑖𝑛 . ­ 𝑛 2𝑛+1 𝑛!2 |𝑥| (2𝑛)!(𝑛 − 𝑠)!𝑠! « 𝑠=0 ¬ 𝑛

In other words, we find (𝜀 𝑖 · ∇ 𝑥 ) 𝑛

1 |𝑥|

(4.50) 𝑛

⌊2⌋
𝑛 (2𝑛)! 1 ©∑︁ 𝑛!(2𝑛 − 2𝑠)! ª = (−1) |𝑥| 2𝑠 Δ𝑠 ® 𝜀 𝑖 · 𝑥 , ­ (−1) 𝑠 𝑛 2𝑛+1 𝑛!2 |𝑥| (2𝑛)!(𝑛 − 𝑠)!𝑠! « 𝑠=0 ¬ 𝑛

𝑖 = 1, 2, 3. Since the differential operator Δ is invariant with respect to orthogonal transformations, it is easy to see that (𝑦 · ∇ 𝑥 ) 𝑛

1 |𝑥|

(4.51) ⌊𝑛⌋

2 (2𝑛)! 1 𝑛!(2𝑛 − 2𝑠)! ©∑︁ ª = (−1) (−1) 𝑠 |𝑥| 2𝑠 Δ𝑠 ® (𝑦 · 𝑥) 𝑛 ­ 𝑛 2𝑛+1 𝑛!2 |𝑥| (2𝑛)!(𝑛 − 𝑠)!𝑠! « 𝑠=0 ¬

𝑛

is valid for every 𝑦 ∈ R3 . Now, as we have seen in Theorem 4.4, each 𝐻𝑛 ∈ Hom𝑛 may be represented in the form 𝐻𝑛 (𝑥) =

𝑑 (Hom ∑︁ 𝑛 )

𝑐 𝑗 (𝑥 𝑗 · 𝑥) 𝑛 ,

𝑥 ∈ R3 ,

(4.52)

𝑗=1

where 𝑐 𝑗 , 𝑗 = 1, ..., 𝑑 (Hom𝑛 ), are suitable coefficients and 𝑥 1 , ..., 𝑥 𝑑 (Hom𝑛 ) is a fundamental system relative to Hom𝑛 .

4.2 Addition Theorem

93

Consequently, we have the following result: Theorem 4.10 Let 𝐻𝑛 be a homogeneous polynomial of degree 𝑛. Then, for each 𝑥 ∈ R3 , |𝑥| ≠ 0, (𝐻𝑛 (∇))

1 |𝑥| 𝑛

⌊2⌋ 1 ©∑︁ 𝑛!(2𝑛 − 2𝑠)! ª 𝑛 (2𝑛)! = (−1) |𝑥| 2𝑠 Δ𝑠 ® 𝐻𝑛 (𝑥). ­ (−1) 𝑠 𝑛!2𝑛 |𝑥| 2𝑛+1 𝑠=0 (2𝑛)!(𝑛 − 𝑠)!𝑠! « ¬

Using the decomposition (4.40) as in Theorem 4.6, it follows that (𝐻𝑛 (∇))

1 1 1 = (𝐾𝑛 (∇)) + (𝐻𝑛−2 (∇))Δ , |𝑥| |𝑥| |𝑥|

|𝑥| ≠ 0.

(4.53)

Thus, in connection with 1 = 0, |𝑥|

|𝑥| ≠ 0,

(4.54)

Δ𝐾𝑛 (𝑥) = 0,

𝑥 ∈ R3 ,

(4.55)

Δ

we obtain for |𝑥| ≠ 0 (𝐻𝑛 (∇))

1 1 (2𝑛)! 1 = (𝐾𝑛 (∇)) = (−1) 𝑛 𝐾𝑛 (𝑥). |𝑥| |𝑥| 𝑛!2𝑛 |𝑥| 2𝑛+1

(4.56)

By solving (4.56) for 𝐾𝑛 (𝑥), we obtain from Theorem 4.10 the following lemma. Lemma 4.11 Let 𝐻𝑛 be a homogeneous polynomial of degree n. Then ⌊𝑛⌋

2 𝑛!(2𝑛 − 2𝑠)! ©∑︁ ª ProjHarm𝑛 𝐻𝑛 (𝑥) = ­ (−1) 𝑠 |𝑥| 2𝑠 Δ𝑠 ® 𝐻𝑛 (𝑥) (2𝑛)!(𝑛 − 𝑠)!𝑠! « 𝑠=0 ¬

(4.57)

such that ProjHarm⊥𝑛 𝐻𝑛 (𝑥) = 𝐻𝑛 (𝑥) − ProjHarm𝑛 𝐻𝑛 (𝑥) 𝑛 2

=

⌊ ⌋ ∑︁

(−1) 𝑠−1

(4.58)

𝑛!(2𝑛 − 2𝑠)! |𝑥| 2𝑠 Δ𝑠 𝐻𝑛 (𝑥). (2𝑛)!(𝑛 − 𝑠)!𝑠!

𝑠=1

The differential operator ProjHarm⊥𝑛 given by (4.58) is called the Clebsch projection (see E.W. Hobson (1955)). It forms a mapping from 𝐻𝑛 ∈ Hom𝑛 to 𝐾𝑛 ∈ Harm𝑛 such that

94

4 Scalar Spherical Harmonics 𝑛

𝐾𝑛 (𝑥) = 𝐻𝑛 (𝑥) −

⌊2⌋ ∑︁

(−1) 𝑠−1

𝑠=1

𝑛!(2𝑛 − 2𝑠)! |𝑥| 2𝑠 Δ𝑠 𝐻𝑛 (𝑥). (2𝑛)!(𝑛 − 2)!𝑠!

(4.59)

Remark 4.12 Note that 𝑛

2

|𝑥| 𝐻𝑛−2 (𝑥) =

⌊2⌋ ∑︁

(−1) 𝑠−1

𝑛!(2𝑛 − 2𝑠)! |𝑥| 2𝑠 Δ𝑠 𝐻𝑛 (𝑥), (2𝑛)!(𝑛 − 𝑠)!𝑠!

(4.60)

𝑠=1

hence, the Clebsch projection can be regarded as a mechanism for the division by |𝑥| 2 . Observing Δ 𝑥 (𝑥 · 𝑦) 𝑛 = 𝑛(𝑛 − 1)|𝑦| 2 (𝑥 · 𝑦) 𝑛−2 ,

𝑦 ∈ R3 ,

(4.61)

we obtain  ProjHarm𝑛

(𝑥 · 𝑦) 𝑛 𝑛!

 (4.62)

⌊𝑛⌋

2 1 ∑︁ (2𝑛 − 2𝑠)!(𝑛!) 2 = (−1) 𝑠 |𝑥| 2𝑠 |𝑦| 2𝑠 (𝑥 · 𝑦) 𝑛−2𝑠 . 𝑛! 𝑠=0 (𝑛 − 2𝑠)!(𝑛 − 𝑠)!𝑠!(2𝑛)!

Thus, we find by using 𝑥 = |𝑥|𝜉, 𝑦 = |𝑦|𝜂, 𝜉, 𝜂 ∈ Ω, the equation   (𝑥 · 𝑦) 𝑛 ProjHarm𝑛 𝑛!

(4.63)

⌊𝑛⌋

2 ∑︁ (2𝑛 + 1)2𝑛 · 𝑛! (2𝑛 − 2𝑠)! (|𝑥| |𝑦|) 𝑛 = (−1) 𝑠 𝑛 (𝜉 · 𝜂) 𝑛−2𝑠 . (2𝑛 + 1)! 2 (𝑛 − 2𝑠)!(𝑛 − 𝑠)!𝑠! 𝑠=0

Suppose that {𝐻𝑛, 𝑗 } 𝑗=1,...,𝑑 (Harm𝑛 ) is an orthonormal system in Harm𝑛 with respect to (·, ·)Hom𝑛 . Let {𝑈𝑛, 𝑗 } 𝑗=1,...,𝑑 (Hom𝑛 ) −𝑑 (Harm𝑛 ) be an orthonormal system in Harm⊥ 𝑛. Then, the union of both systems {𝐻𝑛, 𝑗 } 𝑗=1,...,𝑑 (Harm𝑛 ) ∪ {𝑈𝑛, 𝑗 } 𝑗=1,...,𝑑 (Hom𝑛 ) −𝑑 (Harm𝑛 )

(4.64)

forms an orthonormal system in Hom𝑛 . Therefore, it follows that (𝑥 · 𝑦) 𝑛 = 𝑛!

𝑑 (Harm ∑︁ 𝑛 ) 𝑗=1

𝐻𝑛, 𝑗 (𝑥) 𝐻𝑛, 𝑗 (𝑦) +

𝑑 (Hom𝑛 )∑︁ −𝑑 (Harm𝑛 )

𝑈𝑛, 𝑗 (𝑥) 𝑈𝑛, 𝑗 (𝑦)

(4.65)

𝑗=1

for any pair 𝑥, 𝑦 ∈ R3 . On the one hand, in view of the definition of the projection operator ProjHarm𝑛 , we obtain

4.2 Addition Theorem

95 𝑑 (Harm𝑛 )

𝑑 (Hom𝑛 )∑︁ −𝑑 (Harm𝑛 )

© ∑︁ ProjHarm𝑛 ­ 𝐻𝑛, 𝑗 (𝑥)𝐻𝑛, 𝑗 (𝑦) + 𝑗=1 « 𝑑 (Harm ∑︁ 𝑛 ) = 𝐻𝑛, 𝑗 (𝑥)𝐻𝑛, 𝑗 (𝑦).

𝑗=1

ª 𝑈𝑛, 𝑗 (𝑥)𝑈𝑛, 𝑗 (𝑦) ® ¬ (4.66)

𝑗=1

On the other hand, as we have shown above,   (𝑥 · 𝑦) 𝑛 ProjHarm𝑛 𝑛!

(4.67)

⌊𝑛/2⌋ (2𝑛 + 1)2𝑛 𝑛! 𝑛 𝑛 ∑︁ (2𝑛 − 2𝑠)! = |𝑥| |𝑦| (−1) 𝑠 𝑛 (𝜉 · 𝜂) 𝑛−2𝑠 . (2𝑛 + 1)! 2 (𝑛 − 2𝑠)!(𝑛 − 𝑠)!𝑠! 𝑠=0

By comparison of (4.66) and (4.67), we obtain the addition theorem of homogeneous harmonic polynomials in R3 . Theorem 4.13 Let {𝐻𝑛, 𝑗 } 𝑗=1,...,𝑑 (Harm𝑛 ) , 𝑑 (Harm𝑛 ) = 2𝑛 + 1, be an orthonormal system in Harm𝑛 with respect to (·, ·)Hom𝑛 . Then, for 𝑥, 𝑦 ∈ R3 , 𝑥 = |𝑥|𝜉, 𝑦 = |𝑦|𝜂, we have 2𝑛+1 ∑︁ 2𝑛 𝑛! 𝑛 𝑛 𝐻𝑛, 𝑗 (𝑥) 𝐻𝑛, 𝑗 (𝑦) = |𝑥| |𝑦| 𝑃𝑛 (𝜉 · 𝜂), (2𝑛)! 𝑗=1 where we have used the abbreviation 𝑛

𝑃𝑛 (𝑡) =

⌊2⌋ ∑︁ 𝑠=0

(−1) 𝑠

2𝑛 (𝑛

(2𝑛 − 2𝑠)! 𝑡 𝑛−2𝑠 , − 2𝑠)!(𝑛 − 𝑠)!𝑠!

𝑡 ∈ [−1, 1].

(4.68)

Remark 4.14 𝑃𝑛 is known as the Legendre polynomial of degree 𝑛 (see Section 4.5 for a detailed description). Next, we discuss the important question of how, for any pair of elements 𝐻𝑛 ∈ Harm𝑛 , 𝐾𝑛 ∈ Harm𝑛 , the inner product (·, ·)Hom𝑛 defined by (4.9) is related to the (usually used) inner product (·, ·)L2 (Ω) . Theorem 4.15 For 𝐻𝑚 ∈ Harm𝑚 , 𝐾𝑛 ∈ Harm𝑛 , 𝛿 𝑛𝑚 (𝐻𝑚 (∇))𝐾𝑛 (𝑥), 𝜇𝑛

(4.69)

(2𝑛 + 1)! 1 · 3 · . . . · (2𝑛 + 1) = . 4𝜋2𝑛 𝑛! 4𝜋

(4.70)

(𝐻𝑚 , 𝐾𝑛 )L2 (Ω) = where 𝜇 𝑛 is given by 𝜇𝑛 =

96

4 Scalar Spherical Harmonics

Proof By virtue of the Third Green’s Theorem of potential theory, we find  ∫  1 1 𝜕 𝜕 1 𝐾𝑛 (𝑥) = 𝐾𝑛 (𝑦) − 𝐾𝑛 (𝑦) 𝑑𝜔(𝑦) 4𝜋 Ω |𝑥 − 𝑦| 𝜕𝜈 𝑦 𝜕𝜈 𝑦 |𝑥 − 𝑦|

(4.71)

for all 𝑥 ∈ R3 with |𝑥| < 1, where 𝜕/𝜕𝜈 denotes the derivative in the direction of the outer normal to Ω. Therefore, we find (𝐻𝑚 (∇))𝐾𝑛 (𝑥) = 1 4𝜋

∫  Ω

1 𝜕 𝜕 1 (𝐻𝑚 (∇ 𝑥 )) 𝐾𝑛 (𝑦) − 𝐾𝑛 (𝑦) (𝐻𝑚 (∇ 𝑥 )) |𝑥 − 𝑦| 𝜕𝜈 𝑦 𝜕𝜈 𝑦 |𝑥 − 𝑦|

(4.72)  𝑑𝜔(𝑦).

For 𝑥 ≠ 𝑦, we obtain from (4.56) (𝐻𝑚 (∇ 𝑥 ))

1 (2𝑚)! 𝐻𝑚 (𝑥 − 𝑦) = (−1) 𝑚 . |𝑥 − 𝑦| 𝑚!2𝑚 |𝑥 − 𝑦| 2𝑚+1

(4.73)

Because 𝐻𝑚 is homogeneous, this is equivalent to (𝐻𝑚 (∇ 𝑥 ))

1 (2𝑚)! 𝐻𝑚 (𝑦 − 𝑥) = . |𝑥 − 𝑦| 𝑚!2𝑚 |𝑥 − 𝑦| 2𝑚+1

(4.74)

Inserting (4.74) into (4.72) gives (𝐻𝑚 (∇))𝐾𝑛 (𝑥) (4.75)  ∫  (2𝑚)! 1 𝐻𝑚 (𝑦 − 𝑥) 𝜕 𝜕 𝐻𝑚 (𝑦 − 𝑥) = 𝐾𝑛 (𝑦) − 𝐾𝑛 (𝑦) 𝑑𝜔(𝑦). 𝑚 2𝑚+1 (𝑚!)2 4𝜋 Ω |𝑥 − 𝑦| 𝜕𝜈 𝑦 𝜕𝜈 𝑦 |𝑥 − 𝑦| 2𝑚+1 It is easy to see that for 𝑚 ≠ 𝑛 (𝐻𝑚 (∇))𝐾𝑛 (𝑥) | 𝑥=0 = 0,

(4.76)

(𝐻𝑚 (∇))𝐾𝑛 (𝑥) | 𝑥=0 = (𝐻𝑚 (∇))𝐾𝑛 (𝑥) = (𝐻𝑚 , 𝐾𝑛 )Hom𝑛 .

(4.77)

while for 𝑚 = 𝑛

Therefore, we obtain  ∫  1 𝐻𝑚 (𝑦) 𝜕 𝜕 𝐻𝑚 (𝑦) 𝐾𝑛 (𝑦) − 𝐾𝑛 (𝑦) 𝑑𝜔(𝑦) 4𝜋 Ω |𝑦| 2𝑚+1 𝜕𝜈 𝑦 𝜕𝜈 𝑦 |𝑦| 2𝑚+1 ( 0, 𝑚 ≠ 𝑛,  𝑚  = 2 𝑚! (2𝑚)! (𝐻 𝑚 , 𝐾 𝑛 )Hom𝑛 , 𝑚 = 𝑛. Since the normal derivatives of 𝐾𝑛 and 𝐻𝑚 are equal to

(4.78)

4.2 Addition Theorem

97

𝜕 𝐾𝑛 (𝑟𝜉) | 𝑟=1 = 𝑛𝐾𝑛 (𝜉) , 𝜕𝑟

𝜕 𝐻𝑚 (𝑟𝜉) | 𝑟=1 = 𝑚𝐻𝑚 (𝜉), 𝜕𝑟

respectively, it follows that  ∫  1 𝐻𝑚 (𝑦) 𝜕 𝜕 𝐻𝑚 (𝑦) 𝐾𝑛 (𝑦) − 𝐾𝑛 (𝑦) 𝑑𝜔(𝑦) 4𝜋 Ω |𝑦| 2𝑚+1 𝜕𝜈 𝑦 𝜕𝜈 𝑦 |𝑦| 2𝑚+1 ∫ 1 {𝑛𝐻𝑚 (𝜉)𝐾𝑛 (𝜉) + (𝑚 + 1)𝐻𝑚 (𝜉)𝐾𝑛 (𝜉)} 𝑑𝜔(𝜉) = 4𝜋 Ω ∫ 𝑛+𝑚+1 = 𝐻𝑚 (𝜉)𝐾𝑛 (𝜉) 𝑑𝜔(𝜉). 4𝜋 Ω

(4.79)

(4.80)

Thus, by combination of (4.78) and (4.80), we finally obtain the desired result.

□

In other words, to any orthonormal system {𝐻𝑛, 𝑗 } 𝑗=1,...,2𝑛+1 in Harm𝑛 with respect √ to (·, ·)Hom𝑛 there corresponds the L2 (Ω)-orthonormal system { 𝜇 𝑛 𝐻𝑛, 𝑗 } 𝑗=1,...,2𝑛+1 , and vice versa. We summarize the relationship between the two topologies in Harm𝑛 in Table 4.1. Table 4.1 Comparison of inner products. Topologies in Harm𝑛 ∫ (𝐻𝑛 , 𝐾𝑛 ) Hom𝑛 = 𝐻𝑛 (∇) 𝐾𝑛 ( 𝑥 )

(𝐻𝑛 , 𝐾𝑛 ) L2 (Ω) =

𝐻𝑛 ( 𝜉 ) 𝐾𝑛 ( 𝜉 ) 𝑑 𝜔 ( 𝜉 ) Ω

1 (𝐻𝑛 , 𝐾𝑛 ) Hom𝑛 = (𝐻𝑛 , 𝐾𝑛 ) L2 (Ω) 𝜇𝑛

Finally, we are led to the following reformulation of the addition theorem. Theorem 4.16 {𝐻𝑛, 𝑗 } 𝑗=1,...,2𝑛+1 is an orthonormal system in Harm𝑛 with respect √ to (·, ·)Hom𝑛 if and only if { 𝜇 𝑛 𝐻𝑛, 𝑗 } 𝑗=1,...2𝑛+1 is an orthonormal system in Harm𝑛 with respect to (·, ·)L2 (Ω) . For 𝑥, 𝑦 ∈ R3 , we have 2𝑛+1 ∑︁

√

𝜇 𝑛 𝐻𝑛, 𝑗 (𝑥)

𝑗=1

where 𝜇𝑛 =

√

𝜇 𝑛 𝐻𝑛, 𝑗 (𝑦) =

2𝑛 + 1 𝑛 𝑛 |𝑥| |𝑦| 𝑃𝑛 (𝜉 · 𝜂). 4𝜋

(2𝑛 + 1)! 1 · 3 · . . . · (2𝑛 + 1) = . 4𝜋2𝑛 𝑛! 4𝜋

98

4 Scalar Spherical Harmonics

4.3 Exact Computation of Homogeneous Harmonic Polynomials

Our purpose is to explain how a maximal linearly independent system of homogeneous harmonic polynomials of degree 𝑛 can be generated exactly (see W. Freeden, R. Reuter (1984)). The concept is based on the observation that any linearly independent system {𝐻𝑛, 𝑗 } 𝑗=1,...,2𝑛+1 of homogeneous harmonic polynomials of degree 𝑛 ∑︁ 𝐻𝑛,1 (𝑥) = 𝐶 1𝛼 𝑥 𝛼 [ 𝛼]=𝑛

.. .

.. .

.. .. . . ∑︁ 𝐻𝑛,2𝑛+1 (𝑥) = 𝐶 2𝑛+1 𝑥𝛼 𝛼

(4.81)

[ 𝛼]=𝑛 𝑗

can be calculated by exact computation of the coefficients 𝐶 𝛼 , 𝑗 = 1, . . ., 2𝑛 + 1, i.e., 𝑗 𝑛, 𝑗 entirely by integer operations (note that we briefly write 𝐶 𝛼 instead of 𝐶 𝛼 when confusion is not likely to arise). In other words, we want to show that the coefficients 𝑗 𝐶 𝛼 , 𝑗 = 1, ..., 2𝑛 + 1, in (4.81) can be expressed as integers. Let 𝐻𝑛 be a homogeneous polynomial of the form 𝐻𝑛 = Σ [ 𝛼]=𝑛 𝐶 𝛼 𝑥 𝛼 , 𝑥 ∈ R3 , 𝑛 ≥ 2. Assuming that 𝐻𝑛 is harmonic, i.e., Δ𝐻𝑛 (𝑥) = 0 , 𝑥 ∈ R3 , we obtain ∑︁ ∑︁ Δ𝐻𝑛 (𝑥) = Δ 𝐶𝛼𝑥 𝛼 = 𝐶 𝛼 Δ(𝑥 𝛼 ) = 0. (4.82) [ 𝛼]=𝑛

[ 𝛼]=𝑛

Thus, it follows that ∑︁ 𝐶 𝛼 𝛼1 (𝛼1 − 1)𝑥 1𝛼1 −2 𝑥2𝛼2 𝑥 3𝛼3 + 𝛼2 (𝛼2 − 1)𝑥2𝛼2 −2 𝑥 1𝛼1 𝑥3𝛼3 𝛼1 +𝛼2 +𝛼3 =𝑛


+ 𝛼3 (𝛼3 − 1)𝑥3𝛼3 −2 𝑥1𝛼1 𝑥2𝛼2 = 0.

(4.83)

We discuss the terms 𝛼1 (𝛼1 − 1)𝑥1𝛼1 −2 𝑥2𝛼2 𝑥3𝛼3 , 𝛼1 + 𝛼2 + 𝛼3 = 𝑛, 𝛼2 (𝛼2 − 𝛼3 (𝛼3 −

1)𝑥1𝛼1 𝑥2𝛼2 −2 𝑥3𝛼3 , 1)𝑥1𝛼1 𝑥2𝛼2 𝑥3𝛼3 −2 ,

(4.84)

𝛼1 + 𝛼2 + 𝛼3 = 𝑛,

(4.85)

𝛼1 + 𝛼2 + 𝛼3 = 𝑛

(4.86)

in more detail. Every term in (4.84)–(4.86) with index 𝛼 = (𝛼1 , 𝛼2 , 𝛼3 ) 𝑇 satisfying [𝛼] = 𝛼1 + 𝛼2 + 𝛼3 = 𝑛 is a homogeneous polynomial of degree 𝑛 − 2. Hence, the left side of (4.83) is a homogeneous polynomial of degree 𝑛 − 2. Therefore, Δ𝐻𝑛 can be represented in the form

4.3 Exact Computation of Homogeneous Harmonic Polynomials

∑︁

Δ𝐻𝑛 (𝑥) =

99

𝐷𝛽𝑥𝛽.

(4.87)

[𝛽 ]=𝑛−2

The coefficients 𝐷 𝛽 are given by ∑︁

𝐷𝛽 =

(4.88)

𝐶𝛼𝑚𝛽 𝛼,

[ 𝛼]=𝑛

where 𝑚 𝛽 𝛼 is given by

𝑚𝛽 𝛼

 𝛼1 (𝛼1 − 1),     𝛼2 (𝛼2 − 1),  = 𝛼3 (𝛼3 − 1),     0, 

𝛽 − 𝛼 = (−2, 0, 0) 𝑇 , 𝛽 − 𝛼 = (0, −2, 0) 𝑇 , 𝛽 − 𝛼 = (0, 0, −2) 𝑇 , otherwise.

(4.89)

𝐻𝑛 is assumed to be harmonic, i.e., Δ 𝑥 𝐻𝑛 (𝑥) = 0 for all 𝑥 ∈ R3 . However, this means that all numbers 𝐷 𝛽 are equal to 0. Therefore, it follows that ∑︁ 𝐶𝛼𝑚𝛽 𝛼 = 0 (4.90) [ 𝛼]=𝑛 𝑛
2

for all 𝛽 with [𝛽] = 𝑛 − 2. Now, (4.90) is a linear system of unknowns 𝐶 𝛼 , [𝛼] = 𝑛. 𝑛
2

The matrix m = (𝑚 𝛽 𝛼 ) has follows:

m=(

where l = (𝑙 𝛽 𝛿 ) is a

𝑛
2

by

𝑛
2

rows and .. .

l |{z} ( 𝑛2 )

𝑛+2
2

equations in the

𝑛+2
2

columns; m can be partitioned as

r ), |{z} 𝑛 ( 𝑛+2 2 ) − ( 2 ) =2𝑛+1

matrix and r = (𝑟 𝛽 𝛿 ) is a

(4.91)

𝑛
2

by

𝑛+2
2

−

𝑛
2

matrix.

For the set of multi indices of degree 𝑛, we introduce a binary relation (lexicographical order) between its elements ′

′

′

′

′′

′′

′′

′′

𝛼 = (𝛼1 , 𝛼2 , 𝛼3 ) 𝑇 , 𝛼 = (𝛼1 , 𝛼2 , 𝛼3 ) 𝑇

(4.92)

designated by “>” and defined as follows: ′

′′

(4.93)

𝛼 >𝛼 if and only if one of the following relations is satisfied ′

′′

(4.94)

𝛼1 > 𝛼1 or

′

′′

′

′′

𝛼1 = 𝛼1 , 𝛼2 > 𝛼2

(4.95)

100

or

4 Scalar Spherical Harmonics

′

′′

′

′′

′

′′

𝛼1 = 𝛼1 , 𝛼2 = 𝛼2 , 𝛼3 > 𝛼3 .

(4.96)

The binary relation “>” implies an ordering for the multi-indices 𝛼, [𝛼] = 𝑛, according to the mapping

(𝑛, 0, 0) → 1



(𝑛 − 1, 1, 0) → 2 (𝑛 − 1, 0, 1) → 3



(𝑛 − 2, 2, 0) → 4 (𝑛 − 2, 1, 1) → 5 (𝑛 − 2, 0, 2) → 6 .. .. .. ...

   

(0, 𝑛, 0) → .. .. .. ...

𝑛+2
2

(0, 0, 𝑛) →

𝑛+2
2

1 2

3

   −𝑛

    

𝑛+1

.

   

In the same way, the set
of multi-indices 𝛽, [𝛽] = 𝑛−2, may be ordered by increasing integers 𝑖, 1 ≤ 𝑖 ≤ 𝑛2 . Hence, in canonical manner, each pair (𝛽, 𝛼) with [𝛽] =

𝑛 − 2, [𝛼] = 𝑛, corresponds uniquely to a pair (𝑖, 𝑗), 1 ≤ 𝑖 ≤ 𝑛2 , 1 ≤ 𝑗 ≤ 𝑛+2 2 . In this notation, the matrix m = (𝑚 𝛽 𝛼 ), [𝛽] = 𝑛 − 2, [𝛼] = 𝑛

(4.97)

can be rewritten in the ordered form     𝑛 𝑛+2 m = (𝑚 𝑖 𝑗 ), 1 ≤ 𝑖 ≤ , 1≤ 𝑗 ≤ . 2 2

(4.98)

l = 𝑙 𝛽𝛾 , [𝛽] = 𝑛 − 2, [𝛾] = 𝑛 − 2

(4.99)

    𝑛 𝑛 l = (𝑙 𝑖 𝑗 ), 1 ≤ 𝑖 ≤ , 1≤ 𝑗 ≤ . 2 2

(4.100)

Analogously becomes

From (4.89), it can be deduced that
𝑙𝑖 𝑗 = 0 for 𝑖 > 𝑗, 𝑖 = 2, ..., 𝑛2
, 𝑙𝑖 𝑗 ≠ 0 for 𝑖 = 𝑗, 𝑖 = 1, ..., 𝑛2 .

(4.101)

4.3 Exact Computation of Homogeneous Harmonic Polynomials

101

However, this shows that l is non-singular, hence, the matrix m is of maximal rank: 𝑛
𝑛+2
𝑛
. Therefore we are able to find − 2 2 2 , i.e., 2𝑛 + 1 linearly independent solution vectors ( 𝐴1𝛼 ) , . . . , ( 𝐴2𝑛+1 ), [𝛼] = 𝑛, of the homogeneous
linear system (4.90). 𝛼 According to standard arguments of Linear Algebra, the 𝑛+2 by 2𝑛 + 1 matrix a 2 consisting of the vectors ( 𝐴1𝛼 ), ..., ( 𝐴2𝑛+1 ) 𝛼
o 𝑛+2
a = ( 𝐴1𝛼 ), ..., ( 𝐴2𝑛+1 (4.102) 𝛼 ) 2 | {z } 2𝑛+1

may be partitioned in the following form 

a=

 u , −i |{z}

(4.103)

2𝑛+1


where i is the (2𝑛 + 1) by (2𝑛 + 1) unit matrix, and u is a 𝑛+2 2 − (2𝑛 + 1) by (2𝑛 + 1) matrix. Then the linear system m a = 0 can be written as follows: l u = r. Since l is a (2𝑛 + 1) by (2𝑛 + 1) upper triangular matrix, the unknown matrix u can be computed by (2𝑛 + 1)-times backward substitution. The elements of the matrix m = (𝑚 𝛽 𝛼 ) are all integers. Therefore, any solution of the linear system (4.90) is a column vector of rational components. Hence, there exists a matrix c = ((𝐶 1𝛼 ), ..., (𝐶 2𝑛+1 )), [𝛼] = 𝑛, (4.104) 𝛼 the elements of which are all integers (observe that if (𝐶 𝛼 ), [𝛼] = 𝑛, is a solution of (4.90), then 𝑘 (𝐶 𝛼 ), [𝛼] = 𝑛, 𝑘 integer, is a solution, too). In other words, the solution process can be performed strictly in the modulus of integers. Exact computation (without rounding errors) is possible in integer mode by use of integer operations (addition, subtraction, multiplication of integers). When the matrix c has been calculated, the homogeneous harmonic polynomials 𝐻𝑛, 𝑗 given by (4.81) form a (maximal) linearly independent system, i.e., a basis in Harm𝑛 . Finally, it should be emphasized that exact computation, i.e., addition, subtraction, multiplication in integer mode must be performed strictly in the available range of the integer constants. Helpful is an arithmetic for arbitrarily long integers whose implementation on a computer system operates with lists so that there is no restriction on the size of the integers worked with (this is a standard feature of computer algebra packages). Let us demonstrate the technique of calculating the matrix c with an example: Example 4.17 We choose the degree 𝑛 = 3. Then an elementary calculation yields

102

4 Scalar Spherical Harmonics





𝑛+2 = 10, 2

  𝑛 = 3, 2

(4.105)

hence, 

   𝑛+2 𝑛 − = 7. 2 2

(4.106)

Every polynomial 𝐻3 ∈ Hom3 may be represented in the form: 𝐻3 ( 𝑥 ) =

𝐶300 + 𝐶120 + 𝐶030 + 𝐶003

𝑥13 + 𝐶210 𝑥𝑥122 + 𝐶201 𝑥12𝑥3 𝑥𝑥122 + 𝐶111 𝑥𝑥1𝑥23 + 𝐶102 𝑥1𝑥32 𝑥23 + 𝐶021 𝑥𝑥223 + 𝐶012 𝑥𝑥232 𝑥33 ,

𝑥 = (𝑥1 , 𝑥2 , 𝑥3 ) 𝑇 ). 𝐻3 has to fulfill the differential equation Δ 𝑥 𝐻3 (𝑥) = 0, 𝑥 ∈ R3 , i.e., 6 𝐶300 𝑥1 + 2 𝐶210 𝑥 2 + 2 𝐶201 𝑥3

(4.107)

+2 𝐶120 𝑥1 + 6 𝐶030 𝑥 2 + 2 𝐶021 𝑥3 +2 𝐶102 𝑥1 + 2 𝐶012 𝑥 2 + 6 𝐶003 𝑥3 = 0. Since Δ 𝑥 𝐻3 (𝑥) = 0 identically for all 𝑥 ∈ R3 , we obtain coefficients

𝑛
2

= 3 equations for the

6𝐶300 + 2𝐶120 + 2𝐶102 = 0,

(4.108)

2𝐶210 + 6𝐶030 + 2𝐶012 = 0,

(4.109)

2𝐶201 + 2𝐶021 + 6𝐶003 = 0.

(4.110)

Using the introduced order for the coefficients 𝐶 𝛼 , [𝛼] = 3, the equation m c = 0 reads in matrix notation

. © 6 0 0 .. 2 0 2 0 0 0 0 ª ­ ® . ­ ® ­ 0 2 0 .. 0 0 0 6 0 2 0 ® ­ ® .. 0 0 2 . 0 0 0 0 2 0 6 « ¬

𝐶300 © ª ­ 𝐶210 ® ­ ® ­ 𝐶201 ® ­ ® ­ ... ® ­ ® 0 ­ 𝐶120 ® ­ ® © ª ­ 𝐶111 ® = ­ 0 ® , ­ ® ­ 𝐶102 ® « 0 ¬ ­ ® ­ 𝐶030 ® ­ ® ­ 𝐶021 ® ­ ® ­ 𝐶012 ® « 𝐶003 ¬

(4.111)

where we have marked the partitioning of the matrix m and the vector (𝐶 𝛼 ) by dashed lines. If we choose 𝐶120 = −1, 𝐶111 = . . . = 𝐶003 = 0,

(4.112)

4.3 Exact Computation of Homogeneous Harmonic Polynomials

103

the linear system is uniquely solved by the vector . 1 ( , 0, 0 .. − 1, 0, 0, 0, 0, 0, 0) 𝑇 . 3

(4.113)

Multiplying this vector by 3, all components become integers . (𝐶 1𝛼 ) = (1, 0, 0 .. − 3, 0, 0, 0, 0, 0, 0) 𝑇 .

(4.114)

In the same way, we generate a set of 7 linearly independent solutions of the above system the components of which are all integers, viz. . (𝐶 2𝛼 ) = (0, 0, 0 .. 0, −1, 0, 0, 0, 0, 0) 𝑇 , . (𝐶 3𝛼 ) = (1, 0, 0 .. 0, 0, −3, 0, 0, 0, 0) 𝑇 , . (𝐶 4𝛼 ) = (0, 3, 0 .. 0, 0, 0, −1, 0, 0, 0) 𝑇 , . (𝐶 5𝛼 ) = (0, 0, 1 .. 0, 0, 0, 0, −1, 0, 0) 𝑇 , . (𝐶 6𝛼 ) = (0, 1, 0 .. 0, 0, 0, 0, 0, −1, 0) 𝑇 , . (𝐶 7𝛼 ) = (0, 0, 3 .. 0, 0, 0, 0, 0, 0, −1) 𝑇 .

(4.115) (4.116) (4.117) (4.118) (4.119) (4.120)

Thus a linearly independent system {𝐻3, 𝑗 } 𝑗=1,...,7 of homogeneous harmonic polynomials of degree 3 is found by the following functions: 𝐻3,1 (𝑥) =

1 · 𝑥13 − 3 · 𝑥1 𝑥22 ,

𝐻3,2 (𝑥) = −1 · 𝑥1 𝑥2 𝑥3 , 𝐻3,3 (𝑥) =

1·

𝐻3,4 (𝑥) =

3·

𝐻3,5 (𝑥) =

1·

𝐻3,6 (𝑥) =

1·

𝐻3,7 (𝑥) =

3·

𝑥13 − 𝑥12 𝑥2 𝑥12 𝑥3 𝑥12 𝑥2 𝑥12 𝑥3

3 · 𝑥1 𝑥32 , − 1 · 𝑥23 , − 1 · 𝑥22 , 𝑥3 , − 1 · 𝑥2 𝑥32 , − 1 · 𝑥33 .

(4.121) (4.122) (4.123) (4.124) (4.125) (4.126) (4.127)

Let us summarize the solution process once the linear system is given: (i) Choosing the lower part of the vector identically 0 besides one component. (ii) Solving the system by backward substitution. (iii) Multiplying every resulting vector by an appropriate integer.

104

4 Scalar Spherical Harmonics

It is worth mentioning that, corresponding to the linearly independent system {𝐻𝑛, 𝑗 } 𝑗=1,...,2𝑛+1 of homogeneous harmonic polynomials of degree 𝑛, an ortho∗ } gonal system, in {𝐻𝑛, 𝑗 𝑗=1,....,2𝑛+1 with respect to both the topology of Hom𝑛 and 2 L (Ω) can be constructed only by integer operations (according to the well-known ∗ are computed recursively. Gram-Schmidt process). To this end, the functions 𝐻𝑛, 𝑗 We start from ∗ 𝐻𝑛,1 = 𝐻𝑛,1 . (4.128) Then we set ∗ 𝑛 ∗ 𝐻𝑛,2 = 𝑎 2,1 𝐻𝑛,1 + 𝐻𝑛,2 .

(4.129)

𝑛 has to be chosen such that 𝐻 ∗ is orthogonal to 𝐻 ∗ : The coefficient 𝑎 2,1 𝑛,2 𝑛,1 ∗ ∗ (𝐻𝑛,2 , 𝐻𝑛,1 )Hom𝑛 = 0.

It turns out that 𝑛 𝑎 2,1 =−

(4.130)

∗ ) (𝐻𝑛,2 , 𝐻𝑛,1 Hom𝑛

.

∗ , 𝐻∗ ) (𝐻𝑛,1 𝑛,1 Hom𝑛

(4.131)

It should be noted that numerator and denominator may be determined exactly. Now, let ∗ 𝑛 ∗ 𝑛 ∗ 𝐻𝑛,3 = 𝑎 3,1 𝐻𝑛,1 + 𝑎 3,2 𝐻𝑛,2 + 𝐻𝑛,3 . (4.132) The requirements ∗ ∗ (𝐻𝑛,3 , 𝐻𝑛,1 )Hom𝑛 = 0,

(4.133)

∗ ∗ (𝐻𝑛,3 , 𝐻𝑛,2 )Hom𝑛

(4.134)

=0

lead to 𝑛 𝑎 3,1 =−

𝑛 𝑎 3,2

=−

∗ ) (𝐻𝑛,3 , 𝐻𝑛,1 Hom𝑛

,

(4.135)

.

(4.136)

∗ , 𝐻∗ ) (𝐻𝑛,1 𝑛,1 Hom𝑛 ∗ ) (𝐻𝑛,3 , 𝐻𝑛,2 Hom𝑛 ∗ , 𝐻∗ ) (𝐻𝑛,2 𝑛,2 Hom𝑛

Again, the coefficients can be deduced by integer operations. Analogously we obtain, in general, ∗ 𝐻𝑛,1 = 𝐻𝑛,1 , (4.137) ∗ ∗ ∗ 𝐻𝑛,𝑘 = 𝑎 𝑛𝑘,1 𝐻𝑛,1 + ... + 𝑎 𝑛𝑘,𝑘−1 𝐻𝑛,𝑘−1 + 𝐻𝑛,𝑘 ,

where the coefficients 𝑎 𝑛𝑘,𝑠 = −

𝑘 = 2, ..., 2𝑛 + 1,

∗ ) (𝐻𝑛,𝑘 , 𝐻𝑛,𝑠 Hom𝑛 ∗ ∗ (𝐻𝑛,𝑠 , 𝐻𝑛,𝑠 )Hom𝑛

(4.138)

(4.139)

are computable exactly by integer operations, i.e., 𝑎 𝑛𝑘,𝑠 is known exactly as a fraction of integers.

4.3 Exact Computation of Homogeneous Harmonic Polynomials

105

∗ According to this well-known orthogonalization scheme, each function 𝐻𝑛, 𝑗 is a linear combination of the functions 𝐻𝑛,1 , ..., 𝐻𝑛,2𝑛+1 . The coefficients of this linear combination can be obtained exactly as rational numbers, too. Thus, there exists a 𝑗 vector (𝐵 𝛼 ) such that ∑︁ 𝑗 ∗ 𝐻𝑛, 𝐵 𝛼 𝑥 𝛼 , 𝑗 = 1, ..., 2𝑛 + 1. (4.140) 𝑗 (𝑥) = [ 𝛼]=𝑛 𝑗

The vectors (𝐵 𝛼 ), 𝑗 = 1, ..., 2𝑛 + 1, form a matrix b whose elements consist of fractions of integers (provided that all numbers in the course of computation have been calculated in such a way that numerator and denominator are known as integers).

Lemma 4.18 There exists a sequence of homogeneous harmonic polynomials ∗ } {𝐻𝑛, 𝑗 𝑗=1,...,2𝑛+1 of degree 𝑛 with ∗ ∗ (𝐻𝑛, 𝑗 , 𝐻 𝑛,𝑙 )Hom𝑛 = 0,

𝑗 ≠𝑙,

viz. ∗ 𝐻𝑛,1 = 𝐻𝑛,1 ∗ ∗ ∗ 𝐻𝑛,𝑘 = 𝑎 𝑛𝑘,1 𝐻𝑛,1 + ... + 𝑎 𝑛𝑘,𝑘−1 𝐻𝑛,𝑘−1 + 𝐻𝑛,𝑘 ,

𝑘 = 2, ..., 2𝑛 + 1,

where all coefficients 𝑎 𝑛𝑘,𝑠 are computable by integer operations.

Remark 4.19 Provided that the expression

√︃

∗ , 𝐻∗ ) (𝐻𝑛, 𝑗 𝑛, 𝑗 Hom𝑛 has been stored as

the radicant of an integer, a Hom𝑛 -orthonormal system of homogeneous harmonic polynomials of degree 𝑛 can be calculated exactly, i.e., exclusively by integer operations. Lemma 4.20 The system √︃ √︃ ∗ , 𝐻 ∗ ) −1 𝐻 ∗ , . . . , ∗ ∗ −1 𝐻 ∗ (𝐻𝑛,1 (𝐻𝑛,2𝑛+1 , 𝐻𝑛,2𝑛+1 )Hom 𝑛,2𝑛+1 𝑛,1 Hom𝑛 𝑛,1 𝑛 is an orthonormal system of homogeneous harmonic polynomials of degree 𝑛 with respect to (·, ·)Hom𝑛 , while √︃ √︃ ∗ , 𝐻 ∗ ) −1 𝐻 ∗ , . . . , ∗ ∗ −1 𝐻 ∗ 𝜇 𝑛 (𝐻𝑛,1 𝜇 𝑛 (𝐻𝑛,2𝑛+1 , 𝐻𝑛,2𝑛+1 )Hom 𝑛,1 𝑛,2𝑛+1 𝑛,1 Hom𝑛 𝑛 is an orthonormal system of homogeneous harmonic polynomials of degree 𝑛 with ∗ , 𝐻∗ ) respect to (·, ·)L2 (Ω) . The values (𝐻𝑛, 𝑗 𝑛, 𝑗 Hom𝑛 can be determined entirely by integer operations.

106

4 Scalar Spherical Harmonics

Example 4.21 We only deal with the degree 𝑛 = 3 (for a table of higher degrees, see W. Freeden, R. Reuter (1984)). According to our orthonormalization process due to Gram-Schmidt, we are able to deduce from the maximal system of linearly independent homogeneous harmonic polynomials {𝐻3, 𝑗 } 𝑗=1,...,7 an orthogonal system ∗ } {𝐻3, 𝑗 𝑗=1,...,7 . The resulting functions are listed below: ∗ 𝐻3,1 (𝑥) = 𝑥13 − 3𝑥1 𝑥22 , ∗ 𝐻3,2 (𝑥) = 𝑥1 𝑥 2 𝑥 3 , ∗ 𝐻3,3 (𝑥) = 𝑥13 + 𝑥 1 𝑥22 − 4𝑥 1 𝑥 32 , ∗ 𝐻3,4 (𝑥) = 3𝑥12 𝑥2 − 𝑥23 − 𝑥23 , ∗ 𝐻3,5 (𝑥) = 𝑥12 𝑥3 − 𝑥 22 𝑥 3 , ∗ 𝐻3,6 (𝑥) = 𝑥12 𝑥2 + 𝑥23 − 4𝑥 2 𝑥32 , ∗ 𝐻3,7 (𝑥) = 3𝑥12 𝑥3 + 3𝑥22 𝑥3 − 2𝑥33 . 𝑗

That means, all components 𝐵 𝛼 ≠ 0 are decomposed into an integer times a product of the prime numbers 2, 3. An easy calculation gives ∗ , 𝐻∗ ) (𝐻3,1 3,1 Hom3 ∗ , 𝐻∗ ) (𝐻3,2 3,2 Hom3 ∗ , 𝐻∗ ) (𝐻3,3 3,3 Hom3 ∗ , 𝐻∗ ) (𝐻3,4 3,4 Hom3 ∗ , 𝐻∗ ) (𝐻3,5 3,5 Hom3 ∗ , 𝐻∗ ) (𝐻3,6 3,6 Hom3 ∗ , 𝐻∗ ) (𝐻3,7 3,7 Hom3

= 24 = 1 · 23 · 31 , = 1 = 1 · 20 · 30 , = 40 = 5 · 23 · 30 , = 24 = 1 · 23 · 31 , = 4 = 1 · 22 · 30 , = 40 = 5 · 23 · 30 , = 60 = 5 · 22 · 31 .

Thus, the integers are decomposed into a (positive) integer times a product of prime numbers ≤ 3. Consequently, the orthonormal system √︃ ∗ , 𝐻 ∗ ) −1 𝐻 ∗ (𝐻𝑛, 𝑗 𝑛, 𝑗 Hom3 𝑛, 𝑗

(4.141)

∗ } (with respect to (·, ·)Hom3 ). corresponding to {𝐻𝑛, 𝑗 𝑗=1,...,7 may be listed as follows:

4.3 Exact Computation of Homogeneous Harmonic Polynomials

√︃

107

∗ ∗ , 𝐻 ∗ ) −1 (𝐻3,1 3,1 Hom3 𝐻3,1 (𝑥)

√︁ = (1 · 20 · 30 · 𝑥 13 𝑥 20 𝑥30 − 1 · 20 · 31 · 𝑥 11 𝑥 22 𝑥 30 )/ 1 · 23 · 31 , √︃ ∗ ∗ , 𝐻 ∗ ) −1 (𝐻3,2 3,2 Hom3 𝐻3,2 (𝑥) √︁ = (1 · 20 · 30 · 𝑥 11 𝑥 21 𝑥31 )/ 1 · 20 · 30 , √︃ ∗ ∗ , 𝐻 ∗ ) −1 (𝐻3,3 3,3 Hom3 𝐻3,3 (𝑥) = (1 · 20 · 30 · 𝑥 13 𝑥 20 𝑥30 + 1 · 20 · 30 · 𝑥11 𝑥22 𝑥30 √︁ −1 · 22 · 30 · 𝑥 11 𝑥20 𝑥32 )/ 5 · 23 · 30 , √︃ ∗ ∗ , 𝐻 ∗ ) −1 (𝐻3,4 3,4 Hom3 𝐻3,4 (𝑥) √︁ = (1 · 20 · 31 · 𝑥 12 𝑥 21 𝑥30 − 1 · 20 · 30 · 𝑥 10 𝑥 23 𝑥 30 )/ 1 · 23 · 31 , √︃ ∗ ∗ , 𝐻 ∗ ) −1 (𝐻3,5 𝐻3,5 (𝑥) 3,5 Hom3 √︁ = (1 · 20 · 30 · 𝑥 12 𝑥 20 𝑥31 − 1 · 20 · 30 · 𝑥 10 𝑥 22 𝑥 31 )/ 1 · 22 · 30 ,

√︃

∗ ∗ , 𝐻 ∗ ) −1 (𝐻3,6 𝐻3,6 (𝑥) 3,6 Hom3

√︁ = (1 · 20 · 30 · 𝑥12 𝑥21 𝑥30 + 1 · 20 · 30 · 𝑥10 𝑥23 𝑥30 − 5 · 23 · 30 · 𝑥10 𝑥21 𝑥32 )/ 1 · 22 · 30 , √︃ ∗ , 𝐻 ∗ ) −1 𝐻 ∗ (𝑥) (𝐻3,7 3,7 Hom3 3,7 = (1 · 20 · 31 · 𝑥12 𝑥20 𝑥31 + 1 · 20 · 31 · 𝑥10 𝑥22 𝑥31 √︁ −1 · 21 · 30 · 𝑥 10 𝑥 20 𝑥33 )/ 5 · 22 · 31 . Finally, the orthonormal system of homogeneous harmonic polynomials of degree 𝑛 (with respect to (·, ·)L2 (Ω) ) is given as follows √︃

∗ ∗ , 𝐻 ∗ ) −1 𝜇3 (𝐻3, 𝑗 3, 𝑗 Hom3 𝐻3, 𝑗 ,

with 𝜇3 =

𝑗 = 1 ...,7

105 1 · 3 · 5 · 7 = . 4𝜋 4𝜋

(4.142)

(4.143)

Our considerations have shown how a basis of Harm𝑛 can be computed entirely by integer operations from 2𝑛 + 1 systems of linear equations. The basis functions obtained can be orthonormalized exactly by means of the well-known Gram-Schmidt orthonormalization process. As a result, there are 2𝑛 + 1 homogeneous harmonic

108

4 Scalar Spherical Harmonics

polynomials available (orthonormalized in the sense of (·, ·)Hom𝑛 ) . However, the disadvantage in that approach is that the linear systems of equations result in basis functions which are all involved in the computational work of the orthonormalization. Later on (in Section 4.14), when Legendre harmonics come into play, an algorithm will be presented which reduces the amount of computational work by a factor less than 4, but which is close to 4 if the degree 𝑛 becomes large enough.

4.4 Definition of Scalar Spherical Harmonics

We begin by introducing scalar spherical harmonics. Essential tool is the theory of homogeneous harmonic polynomials. Definition 4.22 Let 𝐻𝑛 be a homogeneous harmonic polynomial of degree 𝑛 in R3 , i.e., 𝐻𝑛 ∈ Harm𝑛 (R3 ). The restriction 𝑌𝑛 = 𝐻𝑛 |Ω

(4.144)

is called a spherical harmonic of degree 𝑛. The space of all spherical harmonics of degree 𝑛, i.e., the set of all restrictions 𝑌𝑛 = 𝐻𝑛 |Ω, 𝐻𝑛 ∈ Harm𝑛 (R3 ), is denoted by Harm𝑛 (Ω). More explicitly, Harm𝑛 (Ω) = Harm𝑛 (R3 )|Ω.

(4.145)

Remark 4.23 In what follows, we simply write Harm𝑛 instead of Harm𝑛 (R3 ) (or Harm𝑛 (Ω)) if no confusion is likely to arise. We know already that the linear space Harm𝑛 is of dimension 2𝑛 + 1, that is 𝑑 (Harm𝑛 ) = 2𝑛 + 1. From Theorem 4.15, it follows that spherical harmonics of different orders are orthogonal in the sense of the L2 -inner product ∫ (𝑌𝑛 , 𝑌𝑚 )L2 (Ω) = 𝑌𝑛 (𝜉)𝑌𝑚 (𝜉) 𝑑𝜔(𝜉) = 0, 𝑛 ≠ 𝑚. (4.146) Ω

Using the standard method of separation, we have 𝐻𝑛 (𝑥) = 𝑟 𝑛𝑌𝑛 (𝜉), 𝑥 = 𝑟𝜉, 𝑟 = |𝑥|, 𝜉 ∈ Ω. Observing the identity   1 𝑑 2 𝑑 𝑛 𝑟 𝑟 = 𝑛(𝑛 + 1)𝑟 𝑛−2 (4.147) 𝑑𝑟 𝑟 2 𝑑𝑟 we obtain

4.4 Definition of Scalar Spherical Harmonics

109

0 = Δ𝐻𝑛 (𝑥) = 𝑟 𝑛−2 𝑛(𝑛 + 1)𝑌𝑛 (𝜉) + 𝑟 𝑛−2 Δ∗𝑌𝑛 (𝜉).

(4.148)

Thus, we are able to formulate the following lemma. Lemma 4.24 Any spherical harmonic 𝑌𝑛 , 𝑛 = 0, 1, . . . , is a twice differentiable eigenfunction of the Beltrami operator corresponding to the eigenvalue −𝑛(𝑛 + 1). More explicitly, (Δ∗ − (Δ∗ ) ∧ (𝑛))𝑌𝑛 (𝜉) = 0,

𝜉 ∈ Ω,

𝑌𝑛 ∈ Harm𝑛 ,

where the “spherical symbol” {(Δ∗ ) ∧ (𝑛)} 𝑛=0,1,... of the Beltrami operator Δ∗ is given by (Δ∗ ) ∧ (𝑛) = −𝑛(𝑛 + 1), 𝑛 = 0, 1, . . . . Remark 4.25 Throughout the book, for convenience, the capital letter 𝑌 followed by double indices, for example, 𝑌𝑛, 𝑗 , denotes a member of degree 𝑛 and order 𝑗 within an orthonormal system {𝑌𝑛,1 , . . . , 𝑌𝑛,2𝑛+1 } with respect to (·, ·)L2 (Ω) . A special realization of an L2 (Ω)-orthonormal system is presented in Section 4.12 (where we introduce a system involving associated Legendre functions). In terms of spherical harmonics, the addition theorem allows the following reformulation. Theorem 4.26 Let {𝑌𝑛, 𝑗 } 𝑗=1,...,2𝑛+1 be an L2 (Ω)-orthonormal system in Harm𝑛 . Then, for any pair (𝜉, 𝜂) ∈ Ω × Ω, 2𝑛+1 ∑︁

𝑌𝑛, 𝑗 (𝜉)𝑌𝑛, 𝑗 (𝜂) =

𝑗=1

2𝑛 + 1 𝑃𝑛 (𝜉 · 𝜂). 4𝜋

Proof Theorem 4.26 follows immediately from Theorem 4.16.

□

Suppose that t is an orthogonal transformation. Then 𝜉 ↦→ 𝑌𝑛, 𝑗 (t𝜉), 𝜉 ∈ Ω, is a spherical harmonic of degree 𝑛. Thus, we are able to write this function as linear combination 𝑌𝑛, 𝑗 (t𝜉) =

2𝑛+1 ∑︁ ∫ 𝑟=1

Ω

|

𝑌𝑛, 𝑗 (t𝜂)𝑌𝑛,𝑟 (𝜂) 𝑑𝜔(𝜂) 𝑌𝑛,𝑟 (𝜉). {z } =𝑐 𝑛𝑗,𝑟

Moreover, the addition theorem tells us that, for 𝜉, 𝜂 ∈ Ω,

(4.149)

110

4 Scalar Spherical Harmonics

𝑃𝑛 (t𝜉 · t𝜂) =

2𝑛+1 ∑︁

𝑌𝑛, 𝑗 (t𝜉)𝑌𝑛, 𝑗 (t𝜂)

(4.150)

𝑗=1

=

2𝑛+1 ∑︁ 2𝑛+1 ∑︁

𝑐 𝑛𝑗,𝑟 𝑌𝑛,𝑟 (𝜉)

𝑗=1 𝑟=1

=

2𝑛+1 ∑︁ 2𝑛+1 ∑︁

=

𝑐 𝑛𝑗,𝑠𝑌𝑛,𝑠 (𝜂)

𝑠=1

𝑌𝑛,𝑟 (𝜉)𝑌𝑛,𝑠 (𝜂)

𝑟=1 𝑠=1 2𝑛+1 ∑︁

2𝑛+1 ∑︁

2𝑛+1 ∑︁

𝑐 𝑛𝑗,𝑟 𝑐 𝑛𝑗,𝑠

𝑗=1

𝑌𝑛,𝑟 (𝜉)𝑌𝑛,𝑟 (𝜂)

𝑟=1

= 𝑃𝑛 (𝜉 · 𝜂) such that 𝛿 𝑗𝑙 =

2𝑛+1 ∑︁ ∫ 𝑟=1

∫ 𝑌𝑛, 𝑗 (t𝜂)𝑌𝑛,𝑟 (𝜂) 𝑑𝜔(𝜂)

Ω

𝑌𝑛,𝑙 (t𝜂)𝑌𝑛,𝑟 (𝜂) 𝑑𝜔(𝜂) Ω

∫ 𝑌𝑛, 𝑗 (t𝜂)𝑌𝑛,𝑙 (t𝜂) 𝑑𝜔(𝜂).

=

(4.151)

Ω

Lemma 4.27 If t ∈ 𝑂 (3), then the matrix ∫  𝑌𝑛, 𝑗 (t𝜂)𝑌𝑛,𝑟 (𝜂) 𝑑𝜔(𝜂) Ω

(4.152) 𝑗,𝑟=1,...2𝑛+1

is orthogonal. Because of 𝑃𝑛 (1) = 1, we find 2𝑛+1 ∑︁

(𝑌𝑛, 𝑗 (𝜉)) 2 =

𝑗=1

2𝑛 + 1 , 4𝜋

𝜉 ∈ Ω.

(4.153)

If we remember that every 𝑌𝑛 ∈ Harm𝑛 can be written in the form 𝑌𝑛 =

2𝑛+1 ∑︁

(𝑌𝑛 , 𝑌𝑛, 𝑗 )L2 (Ω) 𝑌𝑛, 𝑗 ,

𝑗=1

we immediately get the following lemma. Lemma 4.28 (Reproducing Kernel in Harm𝑛 ) For every 𝑌𝑛 ∈ Harm𝑛 ∫ 2𝑛 + 1 𝑌𝑛 (𝜂)𝑃𝑛 (𝜉 · 𝜂) 𝑑𝜔(𝜂) = 𝑌𝑛 (𝜉), 𝜉 ∈ Ω, 4𝜋 Ω

(4.154)

4.4 Definition of Scalar Spherical Harmonics

111

that is (𝜉, 𝜂) ↦→ 𝐾Harm𝑛 (𝜉, 𝜂) =

2𝑛 + 1 𝑃𝑛 (𝜉 · 𝜂), 4𝜋

(𝜉, 𝜂) ∈ Ω × Ω,

represents the (uniquely determined) reproducing kernel in Harm𝑛 . Moreover, (Δ∗𝜂 − (Δ∗ ) ∧ (𝑛))𝐾Harm𝑛 (𝜉, 𝜂) = 0,

𝜂 ∈ Ω,

holds for all 𝜉 ∈ Ω. From Lemma 4.28, we easily obtain the following result. Lemma 4.29 Let Harm0,...,𝑚 =

𝑚 Ê

Harm𝑛 .

(4.155)

𝑛=0

Then (𝜉, 𝜂) ↦→ 𝐾Harm0,...,𝑚 (𝜉, 𝜂) =

𝑚 ∑︁ 2𝑛 + 1 𝑛=0

4𝜋

𝑃𝑛 (𝜉 · 𝜂)

(4.156)

is the (uniquely) determined reproducing kernel in Harm0,...,𝑚 , i.e., 𝐾Harm0,...,𝑚 (𝜉, ·), 𝜉 ∈ Ω, is a member of Harm0,...,𝑚 with ∫ 𝑌 (𝜂)𝐾Harm (𝜉, 𝜂) 𝑑𝜔(𝜂) = 𝑌 (𝜉), 𝜉 ∈ Ω, (4.157) Ω

for all 𝑌 ∈ Harm0,...,𝑚 . Observing that ∫

(𝑌𝑛 (𝜉)) 2 𝑑𝜔(𝜉) =

Ω

2𝑛+1 ∑︁

(𝑌𝑛 , 𝑌𝑛, 𝑗 )L2 2 (Ω)

(4.158)

𝑗=1

we find in connection with (4.153) and (4.154) (𝑌𝑛 (𝜉)) 2 ≤

2𝑛+1 ∑︁

(𝑌𝑛 , 𝑌𝑛, 𝑗 )L2 2 (Ω)

𝑗=1

=

2𝑛+1 ∑︁

(𝑌𝑛, 𝑗 (𝜉)) 2

𝑗=1

2𝑛+1 2𝑛 + 1 ∑︁ (𝑌𝑛 , 𝑌𝑛, 𝑗 )L2 2 (Ω) . 4𝜋 𝑗=1

This yields the following lemma. Lemma 4.30 For every 𝑌𝑛 ∈ Harm𝑛

(4.159)

112

4 Scalar Spherical Harmonics

1 2𝑛 + 1 2 = sup |𝑌𝑛 (𝜉)| ≤ ∥𝑌𝑛 ∥ L2 (Ω) 4𝜋 𝜉 ∈Ω   1 ∫  1/2 2𝑛 + 1 2 (𝑌𝑛 (𝜉)) 2 𝑑𝜔(𝜉) = . 4𝜋 Ω 

∥𝑌𝑛 ∥ C(Ω)

(4.160)

In particular,  ∥𝑌𝑛, 𝑗 ∥ C(Ω) = sup |𝑌𝑛, 𝑗 (𝜉)| ≤ 𝜉 ∈Ω

2𝑛 + 1 4𝜋

 12 ,

(4.161)

𝑗 = 1, . . . , 2𝑛 + 1.

4.5 Legendre Polynomials

The function 𝑃𝑛 : [−1, 1] → R, 𝑛 = 0, 1, ..., defined by (4.68) 𝑛

𝑃𝑛 (𝑡) =

⌊2⌋ ∑︁

(−1) 𝑠

𝑠=0

(2𝑛 − 2𝑠)! 𝑡 𝑛−2𝑠 , 2𝑛 (𝑛 − 2𝑠)!(𝑛 − 𝑠)!𝑠!

𝑡 ∈ [−1, 1],

(4.162)

is called the Legendre polynomial. 𝑃𝑛 is uniquely determined by the properties: (i) 𝑃𝑛 is a polynomial of degree 𝑛 on the interval [−1, 1], ∫

1

𝑃𝑛 (𝑡)𝑃𝑚 (𝑡) 𝑑𝑡 = 0 for 𝑛 ≠ 𝑚,

(ii) −1

(iii) 𝑃𝑛 (1) = 1. This is easily seen from the usual process of orthogonalization. In particular, we have for 𝑛 = 0, . . . , 4 𝑃0 (𝑡) = 1, 𝑃3 (𝑡) =

3 2 1 𝑡 − , 2 2 35 4 15 2 3 𝑃4 (𝑡) = 𝑡 − 𝑡 + . 8 4 8

𝑃1 (𝑡) = 𝑡,

5 3 3 𝑡 − 𝑡, 2 2

𝑃2 (𝑡) =

(4.163) (4.164)

A graphical impression of some Legendre polynomials can be found in Fig. 4.1.

4.5 Legendre Polynomials

113

1

0.5

0 𝑃1 𝑃2 𝑃3 𝑃4

−0.5

−1 -1

0

-0.5

0.5

1

Fig. 4.1: Legendre polynomials 𝑡 ↦→ 𝑃𝑛 (𝑡), 𝑡 ∈ [−1, 1], 𝑛 = 1, . . . , 4.

Furthermore, ∫

1

𝑃𝑛 (𝑡)𝑃𝑚 (𝑡) 𝑑𝑡 = −1

2 𝛿 𝑛𝑚 . 2𝑛 + 1

(4.165)

Definition 4.31 For 𝜉 ∈ Ω, the function 𝑃𝑛 (𝜉·) : 𝜂 ↦→ 𝑃𝑛 (𝜉 · 𝜂), 𝜂 ∈ Ω, is called the scalar 𝜉-Legendre kernel of degree 𝑛. Applying the Cauchy-Schwarz inequality to the addition theorem (Theorem 4.26), we obtain for the scalar 𝜉-Legendre kernel 2𝑛+1 ∑︁ 2𝑛 + 1 𝑃 (𝜉 · 𝜂) = 𝑌 (𝜉)𝑌 (𝜂) (4.166) 𝑛 𝑛, 𝑗 𝑛, 𝑗 4𝜋 𝑗=1 v v u u t t 2𝑛+1 ∑︁ ∑︁
2 2𝑛+1
2 ≤ 𝑌𝑛, 𝑗 (𝜉) 𝑌𝑛, 𝑗 (𝜂) 𝑗=1

𝑗=1

2𝑛 + 1 | 𝑃𝑛 (1) | 4𝜋 2𝑛 + 1 = 𝑃𝑛 (1). 4𝜋 =

Therefore, it follows that |𝑃𝑛 (𝑡)| ≤ 𝑃𝑛 (1) = 1 , 𝑡 ∈ [−1, 1].

(4.167)

Moreover, the Legendre polynomial 𝑃𝑛 satisfies the estimate (see, for example, C. M¨uller (1952)) |𝑃𝑛(𝑘 ) (𝑡)| ≤ |𝑃𝑛(𝑘 ) (1)|, (4.168)

114

4 Scalar Spherical Harmonics

where 𝑃𝑛(𝑘 ) (1) = 𝑂 (𝑛2𝑘 ). In particular, we have 𝑛(𝑛 + 1) . 2

𝑃𝑛′ (1) =

(4.169)

Furthermore, for 𝑘 = 2, 3, . . . , 𝑛, 𝑃𝑛(𝑘 ) (1)

 𝑘 1 1 = 𝑛(𝑛 + 1) ((𝑛(𝑛 + 1) − 1 · 2) . . . (𝑛(𝑛 + 1) − 𝑘 (𝑘 − 1))) . 2 𝑘!

From Lemma 4.24 in combination with Theorem 4.26, it follows that 

(1 − 𝑡 2 )



𝑑 𝑑𝑡

2 − 2𝑡

 𝑑 + 𝑛(𝑛 + 1) 𝑃𝑛 (𝑡) = 0, 𝑑𝑡

where 

𝑑 𝐿 𝑡 = (1 − 𝑡 2 ) 𝑑𝑡

2

𝑑 − 2𝑡 𝑑𝑡

𝑡 ∈ [−1, 1],

(4.170)

! (4.171)

is the Legendre operator, i.e., the part of the Beltrami operator that depends only on the polar distance 𝑡. We therefore obtain the following lemma. Lemma 4.32 The Legendre polynomial 𝑃𝑛 is the only twice differentiable eigenfunction of the “Legendre operator” (4.171) on [−1, 1], corresponding to the eigenvalues −𝑛(𝑛 + 1), 𝑛 = 0, 1, ..., and bounded on [−1, 1] with 𝑃𝑛 (1) = 1. The differential equation (4.170) shows that 𝑃𝑛 and 𝑃𝑛′ cannot vanish simultaneously such that 𝑃𝑛 has no multiple zeros. The orthogonality relation for Legendre polynomials implies that 𝑃𝑛 has, at most, 𝑘 different zeros, 𝑧 1 , . . . 𝑧 𝑘 , 𝑘 ≤ 𝑛, in the interval (−1, 1). Letting 𝐼 𝑘 (𝑡) = (𝑡 − 𝑧 1 ) · · · (𝑡 − 𝑧 𝑘 )

(4.172)

we obtain 𝐼 𝑘 (1) > 0 and 𝑃𝑛 = 𝐽𝑛−𝑘 𝐼 𝑘 . The polynomial 𝐽𝑛−𝑘 is positive in [−1, 1], and we have ∫ 1 ∫ 1 𝑃𝑛 (𝑡)𝐼 𝑘 (𝑡) 𝑑𝑡 = 𝐽𝑛−𝑘 (𝑡)𝐼 𝑘2 (𝑡) 𝑑𝑡 > 0. (4.173) −1

−1

As 𝑃𝑛 is orthogonal to all polynomials of degree < 𝑛, this is possible only for the case 𝑘 = 𝑛. Thus, we can conclude that 𝑃𝑛 has 𝑛 different zeros in the interval (−1, 1). The zeros of the Legendre polynomial for 𝑛 = 1, 2, 3, 4 are listed in Table 4.2. The zeros of the Legendre polynomials 𝑃𝑛 are interlaced (cf. Fig. 4.1).

4.5 Legendre Polynomials

115

Table 4.2 Zeros of the Legendre polynomial. 𝑛=1 𝑛=2 𝑛=3

𝑧1 𝑧2 𝑧3 𝑧2 𝑧4 𝑧3

𝑛=4

=0 = −𝑧1 = −𝑧1 =0 = −𝑧1 = −𝑧2

= 0.5773502692... = 0.7745966692... = 0.8611363116... = 0.3399810436...

Lemma 4.33 The Legendre polynomial 𝑃𝑛 has 𝑛 different zeros in the interval (−1, 1). From the binomial theorem, it follows that (𝑡 2 − 1) 𝑛 =

𝑛 ∑︁

(−1) 𝑠

𝑛! 𝑡 2𝑛−2𝑠 , (𝑛 − 𝑠)!𝑠!

𝑛 = 0, 1, . . . .

(4.174)

𝑠=0

For all 𝑠 ≤ [𝑛/2] we find 

𝑑 𝑑𝑡

𝑛

𝑡 2𝑛−2𝑠 =

(2𝑛 − 2𝑠)! 𝑛−2𝑠 𝑡 , (𝑛 − 2𝑠)!

while for [𝑛/2] < 𝑠 ≤ 𝑛 we obtain  𝑛 𝑑 𝑡 2𝑛−2𝑠 = 0. 𝑑𝑡

(4.175)

(4.176)

Therefore, we see that 𝑛



𝑑 𝑑𝑡

𝑛

2

(𝑡 − 1)

𝑛

=

⌊2⌋ ∑︁ 𝑠=0

(−1) 𝑠

𝑛! (2𝑛 − 2𝑠)! 𝑛−2𝑠 𝑡 . (𝑛 − 𝑠)!𝑠! (𝑛 − 2𝑠)!

(4.177)

By comparison with the definition of the Legendre polynomial (4.68), we obtain the Rodriguez formula. Lemma 4.34 For 𝑛 = 0, 1, . . . ,  𝑛 1 𝑑 𝑃𝑛 (𝑡) = 𝑛 (𝑡 2 − 1) 𝑛 , 2 𝑛! 𝑑𝑡

𝑡 ∈ [−1, 1].

(4.178)

Integrating by parts, we obtain the Rodriguez rule ∫

1

−1

1 𝐹 (𝑡)𝑃𝑛 (𝑡) 𝑑𝑡 = 𝑛 2 𝑛!

∫

1

−1

𝐹 (𝑛) (𝑡)(1 − 𝑡 2 ) 𝑛 𝑑𝑡

(4.179)

116

4 Scalar Spherical Harmonics

for every 𝐹 ∈ C (𝑛) [−1, 1]. As an application of this formula, we discuss the integrals ∫

1

′ 𝑃 𝑘 (𝑡)(𝑃𝑛′ (𝑡) − 𝑃𝑛−2 (𝑡)) 𝑑𝑡 = −

−1

∫

1

(𝑃𝑛 (𝑡) − 𝑃𝑛−2 (𝑡))𝑃′𝑘 (𝑡) 𝑑𝑡.

(4.180)

−1

′ 𝑃𝑛′ − 𝑃𝑛−2 is a polynomial of degree 𝑛 − 1 on [−1, 1]. Consequently, the integral (4.180) vanishes for 𝑘 ≥ 𝑛. On the other hand, 𝑃′𝑘 is a polynomial of degree 𝑘 − 1. This means that the integral vanishes also for 𝑘 < 𝑛 − 1 so that (4.180) differs from zero only for 𝑘 = 𝑛 − 1. Thus ′ 𝑃𝑛′ (𝑡) − 𝑃𝑛−2 (𝑡) = 𝑐 𝑛 𝑃𝑛−1 (𝑡),

𝑡 ∈ [−1, 1].

(4.181)

In connection with (4.169), we find ′ 𝑃𝑛′ (1) − 𝑃𝑛−2 (1) = 2𝑛 − 1.

(4.182)

Therefore, it follows that ′ 𝑃𝑛′ (𝑡) − 𝑃𝑛−2 (𝑡) = (2𝑛 − 1)𝑃𝑛−1 (𝑡) ,

𝑡 ∈ [−1, 1].

(4.183)

𝑛 ≥ 2,

(4.184)

Equivalently, we have ∫ (2𝑛 − 1)

1

𝑃𝑛−1 (𝑡) 𝑑𝑡 = 𝑃𝑛−2 (𝑠) − 𝑃𝑛 (𝑠), 𝑠

for all 𝑠 ∈ [−1, 1]. By similar arguments, we are able to show that ′ 𝑃𝑛+1 (𝑡) − 𝑡𝑃𝑛′ (𝑡) = (𝑛 + 1)𝑃𝑛 (𝑡),

(4.185)

(𝑡 2 − 1)𝑃𝑛′ (𝑡) = 𝑛𝑡𝑃𝑛 (𝑡) − 𝑛𝑃𝑛−1 (𝑡),

(4.186)

(𝑛 + 1)𝑃𝑛+1 (𝑡) + 𝑛𝑃𝑛−1 (𝑡) − (2𝑛 + 1)𝑡𝑃𝑛 (𝑡) = 0.

(4.187)

The formulas (4.185)-not-(4.187) are known as recurrence formulas for the Legendre polynomials. Moreover, we have the following result. Lemma 4.35 For 𝑛 = 1, 2, . . ., 𝑡 ∈ [−1, 1], (𝑡 2 − 1)𝑃𝑛′ (𝑡) =

𝑛(𝑛 + 1) (𝑃𝑛+1 (𝑡) − 𝑃𝑛−1 (𝑡)). 2𝑛 + 1

Proof Inserting (4.187) into (4.186) we find

(4.188)

4.5 Legendre Polynomials

117

(𝑡 2 − 1)𝑃𝑛′ (𝑡) = 𝑛(𝑡𝑃𝑛 (𝑡) − 𝑃𝑛−1 (𝑡))   𝑛+1 𝑛 =𝑛 𝑃𝑛+1 (𝑡) + 𝑃𝑛−1 (𝑡) − 𝑃𝑛−1 (𝑡) 2𝑛 + 1 2𝑛 + 1   𝑛+1 𝑛 − (2𝑛 + 1) =𝑛 𝑃𝑛+1 (𝑡) + 𝑃𝑛−1 (𝑡) 2𝑛 + 1 2𝑛 + 1 𝑛(𝑛 + 1) (𝑃𝑛+1 (𝑡) − 𝑃𝑛−1 (𝑡)) . = 2𝑛 + 1

(4.189)

□

This is the desired result.

In addition, we mention the following results involving derivatives of the Legendre polynomial. Lemma 4.36 The following identities are valid: (i) For 𝑛 = 0, 1, . . . ′ 𝑃2𝑛+1 (𝑡) =

𝑛 ∑︁

(4𝑘 + 1)𝑃2𝑘 (𝑡).

(4.190)

(4𝑘 + 3)𝑃2𝑘+1 (𝑡).

(4.191)

(2𝑘 + 1)𝑃 𝑘 (𝑡) + 𝑛𝑃𝑛 (𝑡).

(4.192)

𝑘=0

(ii) For 𝑛 = 1, 2, . . . 𝑛−1 ∑︁

′ 𝑃2𝑛 (𝑡) =

𝑘=0

(iii) For 𝑛 = 1, 2, . . . (1 + 𝑡)𝑃𝑛′ (𝑡) =

𝑛−1 ∑︁ 𝑘=0

Proof We prove statement (iii) only. It is clear that there exist coefficients 𝑎 𝑛,0 , . . . , 𝑎 𝑛,𝑛 such that 𝑛 ∑︁ (1 + 𝑡)𝑃𝑛′ (𝑡) = 𝑎 𝑛,𝑘 𝑃 𝑘 (𝑡). (4.193) 𝑘=0

Now, by virtue of (4.165), ∫

1

𝑃𝑛′ (𝑡)(1 + 𝑡)𝑃𝑙 (𝑡) 𝑑𝑡 = 𝑎 𝑛,𝑙

−1

2 2𝑙 + 1

(4.194)

for 𝑙 = 0, . . . , 𝑛. Integration by parts yields ∫

1

−1

𝑃𝑛′ (𝑡)(1 + 𝑡)𝑃𝑙 (𝑡) 𝑑𝑡 = 2 − 𝛿 𝑛𝑙

2 − 2𝑙 + 1

∫

1

−1

𝑃𝑛 (𝑡)(1 + 𝑡)𝑃𝑙′ (𝑡) 𝑑𝑡.

(4.195)

118

4 Scalar Spherical Harmonics

For 𝑙 = 0, . . . , 𝑛−1 the last integral on the right-hand side vanishes, since (1+𝑡)𝑃𝑙′ (𝑡) is of degree ≤ 𝑛 − 1. Thus, it follows that 𝑎 𝑛,𝑙 = 2𝑙 + 1 for 𝑙 = 0, . . . , 𝑛 − 1. An easy calculation shows that 1 ∫ 1 ∫ 1 2 1 1 2 ′ 𝑃𝑛 (𝑡)𝑃𝑛 (𝑡)(1 + 𝑡) 𝑑𝑡 = 𝑃𝑛 (𝑡)(1 + 𝑡) − 𝑃 (𝑡) 𝑑𝑡 2 2 −1 𝑛 −1 −1 1 =1− , (4.196) 2𝑛 + 1 □

hence, 𝑎 𝑛,𝑛 = 𝑛. This gives the required result. The power series 𝜙(ℎ) =

∞ ∑︁

𝑃𝑛 (𝑡) ℎ 𝑛 ,

𝑡 ∈ [−1, 1],

(4.197)

𝑛=0

is absolutely and uniformly convergent for all ℎ with |ℎ| ≤ ℎ0 , ℎ0 ∈ [0, 1). By differentiation with respect to ℎ and comparing coefficients according to (4.197), we find (1 + ℎ2 − 2ℎ𝑡)𝜙′ (ℎ) = (𝑡 − ℎ)𝜙(ℎ). (4.198) This differential equation is uniquely solvable under the initial condition 𝜙(0) = 1. Since it is not hard to show that ℎ ↦→ (1 + ℎ2 − 2ℎ𝑡) −1/2 , ℎ ∈ (−1, 1),

(4.199)

solves this initial value problem, we have the following generating series expansion of the Legendre polynomials. Lemma 4.37 For 𝑡 ∈ [−1, 1] and all ℎ ∈ (−1, 1) ∞ ∑︁ 𝑛=0

𝑃𝑛 (𝑡)ℎ 𝑛 = √

1 . 1 + ℎ2 − 2ℎ𝑡

Among other areas of application, the subject of potential theory is concerned with forces of attraction due to the presence of a gravitational field. Central to the discussion of gravitational attraction is Newton’s law of gravitation for the force field generated by a single particle (cf. Chapter 12): the gravitational force 𝑓 in free space (i.e., free of point masses) is related to the potential function 𝐹 according to 𝑓 (𝑥) = −∇𝐹 (𝑥), 𝑥 ≠ 𝑦,

(4.200)

when the potential between a mass point 𝑦 and a point of free space 𝑥 has the form 𝐹 (𝑥) = 𝑘 |𝑥 − 𝑦| −1 , 𝑥 ≠ 𝑦

(4.201)

4.5 Legendre Polynomials

119

(𝑘 is the gravitational constant). Because of spherical symmetry of the gravitational field, the potential function of a single particle depends only upon the radial distance, i.e., the inner product of the direction vectors of 𝑥 and 𝑦. In order to obtain this result, let us suppose for the sake of definition 𝑥 = |𝑥|𝜉, 𝑦 = |𝑦|𝜂, 𝜉, 𝜂 ∈ Ω, |𝑥| < |𝑦|. Then we find ! −1/2  2 1 1 |𝑥| |𝑥| = 1+ −2 𝜉 ·𝜂 . (4.202) |𝑥 − 𝑦| |𝑦| |𝑦| |𝑦| Returning now to Lemma 4.37 with 𝑡 = 𝜉 ·𝜂 and ℎ = |𝑥|/|𝑦|, we find that the potential function has the series expansion 𝑛 ∞  1 1 ∑︁ |𝑥| = 𝑃𝑛 (𝜉 · 𝜂). |𝑥 − 𝑦| |𝑦| 𝑛=0 |𝑦|

(4.203)

Moreover, our considerations have shown that ∞

∑︁ (−1) 𝑛 1 1 = |𝑥| 𝑛 (𝜉 · ∇ 𝑦 ) 𝑛 , |𝑥 − 𝑦| 𝑛=0 𝑛! |𝑦| where

(−1) 𝑛 1 𝑃𝑛 (𝜉 · 𝜂) (𝜉 · ∇ 𝑦 ) 𝑛 = , 𝑛! |𝑦| |𝑦| 𝑛+1

𝑛 = 0, 1, . . .

(4.204)

(4.205)

Identity (4.205) is known as Maxwell’s representation formula. It shows that the Legendre polynomials may be obtained by repeated differentiations of the “fundamental solution” 𝑦 → |𝑦| −1 , 𝑦 ≠ 0, of the Laplace equation in the direction of the unit vector 𝜉. Thus, the potential on the right-hand side of Maxwell’s representation formula may be regarded as the potential of a pole of order 𝑛 with the axis 𝜉 at the origin. The power series in Lemma 4.37 can be differentiated for all ℎ ∈ (−1, 1). Thus, it follows that ∞ ∑︁ ℎ−𝑡 − = 𝑛𝑃𝑛 (𝑡)ℎ 𝑛−1 . (4.206) (1 + ℎ2 − 2ℎ𝑡) 3/2 𝑛=1 Now, it is easy to see that 1 2ℎ2 − 2ℎ𝑡 1 − ℎ2 − = . √ (1 + ℎ2 − 2ℎ𝑡) 3/2 1 + ℎ2 − 2ℎ𝑡 (1 + ℎ2 − 2ℎ𝑡) 3/2 This gives us the following result. Lemma 4.38 For all 𝑡 ∈ [−1, 1] and ℎ ∈ (−1, 1) ∞ ∑︁ 1 − ℎ2 = (2𝑛 + 1)ℎ 𝑛 𝑃𝑛 (𝑡). (1 + ℎ2 − 2ℎ𝑡) 3/2 𝑛=0

(4.207)

120

4 Scalar Spherical Harmonics

Lemma 4.37 can be used to prove an integral representation for the Legendre polynomial. To this end, we start from the well known elementary integral ∫ 𝜋 𝑑𝜑 𝜋 = √︁ (|𝛾| < 1). (4.208) 1 + 𝛾 cos 𝜑 0 1 − 𝛾2 We set

√ ℎ 𝑡2 − 1 𝛾 = − . 1 − ℎ𝑡

(4.209)

On the one hand, it follows that ∫ 𝜋 ∫ 𝜋 √︁ 1 𝑑𝜑 = (1 − ℎ𝑡) [1 − ℎ(𝑡 + 𝑡 2 − 1 cos 𝜑)] −1 𝑑𝜑 1 + 𝛾 cos 𝜑 0 0 ∫ 𝜋 ∑︁ ∞ √︁ = (1 − ℎ𝑡) (𝑡 + 𝑡 2 − 1 cos 𝜑) 𝑛 ℎ 𝑛 𝑑𝜑. 0

(4.210)

𝑛=0

On the other hand, we obtain 𝜋 √︁

1−

𝛾2

𝜋 𝜋(1 − ℎ𝑡) = √︁ = √ . 2 2 2 1 + ℎ2 − 2ℎ𝑡 1 − ℎ (𝑡 − 1)/(1 − ℎ𝑡)

(4.211)

In connection with Lemma 4.37, this yields 𝜋 √︁

1−

𝛾2

= (1 − ℎ𝑡)𝜋

∞ ∑︁

𝑃𝑛 (𝑡)ℎ 𝑛 .

(4.212)

𝑛=0

By comparison, we therefore obtain ∞ ∫ ∑︁ 𝑛=0

𝜋

∞ √︁ ∑︁ 𝑛 𝑛 2 (𝑡 + 𝑡 − 1 cos 𝜑) 𝑑𝜑 ℎ = 𝜋 𝑃𝑛 (𝑡)ℎ 𝑛 .

0

(4.213)

𝑛=0

This gives us the Laplace representation of Legendre polynomials . Lemma 4.39 For 𝑡 ∈ [−1, 1] and 𝑛 = 0, 1, . . . ∫ 𝜋 √︁ 1 𝑃𝑛 (𝑡) = (𝑡 + 𝑡 2 − 1 cos 𝜑) 𝑛 𝑑𝜑. 𝜋 0 √ By the representation (𝑖 = −1) 𝑃𝑛 (𝑡) =

1 2𝜋

∫

2𝜋



𝑛 √︁ 𝑡 + 𝑖 1 − 𝑡 2 cos 𝜑 𝑑𝜑

0

we obtain an estimate valid for arbitrary 𝑡 ∈ [−1, 1] (cf. C. M¨uller (1969)):

(4.214)

4.5 Legendre Polynomials

121

|𝑃𝑛 (𝑡)| ≤

1 2𝜋

∫

2𝜋

√︁ |𝑡 + 𝑖 1 − 𝑡 2 cos 𝜑| 𝑛 𝑑𝜑

(4.215)

0

√︁ ≤ (|𝑡| + | 1 − 𝑡 2 |) 𝑛 √︁ ≤ (|𝑡| + 1 + |𝑡| 2 ) 𝑛 ≤ 2𝑛 (1 + |𝑡| 2 ) 𝑛/2 . Moreover, it follows by the substitution 𝑠 = cos 𝜑 that (4.214) is equivalent to 1

∫

1 𝑃𝑛 (𝑡) = 𝜋

√︁ (𝑡 + 𝑖𝑠 1 − 𝑡 2 ) 𝑛 (1 − 𝑠2 ) −1/2 𝑑𝑠,

(4.216)

−1

√ so that we obtain from |𝑡 + 𝑖𝑠 1 − 𝑡 2 | = (1 − (1 − 𝑠2 )(1 − 𝑡 2 )) 1/2 the estimate |𝑃𝑛 (𝑡)| ≤

1 𝜋

∫

1

𝑛

𝑒 2 log(1− (1−𝑡

2 ) (1−𝑠 2 ) )

(1 − 𝑠2 ) −1/2 𝑑𝑠.

(4.217)

−1

Since log(1 − (1 − 𝑠2 )(1 − 𝑡 2 )) ≤ −(1 − 𝑠2 )(1 − 𝑡 2 ) we obtain 2 |𝑃𝑛 (𝑡)| ≤ 𝜋

∫

1

𝑛

𝑒 − 2 (1−𝑠

2 ) (1−𝑡 2 )

(1 − 𝑠2 ) −1/2 𝑑𝑠.

(4.218)

(4.219)

0

By the substitution 𝑠 = 1 − 𝑢, we find in connection with 𝑢 ≤ 1 − 𝑠2 ≤ 2𝑢 ∫ 1 𝑛 2 2 𝑒 − 2 𝑢(1−𝑡 ) 𝑢 −1/2 𝑑𝑢 𝜋 0 √ ∫∞ 𝑛 2 2 ≤ 𝑒 − 2 𝑢(1−𝑡 ) 𝑢 −1/2 𝑑𝑢 𝜋 0 √ 𝜋 2 = 𝜋 (𝑛(1 − 𝑡 2 )) 1/2   1/2 1 4 =√ . 𝜋 𝑛(1 − 𝑡 2 )

|𝑃𝑛 (𝑡)| ≤

(4.220)

Lemma 4.40 (Estimate of the Legendre polynomial) For 𝑛 = 1, 2, . . . and 𝑡 ∈ (−1, 1),   1/2 1 4 |𝑃𝑛 (𝑡)| ≤ √ . (4.221) 𝜋 𝑛(1 − 𝑡 2 ) Next, we discuss the circular average of a function on the unit sphere Ω. The problem is equivalent to the investigation of the spherical counterpart 𝑇ℎ of the so-called translation operator

122

4 Scalar Spherical Harmonics

𝑇ℎ (𝐹)(𝜉) =

√

1

2𝜋 1 − ℎ2

∫

𝐹 ∈ L2 (Ω),

𝐹 (𝜂) 𝑑𝜎(𝜂), 𝜉 · 𝜂=ℎ 𝜂 ∈Ω

ℎ ∈ (−1, 1), (4.222)

where 𝑑𝜎 is the line element in R3 . 𝑇ℎ is a bounded positive linear operator mapping L2 (Ω) into L2 (Ω) with the following properties: ||𝑇ℎ (𝐹)|| L2 (Ω) ≤ ||𝐹 || L2 (Ω) , ∫ ∫ 𝑇ℎ (𝐹)(𝜉)𝑌𝑛 (𝜉) 𝑑𝜔(𝜉) = 𝑃𝑛 (ℎ) 𝐹 (𝜉)𝑌𝑛 (𝜉) 𝑑𝜔(𝜉), Ω

(4.223) (4.224)

Ω

for 𝑛 = 0, 1, . . . , ℎ ∈ (−1, 1), 𝐹 ∈ L2 (Ω). In particular, 𝑇ℎ (𝑌𝑛 )(𝜉) = 𝑃𝑛 (ℎ)𝑌𝑛 (𝜉),

𝜉 ∈ Ω, 𝑌𝑛 ∈ Harm𝑛 .

(4.225)

Finally, lim ||𝐹 − 𝑇ℎ (𝐹)|| L2 (Ω) = 0.

(4.226)

ℎ→1 ℎ0

such that |𝐹 (𝜉) − 𝐹 (𝜂)| ≤ 𝜇(𝛾)

(4.245)

for all 𝜂 ∈ Ω with 1 − 𝛾 ≤ 𝜉 · 𝜂 ≤ 1. Thus, it follows that ∫ ∫ ≤ 𝜇(𝛾) 𝐵 (𝜉 · 𝜂)(𝐹 (𝜂) − 𝐹 (𝜉)) 𝑑𝜔(𝜂) 𝐵𝑛 (𝜉 · 𝜂) 𝑑𝜔(𝜂) 𝑛 1−𝛾 ≤ 𝜉 · 𝜂 ≤1

Ω

= 𝜇(𝛾). Summarizing our results, we obtain the following estimate

(4.246)

126

4 Scalar Spherical Harmonics

∫ sup 𝐵𝑛 (𝜉 · 𝜂)𝐹 (𝜂) 𝑑𝜔(𝜂) − 𝐹 (𝜉) 𝜉 ∈Ω Ω ∫ = sup 𝐵𝑛 (𝜉 · 𝜂) (𝐹 (𝜂) − 𝐹 (𝜉)) 𝑑𝜔(𝜂) 𝜉 ∈Ω Ω ∫ ∫ ≤ sup 𝐵𝑛 (𝜉 · 𝜂)𝐹 (𝜂) 𝑑𝜔(𝜂) − 𝐹 (𝜉) 𝐵𝑛 (𝜉 · 𝜂) 𝑑𝜔(𝜂) 𝜉 ∈Ω · 𝜂 ≤1−𝛾, −1≤ 𝜉 · 𝜂 ≤1−𝛾, −1≤ 𝜉| 𝜂|=1 | 𝜂|=1 ∫ + sup 𝐵𝑛 (𝜉 · 𝜂) (𝐹 (𝜂) − 𝐹 (𝜉)) 𝑑𝜔(𝜂) 𝜉 ∈Ω 𝜉 · 𝜂 ≤1, 1−𝛾|≤𝜂|=1 ∫ ≤ sup 𝐵𝑛 (𝜉 · 𝜂)𝐹 (𝜂) 𝑑𝜔(𝜂) 𝜉 ∈Ω · 𝜂 ≤1−𝛾, −1≤ 𝜉| 𝜂|=1 ∫ + ∥𝐹 ∥ C(Ω) 𝐵𝑛 (𝜉 · 𝜂) 𝑑𝜔(𝜂) + 𝜇(𝛾) −1≤ 𝜉 · 𝜂 ≤1−𝛾, | 𝜂|=1

 𝛾  𝑛+1 ≤ 2∥𝐹 ∥ C(Ω) 1 − + 𝜇(𝛾). 2 This shows us that ∫ lim sup 𝐵𝑛 (𝜉 · 𝜂)𝐹 (𝜂)𝑑𝜔(𝜂) − 𝐹 (𝜉) ≤ 𝜇(𝛾) 𝑛→∞ 𝜉 ∈Ω

(4.247)

Ω

for every 𝛾 ∈ (0, 1). Since 𝜇(𝛾) → 0 as 𝛾 → 0, we obtain the following result. Theorem 4.43 For 𝐹 ∈ C(Ω) ∫ lim sup 𝐵𝑛 (𝜉 · 𝜂)𝐹 (𝜂) 𝑑𝜔(𝜂) − 𝐹 (𝜉) = 0. 𝑛→∞ 𝜉 ∈Ω

Ω

Now, it can be readily seen that 𝐵𝑛 (𝑡) =

𝑛 ∑︁ 𝑘=0

for all 𝑡 ∈ [−1, 1]. This shows us that

𝐵∧𝑛 (𝑘)

2𝑘 + 1 𝑃 𝑘 (𝑡) 4𝜋

(4.248)

4.6 Orthogonal (Fourier) Expansions

127

∫ 𝐵𝑛 (𝜉 · 𝜂)𝐹 (𝜂) 𝑑𝜔(𝜂)

(4.249)

Ω

= =

𝑛 ∑︁ 𝑘=0 𝑛 ∑︁

𝐵∧𝑛 (𝑘)

2𝑘 + 1 4𝜋

∫ 𝑃 𝑘 (𝜉 · 𝜂)𝐹 (𝜂) 𝑑𝜔(𝜂) Ω

𝐵∧𝑛 (𝑘) ProjHarm𝑘 (𝐹)(𝜉).

𝑘=0

Thus, we finally have the “Bernstein summability” of a Fourier series expansion in terms of spherical harmonics. Theorem 4.44 For 𝐹 ∈ C(Ω), ∑︁ 2𝑘+1 ∑︁ 𝑛 ∧ ∧ lim sup 𝐵𝑛 (𝑘) 𝐹 (𝑘, 𝑗)𝑌𝑘, 𝑗 (𝜉) − 𝐹 (𝜉) = 0. 𝑛→∞ 𝜉 ∈Ω 𝑘=0 𝑗=1

(4.250)

Theorem 4.44 enables us to prove the closure of the system of spherical harmonics in the space C(Ω). Corollary 4.45 The system {𝑌𝑛, 𝑗 } 𝑛=0,1,..., 𝑗=1,...,2𝑛+1 is closed in C(Ω), that is for any given 𝜀 > 0 and each 𝐹 ∈ C(Ω), there exists a linear combination 𝑁 2𝑘+1 ∑︁ ∑︁

𝑑 𝑘, 𝑗 𝑌𝑘, 𝑗

𝑘=0 𝑗=1

such that



𝑁 2𝑘+1 ∑︁ ∑︁

𝐹 −

𝑑 𝑌 𝑘, 𝑗 𝑘, 𝑗

𝑘=0 𝑗=1

≤ 𝜀.

C(Ω)

Proof Given 𝐹 ∈ C(Ω). Then, for any given 𝜀 > 0, there exists an integer 𝑁 = 𝑁 (𝜀) such that ∑︁ 𝑁 2𝑘+1 ∑︁ sup 𝐵∧𝑁 (𝑘)𝐹 ∧ (𝑘, 𝑗) 𝑌𝑘, 𝑗 (𝜉) − 𝐹 (𝜉) ≤ 𝜀, (4.251) 𝜉 ∈Ω 𝑘=0 𝑗=1 | {z } =𝑑 𝑘, 𝑗

which proves Corollary 4.45.

□

The point of departure for the Abel-Poisson summability is Lemma 4.38 from which we obtain ∫ 1 ∫ 1 ∞ ∑︁ 1 − ℎ2 𝑛 𝑑𝑡 = (2𝑛 + 1)ℎ 𝑃𝑛 (𝑡) 𝑑𝑡 (4.252) 2 3/2 −1 (1 + ℎ − 2ℎ𝑡) −1 𝑛=0

128

4 Scalar Spherical Harmonics

for all ℎ ∈ (−1, 1). Since

∫1 −1

1 2

𝑃𝑛 (𝑡)𝑃0 (𝑡) 𝑑𝑡 = 0 for 𝑛 ≥ 1, we finally have

∫

1

−1

1 − ℎ2 𝑑𝑡 = 1. (1 + ℎ2 − 2ℎ𝑡) 3/2

(4.253)

Theorem 4.46 (Abel-Poisson Integral Formula) If 𝐹 is continuous on Ω, then ∫ 1 (1 − ℎ2 )𝐹 (𝜂) lim sup 𝑑𝜔(𝜂) − 𝐹 (𝜉) = 0. ℎ→1,ℎ 0 and any given 𝐹 ∈ C(Ω), there exists a linear combination 𝑁 2𝑛+1 ∑︁ ∑︁

𝑏 𝑛, 𝑗 𝑌𝑛, 𝑗

𝑛=0 𝑗=1

such that



𝑁 2𝑛+1 ∑︁ ∑︁

𝐹 −

𝑏 𝑌 𝑛, 𝑗 𝑛, 𝑗

𝑛=0 𝑗=1

≤ 𝜀. L2 (Ω)

Proof Corollary 4.49 follows immediately from Corollary 4.48 by using the norm estimate (2.92). □ Theorem 4.50 The system {𝑌𝑛, 𝑗 } 𝑛=0,1,..., with respect to ∥ · ∥ L2 (Ω) .

𝑗=1,...,2𝑛+1

is closed in the space L2 (Ω)

132

4 Scalar Spherical Harmonics

Proof C(Ω) is dense in L2 (Ω), that is for every 𝐹 ∈ L2 (Ω) there exists a function 𝐺 ∈ C(Ω) with ∥𝐹 − 𝐺 ∥ L2 (Ω) ≤ 𝜀/2. The function 𝐺 ∈ C(Ω) admits an arbitrarily close approximation by finite linear combinations of spherical harmonics. Therefore, the proof of the closure is clear. □ Truncated spherical harmonic expansions admit the following minimum property which should be mentioned for the convenience of the reader. Lemma 4.51 Assume that 𝐹 ∈ L2 (Ω). Then



𝑚 2𝑛+1 ∑︁ ∑︁

∧

𝐹 −

𝐹 (𝑛, 𝑗)𝑌 = inf ∥𝐹 − 𝑌 ∥ L2 (Ω) , 𝑛, 𝑗

𝑌 ∈Harm0,...,𝑚

2 𝑛=0 𝑗=1 L (Ω)

i.e., the problem of finding the linear combination in Harm0,...,𝑚 which is minimal in the L2 (Ω)-norm is solved by the orthogonal projection ProjHarm0,...,𝑚 (𝐹) of 𝐹 onto Harm0,...,𝑚 . Proof An easy calculation shows that

2

𝑚 2𝑛+1 ∑︁ ∑︁

𝐹 −

𝐿 𝑌 𝑛, 𝑗 𝑛, 𝑗

2 𝑛=0 𝑗=1

(4.265)

L (Ω)

= (𝐹, 𝐹)L2 (Ω) −

𝑚 2𝑛+1 ∑︁ ∑︁ 𝑛=0 𝑗=1

| 𝐹 ∧ (𝑛, 𝑗) | 2 +

𝑚 2𝑛+1 ∑︁ ∑︁

| 𝐿 𝑛, 𝑗 − 𝐹 ∧ (𝑛, 𝑗) | 2

𝑛=0 𝑗=1

for arbitrarily given coefficients 𝐿 𝑛, 𝑗 ∈ R. Therefore, the minimum of the righthand side of the Eq. (4.265) is achieved if and only if 𝐿 𝑛, 𝑗 = 𝐹 ∧ (𝑛, 𝑗) for 𝑛 = 0, . . . , 𝑚, 𝑗 = 1, . . . , 2𝑛 + 1. Moreover,

2

𝑚 2𝑛+1 ∑︁ ∑︁

∧

𝐹 − 𝐹 (𝑛, 𝑗)𝑌𝑛, 𝑗



2 𝑛=0 𝑗=1

L (Ω)

= (𝐹, 𝐹)L2 (Ω) −

𝑚 2𝑛+1 ∑︁ ∑︁

|𝐹 ∧ (𝑛, 𝑗)| 2 .

(4.266)

𝑛=0 𝑗=1

We summarize our results in the fundamental theorem of orthogonal (spherical harmonic) expansions. Theorem 4.52 The closure of the system {𝑌𝑛, 𝑗 } 𝑛=0,1,2.... in L2 (Ω) is equivalent to 𝑗=1,...,2𝑛+1 each of the following statements: (i) The orthogonal expansion of any element 𝐻 ∈ L2 (Ω) converges in norm to 𝐻, i.e.,

4.6 Orthogonal (Fourier) Expansions

133



𝑚 2𝑛+1 ∑︁ ∑︁

lim

𝐻 − (𝐻, 𝑌𝑛, 𝑗 )L2 (Ω) 𝑌𝑛, 𝑗

𝑚→∞

𝑛=0 𝑗=1

= 0. L2 (Ω)

(ii) Parseval’s identity holds. That is, for any 𝐻 ∈ L2 (Ω), ∥𝐻 ∥ 2𝐿 2 (Ω) = (𝐻, 𝐻)L2 (Ω) =

∞ 2𝑛+1 ∑︁ ∑︁

| (𝐻, 𝑌𝑛, 𝑗 )L2 (Ω) | 2 .

𝑛=0 𝑗=1

(iii) The extended Parseval identity holds. That is for any 𝐻, 𝐾 ∈ L2 (Ω), (𝐻, 𝐾) 𝐿 2 (Ω) =

∞ 2𝑛+1 ∑︁ ∑︁

(𝐻, 𝑌𝑛, 𝑗 )L2 (Ω) (𝐾, 𝑌𝑛, 𝑗 )L2 (Ω) .

𝑛=0 𝑗=1

(iv) There is no strictly larger orthonormal system containing the orthonormal system {𝑌𝑛, 𝑗 } 𝑛=0,1,..., . 𝑗=1,...,2𝑛+1

(v) The system {𝑌𝑛, 𝑗 }

𝑛=0,1,...; 𝑗=1,...,2𝑛+1

has the completeness property. That is, 𝐻 ∈ L2 (Ω)

and (𝐻, 𝑌𝑛, 𝑗 )L2 (Ω) = 0 for all 𝑛 = 0, 1, ..., 𝑗 = 1, ..., 2𝑛 + 1, implies 𝐻 = 0. (vi) An element 𝐻 of L2 (Ω) is determined uniquely by its orthogonal coefficients. That is, if (𝐻, 𝑌𝑛, 𝑗 )L2 (Ω) = (𝐾, 𝑌𝑛, 𝑗 )L2 (Ω) , 𝑛 = 0, 1, ..., 𝑗 = 1, ..., 2𝑛 + 1,

(4.267)

then 𝐻 = 𝐾. The proof of Theorem 4.52 is omitted (see, for example, P.J. Davis (1963)). The property (i) is of great importance for practical purposes. In particular, it tells us that any continuous function may be approximated (in the L2 (Ω)-sense) by finite truncations of its Fourier (orthogonal) expansion in terms of any L2 (Ω)-orthonormal system of spherical harmonics {𝑌𝑛, 𝑗 } 𝑛=0,1,2.... . 𝑗=1,...,2𝑛+1

Finally, we are interested in pointwise approximation (see, e.g., C. M¨uller (1998)). Theorem 4.53 Let 𝐹 be continuous in the point 𝜉 ∈ Ω. Moreover, assume that 𝐹 is bounded on Ω. Furthermore, suppose that the sequence {𝑆 𝑛 (𝐹)} 𝑛=0,1,... 𝑆 𝑛 (𝐹)(𝜉) =

𝑛 2𝑘+1 ∑︁ ∑︁ 𝑘=0 𝑗=1

𝐹 ∧ (𝑘, 𝑗) 𝑌𝑘, 𝑗 (𝜉)

134

4 Scalar Spherical Harmonics

converges in 𝜉 ∈ Ω. Then 𝐹 (𝜉) = lim 𝑆 𝑛 (𝐹)(𝜉) =

∞ 2𝑘+1 ∑︁ ∑︁

𝐹 ∧ (𝑘, 𝑗)𝑌𝑘, 𝑗 (𝜉).

(4.268)

𝑛→∞ 𝑘=0 𝑗=1

Proof Observing that 𝑆 𝑛 (𝐹)(𝜉) =

∫ 𝑛 ∑︁ 2𝑘 + 1 𝑃 𝑘 (𝜉 · 𝜂)𝐹 (𝜂) 𝑑𝜔(𝜂) 4𝜋 Ω 𝑘=0

(4.269)

we obtain from the Abel-Poisson summability 𝐹 (𝜉) = lim ℎ→1 ℎ 1. 2

=

(9.77)

8 (1)

Δ𝑜𝑄˜ ℎ −Δ∗ Δ𝑄 ˜

6

ℎ (1)

−Δ Δ𝑜𝑄˜ Δ𝑄 ˜ ℎ

∗

ℎ

4

2

0

0

0.2

0.4

0.6

0.8

1

Fig. 9.5: Abel-Poisson kernel uncertainty classification. Shown are the functions (1) −Δ∗ , and ℎ ↦→ Δ𝑜 (1) Δ −Δ∗ . ℎ ↦→ Δ𝑜𝑄˜ , ℎ ↦→ Δ𝑄 ˜ 𝑄˜ 𝑄˜ ℎ

ℎ

ℎ

ℎ

(1)

∗

−Δ is independent of ℎ. All intermediate Note that, in this case, the value Δ𝑜𝑄˜ Δ𝑄 ˜ℎ ℎ cases of “space-frequency localization” are realized by the Abel-Poisson kernel, but it should be emphasized that the Abel-Poisson kernel does not satisfy a minimum uncertainty state.

9.6 Localization of Representative Zonal Kernels

481

Localization of the Dirac Kernel. Letting ℎ formally tend to 1 in the results provided by the uncertainty principle for the Abel-Poisson kernel function we are able to interpret the localization properties of the Dirac kernel on Ω satisfying 𝛿∧ (𝑛) = 1 for all 𝑛 ∈ N0 , so that 𝛿(𝜉 · 𝜂) =

∞ ∑︁ 2𝑛 + 1 𝑛=0

4𝜋

𝑃𝑛 (𝜉 · 𝜂),

𝜉, 𝜂 ∈ Ω,

(9.78)

where the convergence is understood in distributional sense. As a matter of fact, letting ℎ tend to 1 shows us that the variances in the space domain take the constant value 0. On the other hand, the variances in the frequency domain converge to ∞. Hence, the Dirac kernel shows ideal space localization, but no frequency localization. Localization of Non-Bandlimited/Non-Spacelimited Gaussian Kernels. The minimum uncertainty state within the uncertainty relation is provided by the Gaussian probability density function (see W. Freeden et al. (1998), N. La´ın Fern´andez (2003), N. La´ın Fern´andez, J. Prestin (2003)). Consider the function 𝐺 𝜆 given by 𝐺 𝜆 (𝑡) = 𝑒 − (𝜆/2) (1−𝑡 ) ,

𝑡 ∈ [−1, 1],

𝜆 > 0.

(9.79)

An elementary calculation shows us that 𝐺˜ 𝜆 (𝑡) = 𝛾(𝜆)𝑒 − (𝜆/2) (1−𝑡 ) , with √ 𝛾(𝜆) = (1/ 4𝜋)



1 (1 − 𝑒 −2𝜆 ) 2𝜆

(9.80)

 −1/2 ,

(9.81)

𝑜 (1) Δ −Δ∗ → 1 as 𝜆 → ∞ satisfies ∥ 𝐺˜ 𝜆 ∥ L2 [−1,1] = 1. It is not difficult to deduce that Δ𝐺 ˜ 𝜆 𝐺˜ 𝜆 (note that the best value of the uncertainty principle (Theorem 9.11) is 1.

Localization of Non-Bandlimited/Spacelimited Haar Kernels. The Haar kernel is given by 𝐵 ℎ(0) (𝑡) =



0, 𝑡 ∈ [−1, ℎ), 1, 𝑡 ∈ [ℎ, 1].

(9.82)

The Haar kernel is discontinuous so that we also think of smoothed versions of it. More explicitly, for 𝑘 ∈ N, the smoothed Haar kernels are defined by 𝐵 ℎ(𝑘 ) (𝑡)

 𝑡 ∈ [−1, ℎ),   0,  = (𝑡 − ℎ) 𝑘   (1 − ℎ) 𝑘 , 𝑡 ∈ [ℎ, 1]. 

(9.83)

482

9 Scalar Zonal Kernel Functions

For 𝑘 ∈ N0 , an elementary calculation gives ∥𝐵 ℎ(𝑘 ) ∥ 2L2 (Ω) = 2𝜋

∫

1

−1

[𝐵 ℎ(𝑘 ) (𝑡)] 2 𝑑𝑡 = 2𝜋

1−ℎ . 2𝑘 + 1

(9.84)

This leads us to the normalized Haar kernels √︄ 2𝑘 + 1 (𝑘 ) 𝐵˜ ℎ = 𝐵 (𝑘 ) , 2𝜋(1 − ℎ) ℎ

(9.85)

since our uncertainty properties are formulated for kernels with norm 1. 1.5

ℎ = 0.3 ℎ = 0.7 ℎ = 0.9

1

1

0.5

0.5

0 −𝜋

ℎ = 0.3 ℎ = 0.7 ℎ = 0.9

0 − 𝜋/2

0

𝜋

𝜋/2

0

10

20

Fig. 9.6: The Haar kernel 𝐵˜ ℎ(0) for ℎ = 0.3, 0.7, 0.9. The illustrations show the space representation 𝜗 ↦→ 𝐵˜ ℎ(0) (cos(𝜗)), 𝜗 ∈ [−𝜋, 𝜋] (left) and frequency representation 𝑛 ↦→ ( 𝐵˜ ℎ(0) ) ∧ (𝑛), 𝑛 ∈ N0 (right).

An elementary calculation yields ∫

(1)

𝑔 𝑜˜ (𝑘)

𝐵ℎ ( · 𝜖 3 )

1

 2 1 + ℎ + 2𝑘 3 𝑡 𝐵˜ ℎ(𝑘 ) (𝑡) 𝑑𝑡 𝜖 3 = 𝜖 . 2 + 2𝑘 −1

= 2𝜋

(9.86)

Consequently, 

(1)

2

𝜎 𝑜˜ (𝑘)

 =1−

𝐵ℎ

1 + ℎ + 2𝑘 2 + 2𝑘

2 =

(1 − ℎ)(ℎ + 4𝑘 + 3) . (2𝑘 + 2) 2

(9.87)

In connection with (9.39) we finally arrive at (1)

Δ𝑜˜ (𝑘) = 𝐵ℎ

√︁ 1 (1 − ℎ)(ℎ + 4𝑘 + 3). 1 + ℎ + 2𝑘

For the localization in frequency, we first assume 𝑘 ≥ 2. We have

(9.88)

9.6 Localization of Representative Zonal Kernels



2

(3)

𝜎 𝑜˜ (𝑘)

𝐵ℎ (

∫

1

= −2𝜋

·𝜖 3)

−1

483

𝐵˜ ℎ(𝑘 ) (𝑡)𝐿 𝑡 𝐵˜ ℎ(𝑘 ) (𝑡) 𝑑𝑡

2𝑘 + 1 −2𝜋 = 2𝜋(1 − ℎ) (1 − ℎ) 2𝑘 𝑘 (ℎ + 2𝑘) = , (1 − ℎ) (2𝑘 − 1) so that

√︄ (3) Δ𝑜˜ (𝑘) 𝐵ℎ

=

1

∫

(𝑡 − ℎ) 𝑘 𝐿 𝑡 (𝑡 − ℎ) 𝑘 𝑑𝑡

(9.89)

ℎ

𝑘 (ℎ + 2𝑘) . (1 − ℎ)(2𝑘 − 1)

(9.90)

Fig. 9.7 gives a graphical impression of these results for the special cases 𝑘 = 1 and 𝑘 = 3. For particular choices of values ℎ, 𝑘, we obtain the expected results. 4

4 (1)

(1)

Δ𝑜𝐵˜ ℎ∗ Δ−Δ 𝐵˜

3

3

ℎ (1)

Δ𝑜𝐵˜ Δ−Δ 𝐵˜ ℎ

ℎ (1)

∗

Δ𝑜𝐵˜ Δ−Δ 𝐵˜

ℎ

ℎ

2

2

1

1

0 −1

−0.5

Δ𝑜𝐵˜ ℎ∗ Δ−Δ 𝐵˜

0

0 −1

1

0.5

∗

ℎ

−0.5

0

0.5

1

Fig. 9.7: Uncertainty classification of the normalized smoothed Haar kernel function (1) (3) (1) (3) 𝐵˜ (𝑘 ) (𝑘 = 1, left; 𝑘 = 3 right). Δ𝑜 (𝑘) , Δ𝑜 (𝑘) and Δ𝑜 (𝑘) Δ𝑜 (𝑘) are shown as functions 𝐵˜ ℎ

ℎ

𝐵˜ ℎ

𝐵˜ ℎ

𝐵˜ ℎ

of ℎ ∈ [−1, 1]. Localization of Bandlimited/Non-Spacelimited Shannon Kernels. The dyadic Shannon scaling function is given by 𝑆𝐻0,...,2 𝐽 −1 (𝜉 · 𝜂) =

𝐽 −1 2∑︁

𝑆𝐻0,...,2 𝐽 −1 ∧ (𝑛)

𝑛=0

where

( 𝑆𝐻0,...,2 𝐽 −1 ∧ (𝑛) =

1, 0,

2𝑛 + 1 𝑃𝑛 (𝜉 · 𝜂), 4𝜋

if 𝑛 ∈ [0, 2 𝐽 ), . else

𝜉, 𝜂 ∈ Ω,

(9.91)

(9.92)

The dyadic Shannon scaling function 𝑆𝐻0,...,2 𝐽 −1 shows undesired oscillation phenomena, which increase heavily with increasing scales 𝐽 (Fig. 9.8). Mathematically spoken, we understand oscillations as sign changes of the slope. In the Shannon case

484

9 Scalar Zonal Kernel Functions

it is not hard to see in a more qualitative way that 𝑆𝐻0,...,2 𝐽 −1 possesses only simple zeros which imply slope changes. In fact, from the alternative representation 𝑆𝐻0,...,2 𝐽 −1 (𝑡) =

𝐽 −1 2∑︁

𝑛=0

2𝑛 + 1 𝑃𝑛 (𝑡) = 4𝜋

(

2 𝐽 𝑃2 𝐽 (𝑡 ) −𝑃2 𝐽 −1 (𝑡 ) , 4𝜋 𝑡 −1 22𝐽 4𝜋 ,

𝑡 ∈ [−1, 1)

(9.93)

𝑡 = 1,

we are able to deduce that the following statement holds true (for the proofs the reader is referred to M.J. Fengler et al. (2006a)). Lemma 9.12 (Zeros of the Dyadic Shannon Scaling Function) For 𝐽 ∈ N0 , the dyadic Shannon scaling function 𝑆𝐻0,...,2 𝐽 −1 possesses exactly 2 𝐽 − 1 distinct, simple zeros in the interval [-1,1]. As an immediate consequence we find that the dyadic Shannon scaling function of scale 𝐽 ≥ 2 possesses 2 𝐽 − 2 sign changes of its slope on (−1, 1), i.e., oscillations. Let 𝐽 ∈ N0 and 𝜂 ∈ Ω be fixed. Then the L2 (Ω)-norm of the dyadic Shannon scaling function 𝑆𝐻0,...,2 𝐽 −1 (·, 𝜂) of scale 𝐽 is given by 2 𝐽 −1 ∥𝑆𝐻0,...,2 𝐽 −1 (·, 𝜂)∥ L2 (Ω) = √ . 𝜋

(9.94)

The normalized dyadic Shannon scaling function is given by g 𝑆𝐻 0,...,2 𝐽 −1 (·𝜂) =

𝑆𝐻0,...,2 𝐽 −1 (·𝜂) . ∥𝑆𝐻0,...,2 𝐽 −1 (·, 𝜂)∥ L2 (Ω)

(9.95)

Theorem 9.13 (Space Uncertainty of the Dyadic Shannon Scaling Function) Assume that 𝐽 ∈ N and 𝜂 ∈ Ω is fixed. Then the space uncertainty of the L2 (Ω)normalized Shannon scaling function g 𝑆𝐻 0,...,2 𝐽 −1 (·, 𝜂) of scale 𝐽 is given by √ (1) Δ𝑜g 𝑆𝐻 0,...,2 𝐽 −1 (·, 𝜂)

2 𝐽+1 − 1 , 2𝐽 − 1

=

(9.96)

where the expected value and the variance in the space domain are given by (1)

𝑔 𝑜g

𝑆𝐻 0,...,2 𝐽 −1 (·, 𝜂)



(1)

𝑜 𝜎g

𝑆𝐻 0,...,2 𝐽 −1 (·, 𝜂)

=

2𝐽 − 1 𝜂, 2𝐽

(9.97)

=

2 𝐽+1 − 1 . 22𝐽

(9.98)

2

9.6 Localization of Representative Zonal Kernels 𝐽 =2 𝐽 =3 𝐽 =4

4

485 𝐽 =2 𝐽 =3 𝐽 =4

0.8 0.6

2 0.4 0.2

0 −𝜋

− 𝜋/2

0

𝜋

𝜋/2

0

0

5

10

15

20

Fig. 9.8: The normalized dyadic Shannon kernel g 𝑆𝐻 0,...,2 𝐽 −1 (cos(𝜗)) for 𝐽 = 2, 𝐽 = 3, and 𝐽 = 4. The illustrations show the space representation 𝜗 ↦→ g 𝑆𝐻 0,...,2 𝐽 −1 (cos(𝜗)), 𝜗 ∈ [−𝜋, 𝜋] (left) and frequency representation 𝑛 ↦→ (g 𝑆𝐻 0,...,2 𝐽 −1 ) ∧ (𝑛), 𝑛 ∈ N0 (right).

Obviously, the last result indicates together with Theorem 9.12 that the quantity for the space uncertainty does not reflect the amount of oscillations. Theorem 9.14 (Frequency Uncertainty of the Dyadic Shannon Scaling Function) Let 𝐽 ∈ N and 𝜂 ∈ Ω be fixed. Then the uncertainty in frequency of the L2 (Ω)normalized dyadic Shannon scaling function g 𝑆𝐻 0,...,2 𝐽 −1 (·, 𝜂) of scale 𝐽 is given by √ 22𝐽 − 1 −Δ∗ 𝑜 (3) Δg = Δg = (9.99) √ 𝑆𝐻 0,...,2 𝐽 −1 (·, 𝜂) 𝑆𝐻 0,...,2 𝐽 −1 (·, 𝜂) 2 where the expected value (which coincides with the variance with respect to the operator 𝐿 ∗ ) and the variance in the frequency domain (with respect to the negative Beltrami operator −Δ∗ ) read as follows ∗ 𝑔 −Δ g 𝑆𝐻 0,...,2 𝐽 −1 (·, 𝜂)



−Δ 𝜎g

∗

𝑆𝐻 0,...,2 𝐽 −1 (·, 𝜂)

 2 22𝐽 − 1 𝑜 (3) = = 𝜎g , 𝑆𝐻 0,...,2 𝐽 −1 (·, 𝜂) 2

2 =

(2 𝐽 − 1) 2 (2 𝐽 + 1) 2 . 12

(9.100)

(9.101)

Finally, using Theorem 9.13 and Theorem 9.14, we are able to formulate the following result for the uncertainty product for the dyadic Shannon kernel (cf. Fig. 9.9).

486

9 Scalar Zonal Kernel Functions

Theorem 9.15 (Uncertainty Products) The uncertainty product of the L2 (Ω)normalized dyadic Shannon scaling function g 𝑆𝐻 0...,2 𝐽 −1 (·, 𝜂) of scale 𝐽 is given as follows (1) Δ𝑜g 𝑆𝐻

(3) Δ𝑜g (·, 𝜂) 𝑆𝐻 𝐽 0...,2 −1 0...,2 𝐽 −1 (·, 𝜂)



(2 𝐽+1 − 1)(2 𝐽 + 1) = 2 𝐽+1 − 2

 12 (9.102)

and this product diverges for 𝐽 −→ ∞.

(1)

106

Δ𝑜g

𝑆𝐻 0,...,2 𝐽 −1 (3)

Δ𝑜g

104

𝑆𝐻 0,...,2 𝐽 −1 (1)

Δ𝑜g

𝑆𝐻 0,...,2 𝐽 −1

(3)

Δ𝑜g

𝑆𝐻 0,...,2 𝐽 −1

102 100 10−2 0

5

10

15

20

Fig. 9.9: Uncertainty classification of the dyadic Shannon kernel. Shown are the (1) (3) (1) (3) functions 𝐽 ↦→ Δ𝑜g , 𝐽 ↦→ Δ𝑜g , and 𝐽 ↦→ Δ𝑜g Δ𝑜g . 𝑆𝐻 0,...,2 𝐽 −1

𝑆𝐻 0,...,2 𝐽 −1

𝑆𝐻 0,...,2 𝐽 −1

𝑆𝐻 0,...,2 𝐽 −1

Note, that the product diverges for 𝐽 → ∞. Localization of Bandlimited/Non-Spacelimited Bernstein Kernels. The dyadic Bernstein scaling function (𝜉, 𝜂) ↦→ 𝐵𝐸 0,...,2 𝐽 −1 (𝜉 · 𝜂), 𝜉, 𝜂 ∈ Ω, is given by   2 𝐽 −1 2 𝐽 −2 1 + 𝜉 · 𝜂 𝐵𝐸 0,...,2 𝐽 −1 (𝜉 · 𝜂) = , 𝜋 2

𝜉, 𝜂 ∈ Ω.

(9.103)

Clearly, we are able to understand the kernel 𝐵𝐸 0,...,2 𝐽 −1 as bandlimited radial basis function with symbol {𝐵𝐸 0,...,2 𝐽 −1 ∧ (𝑛)} 𝑛∈N0 , such that 𝐽 −1   2 𝐽 −1 2∑︁ 2 𝐽 −2 1 + 𝑡 2𝑛 + 1 𝐵𝐸 0,...,2 𝐽 −1 (𝑡) = = 𝐵𝐸 0,...,2 𝐽 −1 ∧ (𝑛)𝑃𝑛 (𝑡), (9.104) 𝜋 2 4𝜋 𝑛=0

𝑡 = 𝜉 · 𝜂, 𝜉, 𝜂 ∈ Ω.

9.6 Localization of Representative Zonal Kernels

487

Even more, we are able to find an explicit representation of the symbol (cf. M.J. Fengler et al. (2006a)). In fact, elementary calculations yield the following lemma. Lemma 9.16 (Symbol of the Bernstein Scaling Function) Suppose that 𝐽 ∈ N0 . Then 𝐵𝐸 0,...,2 𝐽 −1 possesses the symbol ( ∧

(𝐵𝐸 0,...,2 𝐽 −1 ) (𝑛) =

(2 𝐽 )!(2 𝐽 −1)! , (2 𝐽 −𝑛−1)!(2 𝐽 +𝑛)!

𝑛 ∈ [0, 2 𝐽 ),

0,

else.

(9.105)

The above considerations also help us to determine an explicit representation of the L2 (Ω)-norm of the dyadic Bernstein scaling function. Lemma 9.17 (Norm of the Bernstein Scaling Function) Let 𝐽 ∈ N0 and 𝜂 ∈ Ω be fixed. Then the L2 (Ω)-norm of the dyadic Bernstein scaling function 𝐵𝐸 0,...,2 𝐽 −1 is given by 2 𝐽 −1 ∥𝐵𝐸 0,...,2 𝐽 −1 (·𝜂)∥ L2 (Ω) = √︁ . (9.106) 𝜋(2 𝐽+1 − 1) The L2 (Ω)-normalized dyadic Bernstein scaling function is denoted by g 𝐵𝐸 0...,2 𝐽 −1 (·𝜂) =

𝐵𝐸 0,...,2 𝐽 −1 (·𝜂) . ∥𝐵𝐸 0,...,2 𝐽 −1 (·, 𝜂)∥ L2 (Ω)

(9.107)

Obviously, the dyadic Bernstein scaling function features no oscillations, since −1 is a zero of multiplicity 2 𝐽 − 1 (for an illustration of the dydic Bernstein kernel and its symbol, see Fig. 9.10). Theorem 9.18 (Space Uncertainty of the Dyadic Bernstein Scaling Function) Suppose that 𝐽 ∈ N and 𝜂 ∈ Ω is fixed. Then, the space uncertainty of the L2 (Ω)normalized dyadic Bernstein scaling function g 𝐵𝐸 0...,2 𝐽 −1 (·, 𝜂) of scale 𝐽 is given by √ 2 𝐽+1 − 1 𝑜 (1) Δg = , (9.108) 𝐵𝐸 0,...,2 𝐽 −1 (·, 𝜂) 2𝐽 − 1 where the expected value and the variance in the space domain are given by (1)

𝑔 𝑜g

𝐵𝐸 0...,2 𝐽 −1 (·, 𝜂)

=

2𝐽 − 1 𝜂, 2𝐽

(9.109)

488

9 Scalar Zonal Kernel Functions

1.5

𝐽 =2 𝐽 =3 𝐽 =4

1

2

1

0.5

0− 𝜋

𝐽 =2 𝐽 =3 𝐽 =4

− 𝜋/2

0

0

𝜋

𝜋/2

0

Fig. 9.10: The normalized dyadic Bernstein kernel 𝐽 = 2, 𝐽 = 3, and 𝐽 = 4. The illustrations show g 0,...,2 𝐽 −1 (cos(𝜗)), 𝜗 ∈ [−𝜋, 𝜋] (left) and 𝜗 ↦→ 𝐵𝐻 g 0,...,2 𝐽 −1 ) ∧ (𝑛), 𝑛 ∈ N0 (right). 𝑛 ↦→ ( 𝐵𝐻



2

(1)

𝑜 𝜎g

𝐵𝐸 0,...,2 𝐽 −1 (·, 𝜂)

=

10

5

15

20

g 0,...,2 𝐽 −1 (cos(𝜗)) for 𝐵𝐻 the space representation frequency representation

2 𝐽+1 − 1 . 22𝐽

(9.110)

Theorem 9.19 (Frequency Uncertainty of the Dyadic Bernstein Scaling Function) Let 𝐽 ∈ N and 𝜂 ∈ Ω be fixed. Then, the uncertainty in frequency of the L2 (Ω)normalized dyadic Bernstein scaling function g 𝐵𝐸 0...,2 𝐽 −1 (·, 𝜂) of scale 𝐽 is given by √ 2𝐽 − 1 −Δ∗ 𝑜 (3) Δ 𝐵𝐸 = Δg = √ (9.111) g (·, 𝜂) 𝐵𝐸 (·, 𝜂) 0...,2 𝐽 −1 0,...,2 𝐽 −1 2 where the expected value (which coincides with the variance with respect to the 𝑜 (3) = L∗ operator) and the variance in the frequency domain (with respect to the operator −Δ∗ ) are given as follows ∗

−Δ 𝑔 𝐵𝐸 g



0...,2 𝐽 −1 (·, 𝜂) ∗

𝜎Φ−Δ ˜ 𝐵 (·, 𝜂) 𝐽

2

=

 2 2𝐽 − 1 𝑜 (3) = 𝜎g , 𝐵𝐸 0...,2 𝐽 −1 (·, 𝜂) 2

(9.112)

=

(2 𝐽 − 1) 2 3 · 2 𝐽+1 − 1 . 4 2 𝐽+1 − 3

(9.113)

Now we have all tools to discuss the uncertainty products for the two scaling functions. In fact, it is of particular interest whether these products converge to the minimal value 1 as the scale 𝐽 tends to infinity, since a value of 1 indicates that the functions are well localized in space and frequency.

9.6 Localization of Representative Zonal Kernels

489

Theorem 9.20 (Uncertainty Products) The uncertainty product of the L2 (Ω)normalized dyadic Bernstein scaling function g 𝐵𝐸 0...,2 𝐽 −1 (·, 𝜂) of scale 𝐽 is given by  𝐽+1 1 2 −1 2 𝑜 (1) 𝑜 (3) Δg Δg = 𝐽+1 (9.114) 𝐵𝐸 0...,2 𝐽 −1 (·, 𝜂) 𝐵𝐸 2 −2 0...,2 𝐽 −1 (·, 𝜂) and this product tends to 1 for 𝐽 −→ ∞. This result for the Bernstein scaling function of scale 𝐽 can easily be computed from Theorem 9.18 and Theorem 9.19. For an illustration of the uncertainty for the dyadic Bernstein kernel, see Fig. 9.11. (1)

Δ𝑜g

𝐵𝐸 0,...,2 𝐽 −1 (3)

102

Δ𝑜g

𝐵𝐸 0,...,2 𝐽 −1 (1)

Δ𝑜g

(3)

𝐵𝐸 0,...,2 𝐽 −1

Δ𝑜g

𝐵𝐸 0,...,2 𝐽 −1

100

10−2 0

5

10

15

20

Fig. 9.11: Uncertainty classification of the dyadic Bernstein kernel. Shown are the (1) (3) (1) (3) functions 𝐽 ↦→ Δ𝑜g , 𝐽 ↦→ Δ𝑜g , and 𝐽 ↦→ Δ𝑜g Δ𝑜g . Note, 𝐵𝐸 0,...,2 𝐽 −1

𝐵𝐸 0,...,2 𝐽 −1

𝐵𝐸 0,...,2 𝐽 −1

𝐵𝐸 0,...,2 𝐽 −1

that the product converges to 1 for 𝐽 → ∞.

The dyadic Bernstein scaling function constitutes a bandlimited scaling function that is free of oscillations. Moreover, the Bernstein kernel minimizes its uncertainty product in the limit, but unlike the Gaussian kernel which also possess this property (see, e,g, W. Freeden et al. (1998), W. Freeden, M. Schreiner (2009)) this minimizer is bandlimited. In detail, the uncertainty product of the Bernstein scaling function tends to 1. As a consequence, this indicates that the space as well as frequency localization is well balanced. Note that the dyadic arrangement of the scales is not of significance. All results also hold true for other sequences of scales. Finally, it should be emphasized that the Bernstein scaling function possesses a closed representation by elementary expressions. Especially the latter allows a fast and stable evaluation. Using kernels of different scales reflecting the different stages of space/frequency localization (see, e.g., W. Freeden (1998), W. Freeden, V. Michel (1999), W. Freeden

490

9 Scalar Zonal Kernel Functions

et al. (2018a) and the references therein), the modeling process can be adapted to the localization properties of the physical phenomena (see Table 9.2). Table 9.2 Multiscale expansion of scalar (square-integrable) spherical functions 𝐹.

sequence of scale-dependent zonal kernels (i.e., scaling functions) Φ 𝑗

↓

convolutions against Φ 𝑗

lowpass filtered versions of 𝐹 ∫ (Φ 𝑗 ∗ 𝐹 ) ( 𝜉 ) = Φ 𝑗 ( 𝜉 · 𝜂) 𝐹 ( 𝜂) 𝑑 𝜔 ( 𝜂) ,

𝜉 ∈Ω

Ω

↓

continuous “summation” over positions 𝜂∈Ω

“zooming-in” (Φ 𝑗 → 𝛿 as 𝑗 → ∞)

multiscale expansion of 𝐹 involving a Dirac family of zonal scalar kernels ∫ 𝐹 ( 𝜉 ) = lim Φ 𝑗 ( 𝜉 · 𝜂) 𝐹 ( 𝜂) 𝑑 𝜔 ( 𝜂), 𝜉 ∈ Ω 𝑗→∞

Ω

All in all, the uncertainty principle represents a trade off between two “spreads”, one for the space and the other for the frequency. The main statement is that sharp localization in space and in frequency are mutually exclusive. The reason for the validity of the uncertainty relation (Theorem 9.8) is that the operators 𝑜 (1) and 𝑜 (3) do not commute. Thus 𝑜 (1) and 𝑜 (3) cannot be sharply defined simultaneously. Extremal members in the space/momentum relation are the polynomials (i.e., spherical harmonics) and the Dirac function(al)s. Asymptotically optimal kernel are the non-bandlimited Gaussian function and the bandlimited Bernstein function. The estimate (Corollary 9.10) allows us to give a quantitative classification in form of a canonically defined hierarchy of the space/frequency localization properties of kernel functions of the form 𝐾 (𝑡) =

∞ ∑︁ 2𝑛 + 1 𝑛=0

4𝜋

𝐾 ∧ (𝑛)𝑃𝑛 (𝑡),

𝑡 = 𝜉 · 𝜂, 𝜉, 𝜂 ∈ Ω.

(9.115)

In view of the amount of space/frequency localization it is also important to distinguish bandlimited kernels (i.e., 𝐾 ∧ (𝑛) = 0 for all 𝑛 ≥ 𝑁) and non-bandlimited ones. Non-bandlimited kernels usually show a much stronger space localization than bandlimited counterparts. It is not difficult to prove that if 𝐾 ∈ L2 [−1, 1] with ∥𝐾 (𝜉· )∥ L2 (Ω) = 1, 

2

𝑜 (1)

𝜎𝐾 ( 𝜉 ·

)

=1−

∞ ∑︁ 2𝑛 + 1 𝑛=1

4𝜋

!2 𝐾 ∧ (𝑛)𝐾 ∧ (𝑛 + 1)

.

(9.116)

9.7 Dirac Families of Zonal Scalar Kernel Functions

491

Thus, we are really allowed to conclude that, if 𝐾 ∧ (𝑛) ≈ 𝐾 ∧ (𝑛 + 1) ≈ 1 for many successive integers 𝑛, then the local support of (9.115) in space domain is small. Summarizing our considerations we come to the following conclusions: The varieties of the intensity of the localization on the sphere Ω can be also illustrated by considering the kernel function (9.115). By choosing “𝐾 ∧ (𝑛) = 𝛿 𝑛𝑘 ” we obtain a Legendre kernel of degree 𝑘, i.e., we arrive at the left end of our scheme (see Table 9.3). On the other hand, if we formally take 𝐾 ∧ (𝑛) = 1 for 𝑛 ∈ N0 , we obtain the kernel which is the Dirac functional in L2 (Ω). Bandlimited kernels have the property 𝐾 ∧ (𝑛) = 0 for all 𝑛 ≥ 𝑁, 𝑁 ∈ N0 . Non-bandlimited kernels satisfy 𝐾 ∧ (𝑛) ≠ 0 for an infinite number of integers 𝑛 ∈ N0 . Concerning the condition lim𝑛→∞ 𝐾 ∧ (𝑛) = 0 it follows that the slower the sequence {𝐾 ∧ (𝑛)} 𝑛∈N0 converges to zero, the lower is the frequency localization, and the higher is the space localization. Table 9.3 gives a qualitative illustration of the consequences of the uncertainty principle in the theory of zonal kernel functions on the sphere: On the left end of this scheme we have the Legendre kernels with their ideal frequency (momentum) localization. However, they show no space localization, as they are of polynomials nature. Thus the present standard way in applications of increasing the accuracy in spherical harmonic (Fourier) expansions is to increase the maximum degree of the spherical harmonics expansions under consideration. On the right end of the scheme there is the Dirac kernel which maps a function to its value at a certain point. Hence, those functionals have ideal space localization but no frequency localization. All in all, zonal kernel functions exist as bandlimited and non-bandlimited functions. Every bandlimited zonal kernel function refers to a finite number of frequencies. This reduction of the frequency localization allows a finite variance of the space in the uncertainty principle, i.e., this method has both a frequency localization and a space localization. If we move from bandlimited to non-bandlimited zonal kernel functions the frequency localization usually decreases and the space localization increases in accordance to the uncertainty principle. In consequence, if the accuracy has to be increased in zonal kernel approximation (e.g., by splines and wavelets as proposed later on), a denser point grid is required in the (local) region under investigation.

9.7 Dirac Families of Zonal Scalar Kernel Functions

As already pointed out, the spectral representation of a square-integrable function by means of spherical harmonics is essential to solving many problems in today’s applications. In future research,  however, Fourier (orthogonal) expansions in terms of spherical harmonics 𝑌𝑛, 𝑗 will not be the only way of representing a square-

492

9 Scalar Zonal Kernel Functions

integrable function. In order to explain this in more detail, we think of a squareintegrable function as a signal in which the spectrum evolves over space in significant way. We imagine that, at each point on the sphere Ω, the function refers to a certain combination of frequencies, and that these frequencies are continuously changing. This space-evolution of the frequencies, however, is not reflected in the Fourier expansion in terms of non-space localizing spherical harmonics, at least not directly. Therefore, in theory, any member 𝐹 of the space L2 (Ω) can be reconstructed from its Fourier transforms, i.e., the “amplitude spectrum” {𝐹 ∧ (𝑛, 𝑗)} 𝑛=0,1,..., , but the 𝑗=1,...,2𝑛+1 Fourier transform contains information about the frequencies of the function over all positions instead of showing how the frequencies vary in space. In what follows, we present a two-parameter, i.e., scale- and space-dependent method of achieving a reconstruction of a function 𝐹 ∈ L2 (Ω) involving (scalar) zonal kernel functions which we refer to as scaling (kernel) functions converging to the (zonal) Dirac kernel. Roughly speaking, a Dirac family as discussed here is a set of zonal kernels Φ𝜌 : [−1, 1] → R, 𝜌 ∈ (0, ∞), of the form Φ𝜌 (𝜉 · 𝜂) =

=

∞ ∑︁ 𝑛=0 ∞ ∑︁

𝜑𝜌 (𝑛)

2𝑛+1 ∑︁

𝑌𝑛, 𝑗 (𝜉)𝑌𝑛, 𝑗 (𝜂),

(9.117)

𝑗=1

𝜑𝜌 (𝑛)

𝑛=0

2𝑛 + 1 𝑃𝑛 (𝜉 · 𝜂), 4𝜋

𝜉, 𝜂 ∈ Ω,

converging to the “Dirac-kernel” 𝛿 as 𝜌 → 0. As shown in Section 9.5, the distributional Dirac kernel can be formally written as zonal kernel function 𝛿(𝜉 · 𝜂) =

=

∞ 2𝑛+1 ∑︁ ∑︁

𝑌𝑛, 𝑗 (𝜉)𝑌𝑛, 𝑗 (𝜂),

𝑛=0 𝑗=1 ∞ ∑︁ 𝑛=0

2𝑛 + 1 𝑃𝑛 (𝜉 · 𝜂), 4𝜋

(9.118)

𝜉, 𝜂 ∈ Ω.

Consequently, if {Φ𝜌 } 𝜌∈ (0,∞) is a Dirac family of zonal kernels, its “symbol” {𝜑𝜌 (𝑛)} 𝑛=0,1,... constitutes a sequence satisfying the limit relation lim 𝜑𝜌 (𝑛) = 1

(9.119)

𝜌→0 𝜌>0

for each 𝑛 = 0, 1, . . .. Accordingly, if {Φ𝜌 } 𝜌∈ (0,∞) is a Dirac family of zonal kernels, the convolution integrals ∫
Φ𝜌 ∗ 𝐹 (𝜉) = Φ𝜌 (𝜉 · 𝜂)𝐹 (𝜂) 𝑑𝜔(𝜂), 𝜉 ∈ Ω, (9.120) Ω

converge (in a certain topology) to the limit

9.7 Dirac Families of Zonal Scalar Kernel Functions

493

∫ 𝐹 (𝜉) = (𝛿 ∗ 𝐹)(𝜉) =

𝛿(𝜉 · 𝜂)𝐹 (𝜂) 𝑑𝜔(𝜂),

𝜉 ∈ Ω,

(9.121)

Ω

for all 𝜉 ∈ Ω as 𝜌 tends to 0. In more detail, if 𝐹 is a function of class L2 (Ω) and {Φ𝜌 } is a (suitable) Dirac family (tending to the Dirac kernel), then the following limit relation holds true:

lim 𝐹 − Φ𝜌 ∗ 𝐹 L2 (Ω) = 0. (9.122) 𝜌→0,𝜌>0

It should be noted that an approximate convolution identity acts as a space and frequency localization procedure in the following way: As Φ𝜌 , 𝜌 ∈ (0, ∞), is a Dirac family of zonal scalar kernel functions tending to the Dirac kernel, the function Φ𝜌 (𝜂·), is highly concentrated about the point 𝜂 ∈ Ω if the “scale parameter” is a small positive value. Moreover, as 𝜌 tends to infinity, Φ𝜌 (𝜂·) becomes more and more localized in frequency. Correspondingly, the uncertainty principle states that the space localization of Φ𝜌 (𝜂·) becomes more and more decreasing. In conclusion, the products 𝜂 ↦→ Φ𝜌 (𝜉 · 𝜂)𝐹 (𝜂), 𝜂 ∈ Ω, 𝜉 ∈ Ω, for each fixed value 𝜌, display information in 𝐹 ∈ L2 (Ω) at various levels of spatial resolution or frequency ∫ bands. Consequently, as 𝜌 approaches ∞, the convolution integrals Φ𝜌 ∗ 𝐹 = Ω Φ𝜌 (·𝜂)𝐹 (𝜂) 𝑑𝜔(𝜂) display coarser, lower-frequency features. As 𝜌 approaches 0, the integrals give sharper and sharper spatial resolution. In other words, the convolution integrals can measure the space-frequency variations of spectral components, but they have a different space-frequency resolution. Each scale approximation Φ𝜌 ∗ 𝐹 of a function 𝐹 ∈ L2 (Ω) must be made directly by computing the relevant convolution integrals. In doing so, however, it is inefficient to use no information from the approximation Φ𝜌 ∗ 𝐹 within the computation of Φ𝜌′ ∗ 𝐹 provided that 𝜌 ′ < 𝜌. In fact, the efficient construction of multiscale approximation based on Dirac families begins by a multiresolution analysis in terms of wavelets, i.e., a recursive method which is ideal for computation (see W. Freeden et al. (1998), W. Freeden, V. Michel (2005) and the references therein). However, this aspect of constructive approximation will not be discussed here in our approach to spherical functions relevant for geoscientific purposes. A mathematically rigorous formulation of a Dirac family is as follows. Definition 9.21 Let {Φ𝜌 } 𝜌∈ (0,∞) be a family of functions in L2 [−1, 1] satisfying the condition ∫ 1 ∫ 1 (Φ𝜌 ) ∧ (0) = 2𝜋 Φ𝜌 (𝑡)𝑃0 (𝑡) 𝑑𝑡 = 2𝜋 Φ𝜌 (𝑡) 𝑑𝑡 = 1 (9.123) −1

−1

for all 𝜌 ∈ (0, ∞). Then {Φ𝜌 } 𝜌∈ (0,∞) is said to be a Dirac family in L2 (Ω), if

494

9 Scalar Zonal Kernel Functions

lim ∥𝐹 − Φ𝜌 ∗ 𝐹 ∥ L2 (Ω) = 0

(9.124)

𝜌→0

for all 𝐹 ∈ L2 (Ω). Remark 9.22 In the jargon of approximation theory, the family {𝐼𝜌 } 𝜌∈ (0,∞) of operators 𝐼𝜌 given by 𝐼𝜌 𝐹 = Φ𝜌 ∗ 𝐹 is called a (spherical) singular integral, and {Φ𝜌 } 𝜌∈ (0,∞) itself is called the kernel of the singular integral (or, briefly, scaling function). However, we want to point out the convergence of {Φ𝜌 } 𝜌∈ (0,∞) to the Dirac kernel 𝛿 as 𝜌 → 0. This is the reason why we use the notation of the Dirac family in our approach. The convergence of a Dirac family of scalar zonal kernel functions to the scalar Dirac kernel is described in more detail by the following theorem. Theorem 9.23 Let {Φ𝜌 } 𝜌∈ (0,∞) be a family of functions in L2 [−1, 1] satisfying (Φ𝜌 ) ∧ (0) = 1 and ∫

(9.125)

1

|Φ𝜌 (𝑡)| 𝑑𝑡 ≤ 𝑀

2𝜋

(9.126)

−1

for all 𝜌 ∈ (0, ∞) with some constant 𝑀 independent of 𝜌. Then {Φ𝜌 } 𝜌∈ (0,∞) is a Dirac family in L2 (Ω) if and only if lim (Φ𝜌 ) ∧ (𝑛) = 1

(9.127)

𝜌→0

for all 𝑛 ∈ N0 . Proof We have to verify the equivalence (see W. Freeden, K. Hesse (2002)). From the definition of a Dirac family in L2 (Ω), we are able to deduce that lim ∥𝐹 − Φ𝜌 ∗ 𝐹 ∥ L2 (Ω) = 0

(9.128)

𝜌→0

for all 𝐹 ∈ L2 (Ω). Particularly, this holds for all spherical harmonics 𝑌𝑛 of degree 𝑛. The Funk-Hecke formula implies that Φ𝜌 ∗ 𝑌𝑛 = (Φ𝜌 ) ∧ (𝑛)𝑌𝑛 . Thus, it follows that 0 = lim ∥𝑌𝑛 − Φ𝜌 ∗ 𝑌𝑛 ∥ L2 (Ω) 𝜌→0

= lim |1 − (Φ𝜌 ) ∧ (𝑛)| ∥𝑌𝑛 ∥ L2 (Ω) 𝜌→0

and lim𝜌→0 (Φ𝜌 ) ∧ (𝑛) = 1 follows because of ∥𝑌𝑛 ∥ L2 (Ω) ≠ 0 for all spherical harmonics 𝑌𝑛 ≠ 0 of degree 𝑛 ∈ N0 .

9.7 Dirac Families of Zonal Scalar Kernel Functions

495

For the proof of the reverse implication direction, note that the uniform boundedness in the sense of (9.126) imposed on the functions Φ𝜌 , 𝜌 ∈ (0, ∞), and the estimate |𝑃𝑛 (𝑡)| ≤ 1 for all 𝑡 ∈ [−1, 1] and all 𝑛 ∈ N0 imply that (Φ𝜌 ) ∧ (𝑛) ≤ 2𝜋

1

∫

|Φ𝜌 (𝑡)| 𝑑𝑡 ≤ 𝑀,

−1

∫

∧

1

∫ |Φ𝜌 (𝑡)| |𝑃𝑛 (𝑡)| 𝑑𝑡 ≤ 2𝜋

−1

1

(Φ𝜌 ) (𝑛) ≥ −2𝜋

1

∫ |Φ𝜌 (𝑡)| |𝑃𝑛 (𝑡)| 𝑑𝑡 ≥ −2𝜋

|Φ𝜌 (𝑡)| 𝑑𝑡 ≥ −𝑀.

−1

−1

Hence, (Φ𝜌 ) ∧ (𝑛) ∈ [−𝑀, 𝑀] for all 𝑛 ∈ N0 and all 𝜌 ∈ (0, ∞). Therefore, ∥𝐹 − Φ𝜌 ∗ 𝐹 ∥ 2L2 (Ω) =

∞ 2𝑛+1 ∑︁ ∑︁ 

1 − (Φ𝜌 ) ∧ (𝑛)

2 

𝐹 ∧ (𝑛, 𝑙)

2

𝑛=0 𝑙=1

≤ (𝑀 + 1) 2 ∥𝐹 ∥ 2L2 (Ω)

(9.129)

for all 𝜌 ∈ (0, ∞) and all 𝐹 ∈ L2 (Ω). As the upper bound (𝑀 + 1) of the term |1 − (Φ𝜌 ) ∧ (𝑛)| is independent of 𝜌 ∈ (0, ∞), the limit for 𝜌 → 0 and the infinite sum may be interchanged. Hence, lim ∥𝐹 − Φ𝜌 ∗ 𝐹 ∥ L2 (Ω)

(9.130)

𝜌→0

=

∞ 2𝑛+1 ∑︁ ∑︁



∧

lim 1 − (Φ𝜌 ) (𝑛)

2 

∧

𝐹 (𝑛, 𝑙)

2

! 1/2 =0

𝜌→0 𝑛=0 𝑙=1

for all 𝐹 ∈ L2 (Ω), as required.

□

Restricting our attention to non-negative kernels {Φ𝜌 } 𝜌∈ (0,∞) , (i.e., all Φ𝜌 , 𝜌 ∈ (0, ∞), satisfy Φ𝜌 (𝑡) ≥ 0 for almost all 𝑡 ∈ [−1, 1]), more equivalent characterizations of a Dirac family are deducible. The main advantage of non-negative kernels {Φ𝜌 } 𝜌∈ (0,∞) is that the property (Φ𝜌 ) ∧ (0) = 1 implies ∧

∫

1

1 = (Φ𝜌 ) (0) = 2𝜋

∫

1

Φ𝜌 (𝑡) 𝑑𝑡 = 2𝜋 −1

|Φ𝜌 (𝑡)| 𝑑𝑡

(9.131)

−1

i. e., the condition (9.126) is valid with the sharp bound 𝑀 = 1. Theorem 9.24 Let {Φ𝜌 } 𝜌∈ (0,∞) be a family of functions in L2 [−1, 1], which satisfy (Φ𝜌 ) ∧ (0) = 1 and which are non-negative. Then the following properties are equivalent: (i) {Φ𝜌 } 𝜌∈ (0,∞) is a non-negative Dirac family in L2 (Ω), (ii) lim𝜌→0 ∥𝐹 − Φ𝜌 ∗ 𝐹 ∥ L2 (Ω) = 0 for all 𝐹 ∈ L2 (Ω),

496

9 Scalar Zonal Kernel Functions

(iii) lim𝜌→0 (Φ𝜌 ) ∧ (𝑛) = 1 for all 𝑛 ∈ N0 , (iv) lim𝜌→0 (Φ𝜌 ) ∧ (1) = 1, (v) {Φ𝜌 } 𝜌∈ (0,∞) satisfies the “localization property” ∫

𝛿

Φ𝜌 (𝑡) 𝑑𝑡 = 0

lim 𝜌→0

−1

for all 𝛿 ∈ (−1, 1). Proof The statements (i) and (ii) are equivalent by definition and the equivalence of (ii) and (iii) is clear from Theorem 9.24. Obviously, (iii) implies (iv). It remains to show, that (v) follows from (iv) and that (v) implies (iii). Assume (iv) is true. Let 𝛿 ∈ (−1, 1) be arbitrary. Because of the non-negativity of the kernels Φ𝜌 , ∫

∫ 𝛿 1 Φ𝜌 (𝑡) 𝑑𝑡 ≤ (1 − 𝑡) Φ𝜌 (𝑡) 𝑑𝑡 (1 − 𝛿) −1 ∫ 1 1 ≤ (1 − 𝑡) Φ𝜌 (𝑡) 𝑑𝑡 (1 − 𝛿) −1  1 1  = (Φ𝜌 ) ∧ (0) − (Φ𝜌 ) ∧ (1) . 2𝜋 (1 − 𝛿)

𝛿

0 ≤ −1

(9.132)

Taking the limit for 𝜌 → 0 the localization property (v) follows from (vi). Fnally, assume (v) is true, and we want to prove (iii): Property (iii) is equivalent to the following assertion: For every 𝑛 ∈ N and for every 𝜀 > 0, there exists a value 𝜌0 = 𝜌0 (𝜀, 𝑛) ∈ (0, ∞) such that 1 − 𝜀 ≤ (Φ𝜌 ) ∧ (𝑛)) ≤ 1 for all 𝜌 ∈ (0, 𝜌0 ]. By the non-negativity of Φ𝜌 and the estimate |𝑃𝑛 (𝑡)| ≤ 1 for all 𝑛 ∈ N0 , (Φ𝜌 ) ∧ (𝑛) = 2𝜋

∫

1

∫

1

Φ𝜌 (𝑡) 𝑃𝑛 (𝑡) 𝑑𝑡 ≤ 2𝜋 −1

Φ𝜌 (𝑡) 𝑑𝑡 = (Φ𝜌 ) ∧ (0) = 1. (9.133)

−1

Let 𝑛 ∈ N and 𝜀 > 0 be arbitrary. For 𝛿 ∈ (−1, 1), (Φ𝜌 ) ∧ (𝑛) = 2𝜋

∫

1

∫

𝛿

Φ𝜌 (𝑡) 𝑃𝑛 (𝑡) 𝑑𝑡 + 2𝜋 −1

Φ𝜌 (𝑡) 𝑃𝑛 (𝑡) 𝑑𝑡.

As 𝑃𝑛 (1) = 1, 𝛿 ∈ (−1, 1) can be chosen so close to 1 that 𝑃𝑛 (𝑡) ≥ all 𝑡 ∈ [𝛿, 1]. Thus, ∧

∫

𝛿

(Φ𝜌 ) (𝑛) ≥ 2𝜋

√︁

√︁ 1 − (𝜀/2) for

1

∫

Φ𝜌 (𝑡) 𝑃𝑛 (𝑡) 𝑑𝑡 + 2𝜋 1 − (𝜀/2) −1

(9.134)

𝛿

Φ𝜌 (𝑡) 𝑑𝑡. 𝛿

(9.135)

9.7 Dirac Families of Zonal Scalar Kernel Functions

497

As |𝑃𝑛 (𝑡)| ≤ 1 for all 𝛿 ∈ (−1, 1), ∫ 𝛿 ∫ 𝛿 ∫ −2𝜋 Φ𝜌 (𝑡) 𝑑𝑡 ≤ 2𝜋 Φ𝜌 (𝑡) 𝑃𝑛 (𝑡) 𝑑𝑡 ≤ 2𝜋 −1

−1

𝛿

Φ𝜌 (𝑡) 𝑑𝑡.

(9.136)

−1

Therefore, the localization property (v) implies that there exists 𝜌1 , such that the ∫𝛿 estimate 2𝜋 −1 Φ𝜌 (𝑡) 𝑃𝑛 (𝑡) 𝑑𝑡 ≥ −𝜀/2 is valid for all 𝜌 ∈ (0, 𝜌1 ). On the other hand, (Φ𝜌 ) ∧ (0) = 1 for all 𝜌 ∈ (0, ∞), and the localization property implies 1 = lim 2𝜋 𝜌→0

∫

1

Φ𝜌 (𝑡) 𝑑𝑡 ∫

1

∫

𝛿

Φ𝜌 (𝑡) 𝑑𝑡 + lim

= lim 𝜌→0

(9.137)

−1

𝜌→0

−1 1

Φ𝜌 (𝑡) 𝑑𝑡 𝛿

∫

Φ𝜌 (𝑡) 𝑑𝑡.

= lim 𝜌→0

𝛿

∫1 √︁ Hence, there exists 𝜌2 ∈ (0, ∞) such that 2𝜋 𝛿 Φ𝜌 (𝑡) 𝑑𝑡 ≥ 1 − (𝜀/2) for all 𝜌 ∈ (0, 𝜌2 ). The relation (9.135) implies 1 − 𝜀 ≤ (Φ𝜌 ) ∧ (𝑛) ≤ 1 for all 𝜌 ∈ (0, 𝜌0 ) with 𝜌0 = min{𝜌1 , 𝜌2 }. □ Theorem 9.24 immediately leads us to the following notation. Definition 9.25 A family {Φ𝜌 } 𝜌∈ (0,∞) ⊂ L2 [−1, 1] satisfying the conditions (i) (Φ𝜌 ) ∧ (0) = 1, (ii) Φ𝜌 is non-negative on [−1, 1], (iii) lim Φ𝜌 (𝑡) 𝑑𝑡 = 0, 𝛿 ∈ (−1, 1), 𝜌→0

is called a Dirac family of non-negative type in L2 (Ω). Finally, it is worth mentioning that a Dirac family of non-negative type in L2 (Ω), i.e., a family {Φ𝜌 } 𝜌∈ (0,∞) ⊂ L2 [−1, 1], fulfilling the assumptions of Theorem 9.24 satisfies the estimate ∥Φ𝜌 ∗ 𝐹 ∥ L2 (Ω) ≤ ∥𝐹 ∥ L2 (Ω) (9.138) for all 𝜌 ∈ (0, ∞) and for all 𝐹 ∈ L2 (Ω). For a categorization of certain examples of Dirac families of scalar zonal kernel functions, the following definition is helpful (see, e.g., H. Berens et al. (1969)).

498

9 Scalar Zonal Kernel Functions

Definition 9.26 A family {𝐼𝜌 }, 𝐼𝜌 : L2 (Ω) → L2 (Ω), 𝜌 ∈ (0, ∞), is called a semigroup of contraction operators on L2 (Ω), if the following properties are satisfied: (i) For each 𝜌 ∈ (0, ∞), 𝐼𝜌 is a linear bounded operator mapping L2 (Ω) into itself and 𝐼0 = 𝐼 (identity operator). (ii) 𝐼𝜌1 +𝜌2 = 𝐼𝜌1 𝐼𝜌2 , 0 ≤ 𝜌1 ≤ 𝜌2 < ∞ (iii) lim ∥𝐼𝜌 (𝐹) − 𝐹 ∥ L2 (Ω) = 0, 𝐹 ∈ L2 (Ω) 𝜌→0 𝜌>0

(iv) ∥𝐼𝜌 (𝐹)∥ L2 (Ω) ≤ ∥𝐹 ∥ L2 (Ω) , 𝜌 ∈ (0, ∞), 𝐹 ∈ L2 (Ω). Examples of semigroups of contraction operators on L2 (Ω) will be discussed in Section 9.8. We conclude this section by making some words about the spherical wavelet transform (for references, the reader is referred to the list in Section 9.9) The wavelet transform acts as a space and frequency localization operator in the following way: If {Φ𝜌 }, 𝜌 ∈ (0, ∞), is a Dirac family and 𝜌 approaches infinity, the convolution integrals ∫ Φ𝜌 ∗ 𝑈 =

Φ𝜌 (·, 𝜂)𝐹 (𝜂) 𝑑𝜔(𝜂)

(9.139)

Ω

display coarser, lower-frequency features. As 𝜌 approaches zero, the integrals give sharper and sharper spatial resolution. In other words, the convolution integrals can measure the space-frequency variations of spectral components, but they have a different space-frequency resolution. Each scale-space approximation Φ𝜌 ∗ 𝐹 of a function 𝐹 ∈ L2 (Ω) must be made directly by computing the relevant convolution integrals. In doing so, however, it is inefficient to use no information from the approximation Φ𝜌 ∗ 𝐹 within the computation of Φ𝜌′ ∗ 𝐹 provided that 𝜌 ′ < 𝜌. In fact, the efficient construction of wavelets begins by a multiresolution analysis, i.e., a completely recursive method which is therefore ideal for computation. In this context, we observe that ∫ ∞∫ 𝑑𝜎 → 𝐹 ∈ L2 (Ω), 𝑅 → 0, (9.140) Ψ𝜌 (·, 𝜂)𝐹 (𝜂) 𝑑𝜔(𝜂) 𝜎 𝑅 Ω i.e.,

∫

lim 𝐹 − 𝑅→0 𝑅>0

provided that

𝑅

∞∫

𝑑𝜌

Ψ𝜌 (𝜉, 𝜂)𝐹 (𝜂) 𝑑𝜔(𝜂) = 0, 𝜌 L2 (Ω) Ω

(9.141)

9.7 Dirac Families of Zonal Scalar Kernel Functions

Ψ𝜌 (𝜉, 𝜂) =

∞ ∑︁ 𝑛=0

Ψ𝜌∧ (𝑛)

2𝑛+1 ∑︁

499

𝑌𝑛, 𝑗 (𝜉)𝑌𝑛, 𝑗 (𝜂)

(𝜉, 𝜂) ∈ Ω × Ω,

(9.142)

𝑗=1

is given such that 𝜓𝜌∧ (𝑛) = −𝜌

𝑑 ∧ Φ (𝑛) 𝑑𝜌 𝜌

(9.143)

for 𝑛 = 0, 1, . . . and all 𝜌 ∈ (0, ∞). Conventionally, the family {Ψ𝜌 }, 𝜌 ∈ (0, ∞), is called a (scale continuous) wavelet. The (scale continuous) wavelet transform (𝑊𝑇): L2 (Ω) → L2 ((0, ∞) × Ω) is defined by ∫ (𝑊𝑇)(𝐹)(𝜌; 𝜉) = (Ψ𝜌 ) ∗ 𝐹 (𝜉) = Ψ𝜌 (𝜉, 𝜂)𝐹 (𝜂) 𝑑𝜔(𝜂). (9.144) Ω

In other words, the wavelet transform is defined as the L2 (Ω)-inner product of 𝐹 ∈ L2 (Ω) with the set of “rotations” and “dilations” of 𝐹. The (scale continuous) wavelet transform (𝑊𝑇) is invertible on L2 (Ω), i.e., ∫ ∫ ∞ 𝑑𝜌 𝐹= (𝑊𝑇)(𝐹)(𝜌; 𝜂)Ψ𝜌 (·, 𝜂) 𝑑𝜔(𝜂) (9.145) 𝜌 Ω 0 in the sense of ∥ · ∥ L2 (Ω) . From Parseval’s identity in terms of scalar spherical harmonics, it follows that ∫ ∫ ∞ 𝑑𝜌 (Ψ𝜌 ∗ 𝐹)(𝜂) 𝑑𝜔(𝜂) = ∥𝐹 ∥ 2L2 (Ω) (9.146) 𝜌 Ω 0 i.e., (𝑊𝑇) converts a function 𝐹 of one variable into a function of two variables 𝜉 ∈ Ω and 𝜌 ∈ (0, ∞) without changing its total energy. In terms of filtering {Φ𝜌 } and {Ψ𝜌 }, 𝜌 ∈ (0, ∞) may be interpreted as lowpass filter and bandpass filter, respectively. Correspondingly, the convolution operators are given by Φ𝜌 ∗ 𝐹, Ψ𝜌 ∗ 𝐹,

𝐹 ∈ L2 (Ω), 2

𝐹 ∈ L (Ω).

(9.147) (9.148)

The Fourier transforms read as follows: (Φ𝜌 ∗ 𝐹) ∧ (𝑛, 𝑗) = 𝐹 ∧ (𝑛, 𝑗)Φ∧𝜌 (𝑛), ∧

∧

(Ψ𝜌 ∗ 𝐹) (𝑛, 𝑗) = 𝐹 (𝑛,

𝑗)Ψ𝜌∧ (𝑛).

(9.149) (9.150)

These formulas provide the transition from the wavelet transform to the Fourier transform. Since all scales 𝜌 are used, the reconstruction is highly redundant. Of course, the redundancy leads us to the following question which is of particular importance in data analysis:

500

9 Scalar Zonal Kernel Functions

- Given an arbitrary 𝐻 ∈ L2 ((0, ∞) ×Ω), how can we know whether 𝐻 = (𝑊𝑇)(𝐹) for some function 𝐹 ∈ L2 (Ω)? The question amounts to finding the range of the (scale continuous) wavelet transform (𝑊𝑇) : L2 (Ω) → L2 ((0, ∞) × Ω) (see W. Freeden et al. (1998)), i.e., the subspace W = (𝑊𝑇)(L2 (Ω)) ≠ L2 ((0, ∞) × Ω).

(9.151)

Actually, it can be shown that the tendency for minimizing errors by use of the wavelet transform is again expressed in least-squares approximation: Let 𝐻 be an arbitrary element of L2 ((0, ∞) × Ω). Then the unique function 𝐹𝐻 ∈ L2 (Ω) with the property ∥𝐻 − (𝑊𝑇)(𝐹𝐻 )∥ L2 ( (0,∞) ×Ω) = is given by ∫

inf

𝑈 ∈L2 (Ω)

∥𝐻 − (𝑊𝑇)(𝑈)∥ L2 ( (0,∞) ×Ω)

∞∫

𝐻 (𝜌; 𝜂)Ψ𝜌 (·, 𝜂)𝑑𝜔(𝜂)

𝐹𝐻 = 0

Ω

𝑑𝜌 . 𝜌

(9.152)

(9.153)

(𝑊𝑇)(𝐹𝐻 ) is indeed the orthogonal projection of 𝐻 onto W. Another important question in the context of the wavelet transform is: - Given an arbitrary 𝐻 (𝜌; 𝜉) = (𝑊𝑇)(𝐹)(𝜌; 𝜉), 𝜌 ∈ (0, ∞), and 𝜉 ∈ Ω, for some 𝐹 ∈ L2 (Ω), how can we reconstruct 𝐹? The answer is provided by the so-called least-energy representation. It states: Of all possible functions 𝐻 ∈ L2 ((0, ∞) × Ω) for 𝐹 ∈ L2 (Ω), the function 𝐻 = (𝑊𝑇)(𝐹) is unique in that it minimizes the “energy” ∥𝐻 ∥ 2L2 ( (0,∞) ×Ω . More explicitly (see W. Freeden et al. (1998)) ∥(𝑊𝑇)(𝐹)∥ L2 ( (0,∞) ×Ω) =

inf

𝐻 ∈L2 ( (0,∞) ×Ω) (𝑊𝑇 ) −1 (𝐻)=𝐹

∥𝐻 ∥ L2 ( (0,∞) ×Ω) .

9.8 Examples of Dirac Families

Several types of Dirac families can be distinguished which are of basic interest in applications (see Table 9.3).

9.8 Examples of Dirac Families

501

Table 9.3 Three types of kernels: bandlimited, spacelimited, and non-spacelimited/nonbandlimited.

Legendre kernels

zonal kernels

Dirac kernel

𝐾 ∧ (𝑛) = 𝛿𝑛𝑘

general case

𝐾 ∧ (𝑛) = 1, 𝑛 = 0, . . .

bandlimited

spacelimited

𝐾 ∧ (𝑛)

𝐾 ( 𝜉 · 𝜂) = 0, 1− 𝜉 · 𝜂 >ℎ

Shannon

Haar

= 0, 𝑛>𝑁

𝐾 ∧ (𝑛)

= 1, 𝑛≤ 𝑁

𝐾 ( 𝜉 · 𝜂) = 1, 1− 𝜉 · 𝜂 ≤ ℎ

Spacelimited, i.e., locally supported kernel functions are nothing new, having been discussed in one-dimensional Euclidean space already by A. Haar (1910). In what follows, we first present the classical concept initiated by Haar in a generalization to the spherical case (see Fig. 9.6): Example 9.27 The Haar family {𝐻 ℎ } ℎ∈ (0,1) ⊂ L2 [−1, 1], 𝐻 ℎ : [−1, 1] → 𝑅, 𝑡 ↦→ 𝐻 ℎ (𝑡), is given by  0, 𝑡 ∈ [−1, ℎ], 𝐻 ℎ (𝑡) = 1 (9.154) 1 , 2 𝜋 (1−ℎ) 𝑡 ∈ [ℎ, 1]. Obviously, 𝐻 ℎ (𝑡) ≥ 0 for all 𝑡 ∈ [−1, 1] and (𝐻 ℎ ) ∧ (0) = 2𝜋 ∥𝐻 ℎ ∥ L2 [ −1,1] = 1 are fulfilled. Thus {𝐻 ℎ } ℎ∈ (0,1) generates an approximate identity in L2 (Ω). Further properties of the Haar Dirac family follow in the next example by specialization to the case 𝑘 = 0.

Example 9.28 Let 𝑘 be a non-negative integer, i.e., 𝑘 ∈ N0 . The smoothed Haar Dirac family {𝐿 ℎ(𝑘 ) } ℎ∈ (0,1) ⊂ C (𝑘−1) [−1, 1], is defined by 𝐿 ℎ(𝑘 ) : [−1, 1] → R, 𝑡 ↦→ 𝐿 ℎ(𝑘 ) (𝑡), where (cf. (4.42))

502

9 Scalar Zonal Kernel Functions

1.5

ℎ = 0.3 ℎ = 0.7 ℎ = 0.9

1

1

ℎ = 0.3 ℎ = 0.7 ℎ = 0.9

0.5 0.5 0 0− 𝜋

− 𝜋/2

0

𝜋/2

𝜋

0

10

20

Fig. 9.12: The Haar kernel 𝐻 ℎ for ℎ = 0.3, 0.7, 0.9. Space representation 𝜗 ↦→ 𝐻 ℎ (cos(𝜗)) (left) and frequency representation 𝑛 ↦→ (𝐻 ℎ ) ∧ (𝑛) (right). 𝐿 ℎ(𝑘 ) (𝑡) = ((𝐵 ℎ(𝑘 ) ) ∧ (0)) −1 𝐵 ℎ(𝑘 ) (𝑡)

(9.155)

 𝑡 ∈ [−1, ℎ),   0,  = (𝑡 − ℎ) 𝑘   (1 − ℎ) 𝑘 , 𝑡 ∈ [ℎ, 1]. 

(9.156)

with 𝐵 ℎ(𝑘 ) (𝑡)

By definition, 𝐿 ℎ(𝑘 ) is non-negative, has the support [ℎ, 1], and satisfies (𝐿 ℎ(𝑘 ) ) ∧ (0) = 1. Hence, it is a non-negative [ℎ, 1]-locally supported Dirac family. Obviously, the function 𝐿 ℎ(0) , ℎ ∈ (−1, 1), coincides with the Haar function 𝐻 ℎ . The Legendre coefficients of 𝐵 ℎ(𝑘 ) and, hence, 𝐿 ℎ(𝑘 ) , ℎ ∈ (−1, 1), 𝑘 ∈ N0 , can be calculated recursively (cf. W. Freeden et al. (1998)):   1−ℎ (𝐵 ℎ(𝑘 ) ) ∧ (0) = 2𝜋 ≠ 0, (9.157) 𝑘 +1    1−ℎ 1−ℎ (𝑘 ) ∧ (𝐵 ℎ ) (1) = 2𝜋 1− , (9.158) 𝑘 +1 𝑘 +2     2𝑛 + 1 𝑘 +1−𝑛 (𝑘 ) ∧ (𝑘 ) ∧ (𝐵 ℎ ) (𝑛 + 1) = ℎ (𝐵 ℎ ) (𝑛) + (𝐵 ℎ(𝑘 ) ) ∧ (𝑛 − 1). (9.159) 𝑛+𝑘 +2 𝑛+𝑘 +2 It can be shown by use of the estimates for the Legendre polynomials that
|(𝐿 ℎ(𝑘 ) ) ∧ (𝑛)| = 𝑂 (𝑛(1 − ℎ)) − (3/2) −𝑘 for 𝑛 → ∞. The functions 𝐿 ℎ(𝑘 ) , ℎ ∈ (0, 1), 𝑘 ∈ N0 , assume their maximum in 𝑡 = 1. For 𝑘 > 2, the Lipschitz-constant 𝐶ℎ(𝑘 ) for 𝐿 ℎ(𝑘 ) can be chosen as the maximum of the first derivative, which is also taken in the point 𝑡 = 1. Thus, we obtain sup |𝐿 ℎ(𝑘 ) (𝑡)| = 𝐿 ℎ(𝑘 ) (1) = 𝑡 ∈ [−1,1]

and

1 (𝑘 + 1) , 2𝜋 (1 − ℎ)

𝑘 ∈ N0 ,

(9.160)

9.8 Examples of Dirac Families

𝐶ℎ(𝑘 ) =

503

sup |(𝐿 ℎ(𝑘 ) ) (1) (𝑡)| = (𝐿 ℎ(𝑘 ) ) ′ (1) = 𝑡 ∈ [−1,1]

1 𝑘 (𝑘 + 1) , 2𝜋 (1 − ℎ) 2

𝑘 ≥ 2.

(9.161)

The function 𝐿 ℎ(0) is constant on its support. Consequently, Equation (9.161) is also valid for 𝑘 = 0 on supp(𝐿 ℎ(0) ) = [ℎ, 1]. For 𝑘 = 1, the function 𝐿 ℎ(𝑘 ) is continuous and piecewise linear, thus the Lipschitz-constant 𝐶ℎ(1) can be chosen as the first derivative of 𝐿 ℎ(1) on supp(𝐿 ℎ(1) ). Hence, (9.161) is also true for 𝑘 = 1.

ℎ = 0.3 ℎ = 0.7 ℎ = 0.9

4

1

ℎ = 0.3 ℎ = 0.7 ℎ = 0.9

0.5 2 0 0− 𝜋

− 𝜋/2

0

𝜋/2

𝜋

0

10

20

Fig. 9.13: The smoothed Haar kernel 𝐿 ℎ(𝑘 ) for 𝑘 = 2 and ℎ = 0.3, 0.7, 0.9. Space representation 𝜗 ↦→ 𝐿 ℎ(𝑘 ) (cos(𝜗)) (left) and frequency representation 𝑛 ↦→ (𝐿 ℎ(𝑘 ) ) ∧ (𝑛) (right).

𝑘=0 𝑘=1 𝑘=2

0.6 0.4

1

𝑘=0 𝑘=1 𝑘=2

0.5 0.2 0

0 −𝜋

− 𝜋/2

0

𝜋/2

𝜋

0

10

20

Fig. 9.14: The smoothed Haar kernel 𝐿 ℎ(𝑘 ) for ℎ = 0.3 and 𝑘 = 0, 1, 2. Space representation 𝜗 ↦→ 𝐿 ℎ(𝑘 ) (cos(𝜗)) (left) and frequency representation 𝑛 ↦→ (𝐿 ℎ(𝑘 ) ) ∧ (𝑛) (right). In order to discuss the Dirac property for the Haar functions in more detail, we consider the averages

504

9 Scalar Zonal Kernel Functions

𝑀ℎ(𝑘 ) (𝐹)(𝜉) =

∫

𝐿 ℎ(𝑘 ) (𝜉 · 𝜂)𝐹 (𝜂) 𝑑𝜔(𝜂),

𝜉 ∈ Ω, 𝑘 ≥ 0,

(9.162)

Ω

where (see Figs. 9.13 and 9.14 for a graphical illustration)   −1 𝐿 ℎ(𝑘 ) (𝜉 · 𝜂) = (𝐵 ℎ(𝑘 ) ) ∧ (0) 𝐵 ℎ(𝑘 ) (𝜉 · 𝜂),

𝜉, 𝜂 ∈ Ω.

(9.163)

Clearly, the case 𝑘 = 0 describes an equally weighted average over a spherical cap. For 𝜉 ∈ Ω and 𝐹 ∈ C(Ω), we have |𝑀ℎ(0) (𝐹)(𝜉) − 𝐹 (𝜉)| ∫ = ∥Ω(𝜉; ℎ)∥ −1 ≤

(9.164) 𝜉 · 𝜂 ≥ℎ, | 𝜂 |=1

(𝐹 (𝜂) − 𝐹 (𝜉)) 𝑑𝜔(𝜂)

|𝐹 (𝜂) − 𝐹 (𝜉)|.

sup

(9.165)

ℎ≤ 𝜉 · 𝜂 ≤1

Thus, it is easy to see that ∥ 𝑀ℎ(0) (𝐹) − 𝐹 ∥ C(Ω) ≤ 𝜇(𝐹; 1 − ℎ).

(9.166)

Moreover, for ℎ ∈ [0, 1) and 𝐹 ∈ L2 (Ω),

and

∥𝑀ℎ(0) (𝐹)∥ L2 (Ω) ≤ ∥𝐹 ∥ L2 (Ω)

(9.167)

lim ∥𝑀ℎ(0) (𝐹) − 𝐹 ∥ L2 (Ω) = 0.

(9.168)

ℎ→1 ℎ 0 for all 𝜉 ∈ Ω. Moreover, because of the limit relation, lim (1 − ℎ) −1 (ℎ 𝑛 − 1) = −𝑛,

(9.228)

ℎ→1,ℎ0 𝜌 C(Ω) Proof The integral ∫

𝜌

𝐺 𝜏 (Δ∗ 𝐹)(𝜉) 𝑑𝜏

(9.239)

0

exists for all 𝜌 > 0, 𝜉 ∈ Ω and for all 𝐹 ∈ C (2) (Ω). Moreover, it is not difficult to see that ∫ ∫ 𝜌 𝐺 𝜏 (Δ∗ 𝐹)(𝜉) 𝑑𝜏𝑌𝑛, 𝑗 (𝜉) 𝑑𝜔(𝜉) (9.240) Ω 0 ∫ 𝜌∫ = 𝐺 𝜏 (Δ∗ 𝐹)(𝜉)𝑌𝑛, 𝑗 (𝜉) 𝑑𝜔(𝜉) 𝑑𝜏 0 Ω ∫ e−𝑛(𝑛+1)𝜌 − 1 =− Δ∗𝜉 𝐹 (𝜉)𝑌𝑛, 𝑗 (𝜉) 𝑑𝜔(𝜉). 𝑛(𝑛 + 1) Ω Since (Δ∗ ) ∧ (𝑛) = −𝑛(𝑛 + 1) are the eigenvalues of the Beltrami operator Δ∗ , we find for 𝑛 = 0, 1, ...

9.8 Examples of Dirac Families

∫ ∫ Ω

515

𝜌

𝐺 𝜏 (Δ∗ 𝐹)(𝜉) 𝑑𝜏𝑌𝑛, 𝑗 (𝜉) 𝑑𝜔(𝜉) = (e−𝑛(𝑛+1)𝜌 − 1)

0

∫ 𝐹 (𝜉)𝑌𝑛, 𝑗 (𝜉) 𝑑𝜔(𝜉). Ω

(9.241) On the other hand, ∫ ∫ (𝐺 𝜌 (𝐹)(𝜉) − 𝐹 (𝜉))𝑌𝑛, 𝑗 (𝜉) 𝑑𝜔(𝜉) = (e−𝑛(𝑛+1)𝜌 − 1) 𝐹 (𝜉)𝑌𝑛, 𝑗 (𝜉) 𝑑𝜔(𝜉). Ω

Ω

(9.242)

By comparison of (9.241) and (9.242), we obtain ∫ 𝜌 𝐺 𝜌 (𝐹)(𝜉) − 𝐹 (𝜉) = 𝐺 𝜏 (Δ∗ 𝐹)(𝜉) 𝑑𝜏

(9.243)

0

for all 𝜌 > 0 and 𝜉 ∈ Ω. Therefore, it follows that

∫ 𝜌

𝐺 𝜌 (𝐹) − 𝐹

1 ∗ ∗ ∗

− Δ 𝐹 = sup (𝐺 𝜏 (Δ 𝐹)(𝜉) − (Δ 𝐹)(𝜉)) 𝑑𝜏

𝜌 𝜉 ∈Ω 𝜌 0 C(Ω) ∫ 𝜌 1 ≤ sup |𝐺 𝜏 (Δ∗ 𝐹)(𝜉) − Δ∗𝜉 𝐹 (𝜉)| 𝑑𝜏 𝜌 0 𝜉 ∈Ω ≤ sup ||𝐺 𝜏 (Δ∗ 𝐹) − Δ∗ 𝐹 || C(Ω) .

(9.244)

0≤ 𝜏 ≤𝜌

□

Letting 𝜌 tend to 0, we obtain the desired result. In the same way, we obtain the following corollary: Corollary 9.35 For 𝐹, 𝐻 ∈ L2 (Ω) the following statements are equivalent: −𝑛(𝑛 + 1)𝐹 ∧ (𝑛, 𝑗) = 𝐻 ∧ (𝑛, 𝑗), and lim ∥ 𝜌→0 𝜌>0

𝑛 = 0, 1, . . . ,

𝑗 = 1, . . . , 2𝑛 + 1,

𝐺 𝜌 (𝐹) − 𝐹 − 𝐻 ∥ L2 (Ω) = 0 𝜌

If 𝐻 = 0 in L2 (Ω), then 𝐹 = const. Moreover, H. Berens et al. (1969) have shown that the “saturation class” of the Gauss-Weierstrass singular integral operators {𝐺 𝜌 }, 𝜌 ∈ (0, ∞), is given by H(L2 (Ω); −𝑛(𝑛 + 1)) 2

(9.245) 2

∧

∧

= {𝐹 ∈ L (Ω)|∃𝐺 ∈ L (Ω) : 𝐺 (𝑛, 𝑗) = −𝑛(𝑛 + 1)𝐹 (𝑛, 𝑗)}, and the “saturation order” of {𝐺 𝜌 }, 𝜌 ∈ (0, ∞), is 𝑂 (𝜌), 𝜌 → 0. A problem involving the Gauss-Weierstrass kernel (cf. W. Freeden, M. Schreiner (1995a)) is the initial-value problem (heat equation)

516

9 Scalar Zonal Kernel Functions

𝜕 𝑈 (𝑡; 𝜉) = Δ∗𝜉 𝑈 (𝑡; 𝜉), 𝑡 ≥ 0, 𝜕𝑡 𝑈 (0; 𝜉) = 𝐹 (𝜉), 𝜉 ∈ Ω.

𝜉 ∈ Ω,

(9.246) (9.247)

The solution is given by convolution against the Gauss-Weierstrass kernel ∫ 𝑈 (𝑡; 𝜉) = 𝐺 𝑡 (𝐹)(𝜉) = 𝑊𝑡 (𝜉 · 𝜂)𝐹 (𝜂) 𝑑𝜔(𝜂). (9.248) Ω

Formula (9.248) is of fundamental importance in multiscale descriptions of spherical images. All kernels that will be discussed now are chosen in such a way that the support of their spectral generators, i.e., the Legendre symbol is compact. In other words, our interest now is to list bandlimited Dirac families. Example 9.36 The generator of the Shannon Dirac family, Φ𝜌 , 𝜌 ∈ (0, ∞), is given by  1, 𝑛 ∈ [0, 𝜌 −1 ), (Φ𝜌 ) ∧ (𝑛) = (9.249) 0, 𝑛 ∈ [𝜌 −1 , ∞). Its associated kernel (see Fig. 9.17 for a graphical impression) reads Φ𝜌 (𝜉 · 𝜂) =

∑︁ 2𝑛 + 1 𝑃𝑛 (𝜉 · 𝜂), 4𝜋 −1

𝜉, 𝜂 ∈ Ω.

(9.250)

𝑛≤𝜌

As already known, the kernel Φ𝜌 may be interpreted as truncated Dirac kernel. It is not surprising that the Shannon kernel as “finite polynomial kernel” shows strong oscillations in space. This is the price to be paid for the sharp separation in frequency space. To suppress the oscillations, we are led to “smoothed versions” of the Shannon kernel (dependent on an additional parameter 𝛼 ∈ (0, 1)).

Example 9.37 The generator of the smoothed Shannon Dirac family (see Figs. 9.18 and 9.19) reads as follows Φ∧𝜌 (𝑛)

 𝑛 ∈ [0, 𝜌 −1 𝛼),   1,  −1 = (1 − 𝛼) (1 − 𝛼𝑛), 𝑛 ∈ [𝜌 −1 𝛼, 𝜌 −1 ],   0, 𝑛 ∈ [𝜌 −1 , ∞). 

(9.251)

Compared with the Shannon case, there is a linear transition from the value 1 at [0, 𝜌 −1 𝛼] to the value 0 at [𝜌 −1 , ∞). Example 9.38 Of course, many other suitable choices for Φ∧𝜌 (𝑛) can be found for practical purposes. We only mention the CUP-Dirac family (see M. Schreiner (1996)).

9.8 Examples of Dirac Families

517 𝜌 = 1/16 𝜌 = 1/8 𝜌 = 1/4

20

1

10

𝜌 = 1/16 𝜌 = 1/8 𝜌 = 1/4

0.5

0 −𝜋

− 𝜋/2

0

𝜋/2

𝜋

0

0

5

10

20

15

Fig. 9.17: The Shannon scaling function Φ𝜌 for 𝜌 = 1/16, 1/8, 1/4. Space representation 𝜗 ↦→ Φ𝜌 (cos(𝜗)) (left) and frequency representation 𝑛 ↦→ (Φ𝜌 ) ∧ (𝑛) (right). 𝜌 = 1/16 𝜌 = 1/8 𝜌 = 1/4

10

1

𝜌 = 1/16 𝜌 = 1/8 𝜌 = 1/4

0.5

5

0 −𝜋

− 𝜋/2

0

𝜋/2

𝜋

0

0

5

10

20

15

Fig. 9.18: The smoothed Shannon scaling function Φ𝜌 for 𝜌 = 1/16, 1/8, 1/4, and 𝛼 = 0.5. Space representation 𝜗 ↦→ Φ𝜌 (cos(𝜗)) (left) and frequency representation 𝑛 ↦→ (Φ𝜌 ) ∧ (𝑛) (right). 𝜌 = 1/16 𝜌 = 1/8 𝜌 = 1/4

6 4

1

𝜌 = 1/16 𝜌 = 1/8 𝜌 = 1/4

0.5

2 0 −𝜋

− 𝜋/2

0

𝜋/2

𝜋

0

0

5

10

15

20

Fig. 9.19: The CUP-Dirac family Φ𝜌 for 𝜌 = 1/16, 1/8, 1/4. Space representation 𝜗 ↦→ Φ𝜌 (cos(𝜗)) (left) and frequency representation 𝑛 ↦→ (Φ𝜌 ) ∧ (𝑛) (right).

Φ∧𝜌 (𝑛) =



(1 − 𝜌𝑛) 2 (1 + 2𝜌𝑛), 𝑛 ∈ [0, 𝜌 −1 ), 0, [𝜌 −1 , ∞).

(9.252)

518

9 Scalar Zonal Kernel Functions

The illustrations show that the phenomena of oscillation that are still existent for the smoothed Shannon Dirac family can be suppressed by this choice. Next, we are interested in the kernels 𝐵 ℎ(𝑘 ) in view of the uncertainty relation. Using ∥𝐵 ℎ(𝑘 ) ∥ 2

∫

−1

= 2𝜋 we define the kernel

√︄ 𝐵˜ ℎ(𝑘 )

1

= 2𝜋

=

[𝐵 ℎ(𝑘 ) (𝑡)] 2 𝑑𝑡

1−ℎ , 2𝑘 + 1

(9.253)

2𝑘 + 1 𝐵 (𝑘 ) , 2𝜋(1 − ℎ) ℎ

(9.254)

since the uncertainty properties are normally defined for kernels with norm one. We find ∫ 1  2 (1) 1 + ℎ + 2𝑘 3 𝑡 𝐵˜ ℎ(𝑘 ) (𝑡) 𝑑𝑡 𝜀 3 = 𝜀 . (9.255) 𝑔 𝑜˜ (𝑘) 3 = 2𝜋 𝐵ℎ ( · 𝜀 ) 2 + 2𝑘 −1 Consequently, (1)

(𝜎 𝑜˜ (𝑘) ) 2 = 1 −



𝐵ℎ

1 + ℎ + 2𝑘 2 + 2𝑘

2 =

(1 − ℎ)(ℎ + 4𝑘 + 3) . (2𝑘 + 2) 2

(9.256)

Using (9.39), we finally arrive at (1)

Δ𝑜˜ (𝑘) = 𝐵ℎ

√︁ 1 (1 − ℎ)(ℎ + 4𝑘 + 3). 1 + ℎ + 2𝑘

(9.257)

For the localization in frequency, we assume 𝑘 ≥ 2. We have (3) (𝜎 𝑜˜ (𝑘) 3 ) 2 𝐵ℎ ( · 𝜀 )

∫

1

𝐵˜ ℎ(𝑘 ) (𝑡)𝐿 𝑡 𝐵˜ ℎ(𝑘 ) (𝑡) 𝑑𝑡 −1 ∫ 1 2𝑘 + 1 −2𝜋 = (𝑡 − ℎ) 𝑘 𝐿 𝑡 (𝑡 − ℎ) 𝑘 𝑑𝑡 2𝜋(1 − ℎ) (1 − ℎ) 2𝑘 ℎ 𝑘 (ℎ + 2𝑘) = , (1 − ℎ)(2𝑘 − 1) = −2𝜋

so that

√︄ (3) Δ𝑜˜ (𝑘) 𝐵ℎ

=

𝑘 (ℎ + 2𝑘) . (1 − ℎ)(2𝑘 − 1)

(9.258)

(9.259)

The application of 𝐿 𝑡 requires that the kernel is twice differentiable. However, using integration by parts, the results immediately carry over to the case 𝑘 = 1. Fig. 9.20 gives a graphical impression of these results for the special cases 𝑘 = 1 and 𝑘 = 3.

9.8 Examples of Dirac Families

519

3

2

3 (1) Δ𝑜˜ (𝑘) 𝐵ℎ (3) Δ𝑜˜ (𝑘) 𝐵ℎ (1) (3) Δ𝑜˜ (𝑘) Δ𝑜˜ (𝑘) 𝐵 𝐵 ℎ

(1)

Δ𝑜˜ (𝑘) 2

ℎ

ℎ

1

0 −1

𝐵ℎ (3) Δ𝑜˜ (𝑘) 𝐵ℎ (1) (3) Δ𝑜˜ (𝑘) Δ𝑜˜ (𝑘) 𝐵 𝐵 ℎ

1

−0.5

0

0.5

0 −1

1

−0.5

0

1

0.5

Fig. 9.20: Uncertainty classification of the normalized smoothed Haar scaling func(1) (3) (1) (3) tion 𝐵˜ ℎ(𝑘 ) (𝑘 = 1, left; 𝑘 = 3 right). Δ𝑜˜ (𝑘) , Δ𝑜˜ (𝑘) and the product Δ𝑜˜ (𝑘) Δ𝑜˜ (𝑘) are 𝐵ℎ

𝐵ℎ

𝐵ℎ

𝐵ℎ

shown as functions of ℎ.

For the investigation of the uncertainty properties of the Shannon kernels, we start from 2

∥Φ𝜌 ∥ =

−1 ⌊𝜌 ∑︁⌋

𝑛=0

 2𝑛 + 1 1  = (⌊𝜌 −1 ⌋ + 1) + ⌊𝜌 −1 ⌋ ⌊𝜌 −1 + 1⌋ 4𝜋 4𝜋

(9.260)

where, as usual, ⌊𝜌 −1 ⌋ is the largest integer which is less or equal 𝜌 −1 . Observing this result, we define the normalized Shannon kernel by ˜𝜌 = Φ

1 Φ𝜌 . ∥Φ𝜌 ∥

(9.261)

2

−1

(1) (𝜎Φ𝑜˜ ) 2 𝜌

⌊𝜌 −1⌋ 1 © ∑︁ 2𝑛 + 2 ª =1− ­ ® 4𝜋 ∥Φ𝜌 ∥ 2 𝑛=1 « ¬  2 2⌊𝜌 −1 − 1⌋ + ⌊𝜌 −1 ⌋ ⌊𝜌 −1 − 1⌋ =1− , ⌊𝜌 −1 + 1⌋ + ⌊𝜌 −1 ⌋ ⌊𝜌 −1 + 1⌋

so that (1)

𝑜 ΔΦ ˜

Moreover, we find

𝜌

(9.262)

v u  −1 2 u u u −1⌋+⌊𝜌 −1 ⌋ ⌊𝜌 −1 −1⌋ t 1 − 2⌊𝜌−1 −1 −1 =

⌊𝜌

+1⌋+⌊𝜌

⌋ ⌊𝜌

+1⌋

2⌊𝜌 −1 −1⌋+⌊𝜌 −1 ⌋ ⌊𝜌 −1 −1⌋ ⌊𝜌 −1 +1⌋+⌊𝜌 −1 ⌋ ⌊𝜌 −1 +1⌋

.

(9.263)

520

9 Scalar Zonal Kernel Functions −1

(3) (𝜎Φ𝑜˜ ) 2 𝜌

⌊𝜌 ∑︁⌋ 2𝑛 + 1 4𝜋 = −1 𝑛(𝑛 + 1) ⌊𝜌 ⌋ + 1 + ⌊𝜌 −1 ⌋ ⌊𝜌 −1 + 1⌋ 𝑛=0 4𝜋

=

1 ⌊𝜌 −1 ⌋ (1 + ⌊𝜌 −1 ⌋) 2 (2 + ⌊𝜌 −1 ⌋) 2 ⌊𝜌 −1 ⌋ + 1 + ⌊𝜌 −1 ⌋ ⌊𝜌 −1 + 1⌋

such that

√︄ 𝑜 (3) ΔΦ ˜𝜌

=

(9.264)

1 ⌊𝜌 −1 ⌋ (1 + ⌊𝜌 −1 ⌋) 2 (2 + ⌊𝜌 −1 ⌋) . 2 ⌊𝜌 −1 ⌋ + 1 + ⌊𝜌 −1 ⌋ ⌊𝜌 −1 + 1⌋

(9.265)

The results are illustrated in Fig. 9.21.

(1)

103

𝑜 ΔΦ ˜

𝜌 (3)

𝑜 ΔΦ ˜

101

𝜌 (1)

(3)

𝜌

𝜌

𝑜 Δ𝑜 ΔΦ ˜ ˜ Φ

10−1

10−3 10−4

10−3

10−2

10−1

˜ 𝜌. Fig. 9.21: Uncertainty classification of the normalized Shannon Dirac family Φ (1) (3) (1) 𝑜 (3) 𝑜 𝑜 𝑜 Presented are ΔΦ˜ , ΔΦ˜ , and the product ΔΦ˜ ΔΦ˜ as functions of 𝜌 in a double 𝜌 𝜌 𝜌 𝜌 logarithmic setting.

9.9 Bibliographical Notes

There is a long history of zonal kernel functions (also called radial basis functions in the language of approximation theory). First essential results are due to H. Funk (1916), E. Hecke (1918). The investigations involving spherical convolutions lead back to S. Bochner (1954), W. Rudin (1950), A.P. Calderon, A. Zygmund (1955) and many others. H. Berens et al. (1969) presents an overview about the activities in the first half of the last century. F.J. Narcowich, J.D. Ward (1996) introduced an uncertainty principle for the unit sphere Ω ⊂ R3 . Another approach which involves the usage of a differential operator of second order has been made by W. Freeden (1998, 1999). The articles by N. La´ın Fern´andez (2003), N. La´ın Fern´andez, J.

9.9 Bibliographical Notes

521

Prestin (2003) are further significant contributions to the topic of space/frequency localization. The correspondence of the Dirac delta kernel to the so-called Dirac families is well-known in the analysis of symmetries in Euclidean space. Zonal Dirac families of Abel-Poisson, Gauss-Weierstrass type were studied, e.g., by H. Berens et al. (1969), C. M¨uller (1998), W. Freeden et al. (1998, 2018a). Scalar zonal kernel functions are basic tools for constructing spherical splines (compare, e.g., W. Freeden (1981a), G. Wahba (1981), G.E. Backus (1986), W. Freeden (1990), W. Freeden et al. (1996), W. Freeden et al. (1997), W. Freeden, F. Schneider (1999), W. Freeden (1999). In the last decades, wavelets on the sphere have been the focus of several research groups which led to different wavelet approaches. Common to all these proposals is a multiresolution analysis which enables a balanced amount of both frequency (more accurately, angular momentum) and space localization (see e.g., S. Dahlke et al. (1995) and I. Weinreich (2001), D. Potts, M. Tasche (1995), T. Lyche, L. Schumaker (2000), P. Schr¨oder, W. Sweldens (1995)). A group theoretical approach to a continuous wavelet transform on the sphere is followed by J.-P. Antoine, P. Vandergheynst (1999), J.-P. Antoine et al. (2002), and M. Holschneider (1996). The parameter choice of their continuous wavelet transform is the product of 𝑆𝑂 (3) (for the motion on the sphere) and R+ (for the dilations). A continuous wavelet transform approach for analyzing functions on the sphere is presented by Dahlke and Maass (S. Dahlke, P. Maass (1996)). The constructions of the Geomathematics Group in Kaiserslautern on spherical wavelets (W. Freeden, M. Schreiner (1995a), W. Freeden, U. Windheuser (1996), W. Freeden, U. Windheuser (1997), W. Freeden et al. (1998), W. Freeden, K. Hesse (2002), W. Freeden, C. Mayer (2003), W. Freeden et al. (2003), W. Freeden, M. Schreiner (2007)) are intrinsically based on the specific properties concerning the theory of spherical harmonics. W. Freeden, M. Schreiner (2007) are interested in a compromise connecting zonal function expressions and structured grids on the sphere to obtain fast algorithms. W. Freeden, M. Schreiner (2006b), W. Freeden, C. Gerhards (2013), W. Freeden (2021), C. Blick et al. (2021) use zonal functions in mollifier decorrelation of gravitational and geomagnetic signatures (see also Chapter 14).

Chapter 10

Vector Zonal Kernel Functions

In vector theory, the points of the departure to zonal kernel fields are the addition theorems relating vector spherical harmonics to Legendre vector rank-2 tensor fields and the counterparts of the Funk-Hecke formula in the vectorial context. The corresponding kernel functions obtained by summing up the vectorial Legendre kernel functions to certain (bandlimited or non-bandlimited) orthogonal series expansions are called zonal vector kernel functions due to their intimate similarities in definition and structure to scalar zonal functions and their relevance to (geo-)physically motivated applications. Of particular significance in the theory of vector fields is the coordinate-free representation by vector zonal kernel functions. As is well known, coordinate representations of vector spherical harmonics are not calculable without singularities at the poles. Zonal vector functions, i.e., vector Legendre kernel expansions, however, avoid this problem completely, as they are constructed by application of the surface gradient and the surface curl gradient to a scalar zonal kernel function. In fact, zonal vector functions consist of a “directional term” linked to a scalar zonal kernel function (for the normal part) or a one-dimensional derivative (for the tangential parts). Moreover, differential operators of vectorial nature like the surface gradient or the surface curl gradient can completely be treated within an isotropic (vector) framework. It should be pointed out that isotropic vector operators, i.e., operators mapping a scalar function to a vector field (or vice versa) – thereby maintaining their form when subjected to orthogonal transformations – can be expressed by means of convolutions against a vector zonal kernel function. In that sense, vector zonal functions form the canonical bridge between scalar functions and vector fields. In addition, the inherent orthogonal invariance reduces the structural complexity and dimension. It should be mentioned that, in vectorial case, two different techniques can be formulated for representing isotropic operators by convolution. Zonal kernel functions to be used in the vector context can either be formulated as vector fields generated by applying the operator 𝑜 (𝑖) once on scalar zonal functions, or as rank-2 tensor fields by a double application of the operators 𝑜 (𝑖) (see, e.g., M. Bayer et al. (1998), S. Beth (2000), © The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2022 W. Freeden and M. Schreiner, Spherical Functions of Mathematical Geosciences, Geosystems Mathematics, https://doi.org/10.1007/978-3-662-65692-1_10

523

524

10 Vector Zonal Kernel Functions

H. Nutz (2002), C. Mayer (2003)). In the first case, we are led to vector zonal kernel functions, whereas the second case leads to vectorial zonal rank-2 tensor kernel functions (to be used within the vector context (see Table 10.1).

Table 10.1 Overview on (vectorial) zonal rank-2 tensor/vector kernel functions in relation to Legendre kernel functions.

Zonal kernel (generating system)

Linear approach

Bilinear approach

scalar field ({𝑌𝑛, 𝑗 }system)

scalar zonal kernel function ({ 𝑃𝑛 }-system)

vector field (𝑖) ({ 𝑦𝑛, 𝑗 }system)

(vectorial) zonal rank-2 tensor kernel function ({ 𝑣 p𝑛(𝑖,𝑖) }-system)

(vectorial) zonal vector kernel function({ 𝑝𝑛(𝑖) }system)

𝑘 (𝑖)

𝑣 k (𝑖,𝑖)

vector field (𝑖) ({ 𝑦˜ 𝑛, 𝑗 }system)

(vectorial) zonal rank-2 tensor kernel function ({ 𝑣 p˜ 𝑛(𝑖,𝑖) }-system)

(vectorial) zonal vector kernel function({ 𝑝˜ 𝑛(𝑖) }system)

𝑘˜ (𝑖)

𝑣k ˜ (𝑖,𝑖)

𝐾

10.1 Preparatory Material

As already mentioned, two approaches are evident based on the addition theorems (Theorem 6.31 and Theorem 6.46): Zonal rank-2 tensor kernel functions (within the vectorial context) – in this approach called (vectorial) zonal rank-2 tensor kernel function – are defined by a double application of the differential dual operators 𝑜 (𝑖) on scalar zonal kernel functions, whereas the zonal vectorial kernel functions are derived by a single application of these operators to scalar zonal kernel functions. In doing so, we take advantage of the features from the operators 𝑜 (𝑖) , 𝑂 (𝑖) , 𝑖 = 1, 2, 3. Indeed, the operators 𝑜 (𝑖) can be easily applied to scalar zonal kernel functions. For example, for a (sufficiently smooth) scalar zonal kernel function 𝐾, the following identities are well-known: 𝑜 (1) 𝜉 𝐾 (𝜉 · 𝜂) = 𝐾 (𝜉 · 𝜂)𝜂,

(10.1)

10.2 Tensor Zonal Kernel Functions of Rank Two in Vectorial Context

525

′ 𝑜 (2) 𝜉 𝐾 (𝜉 · 𝜂) = 𝐾 (𝜉 · 𝜂)(𝜂 − (𝜉 · 𝜂)𝜉),

(10.2)

𝑜 (3) 𝜉 𝐾 (𝜉

(10.3)

· 𝜂) = 𝐾 ′ (𝜉 · 𝜂)(𝜉 ∧ 𝜂).

Furthermore, we mention (1) 𝑜 (1) 𝜉 𝑜 𝜂 𝐾 (𝜉 · 𝜂) = 𝐾 (𝜉 · 𝜂)𝜉 ⊗ 𝜂,

(10.4)

(2) 𝑜 (2) 𝜉 𝑜 𝜂 𝐾 (𝜉

(10.5)

· 𝜂) = =

∇∗𝜉 ⊗ (𝐾 ′ (𝜉 · 𝜂)(𝜉 − (𝜉 · 𝜂)𝜂)) (∇∗𝜉 𝐾 ′ (𝜉 · 𝜂)) ⊗ (𝜉 − (𝜉 · 𝜂)𝜂) + 𝐾 ′ (𝜉 · 𝜂)∇∗𝜂 ⊗ (𝜉 − (𝜉 · 𝜂)𝜂) ′′

= 𝐾 (𝜉 · 𝜂)(𝜂 − (𝜉 · 𝜂)𝜉) ⊗ (𝜉 − (𝜉 · 𝜂)𝜂) + 𝐾 ′ (𝜉 · 𝜂)(itan (𝜉) − (𝜂 − (𝜉 · 𝜂)𝜉) ⊗ 𝜂), (3) ′ 𝑜 (3) 𝜉 𝑜 𝜂 𝐾 (𝜉 · 𝜂) = 𝐾 (𝜉 · 𝜂)𝜉 ∧ 𝜂 ⊗ 𝜂 ∧ 𝜉

(10.6)

′

+ 𝐾 (𝜉 · 𝜂)((𝜉 · 𝜂)itan (𝜉) − (𝜂 − (𝜉 · 𝜂)𝜉) ⊗ 𝜉), provided that 𝐾 : [−1, 1] → R is sufficiently often differentiable. These formulas show two advantages: First, we only have to calculate the onedimensional derivatives of a scalar zonal kernel function, which reduces the operational effort enormously (note that, in the case of double application of the operators 𝑜 (𝑖) , the two-dimensional derivatives of a scalar zonal kernel function have to be evaluated). Second, no singularities occur when the operators 𝑜 (2) and 𝑜 (3) are applied to scalar zonal kernel functions. Moreover, the basic principles that are governed by the uncertainty relation canonically extend from the scalar to the vector context of zonal kernel functions (even for tangential vector kernel fields).

10.2 Tensor Zonal Kernel Functions of Rank Two in Vectorial Context

We start with the characterization of zonal rank-2 tensor kernel functions (within the vectorial context). This approach arises directly from the scalar theory (see W. Freeden et al. (1998), W. Freeden, M. Schreiner (2009)). To be more concrete, the zonal tensor kernel functions are defined in terms of scalar zonal kernel functions by a double application of the operators 𝑜 (𝑖) .

526

10 Vector Zonal Kernel Functions

Definition 10.1 Assume that 𝐾 (1) ∈ C[−1, 1] and 𝐾 (𝑖) ∈ C (2) [−1, 1], 𝑖 ∈ {2, 3}, are scalar zonal kernel functions. A function 𝑣 k (𝑖,𝑖) : Ω × Ω → R3 ⊗ R3 , 𝑣 (𝑖,𝑖)

k

(𝑖) (𝑖) (𝜉, 𝜂) = 𝑜 (𝑖) (𝜉 · 𝜂), 𝜉 𝑜𝜂 𝐾

𝜉, 𝜂 ∈ Ω,

(10.7)

is called a (vectorial) zonal rank-2 tensor kernel function of type (𝑖, 𝑖) (with respect to { 𝑣 p𝑛(𝑖,𝑖) }). Moreover, 3 ∑︁ 𝑣 𝑣 (𝑖,𝑖) k= k (10.8) 𝑖=1

is called a (vectorial) zonal rank-2 tensor kernel function (with respect to { 𝑣 p𝑛 }). Definition 10.2 A (vectorial) zonal rank-2 tensor kernel function of type (𝑖, 𝑖), 𝑣 k (𝑖) : Ω × Ω → R3 ⊗ R3 , 𝑖 ∈ {1, 2, 3}, is called an l2(𝑖,𝑖) (Ω)-zonal rank-2 tensor kernel function, if 𝑣 k (𝑖,𝑖) (𝜉, ·) is square-integrable on Ω for each 𝜉 ∈ Ω. Furthermore, 𝑣 k = Í3 𝑣 k (𝑖,𝑖) is called an l2 (Ω)-zonal rank-2 tensor kernel function. 𝑖=1 For the explicit representation of the (vectorial) zonal rank-2 tensor kernel functions in terms of the Legendre polynomials, we take advantage of the already known vectorial variants of the Funk-Hecke formula, which should be recapitulated for the convenience of the reader. Theorem 10.3 (Funk-Hecke Formula in Vectorial Context) Let 𝜂 ∈ Ω be fixed. Assume that 𝑔(·, 𝜂) ∈ c (1) (Ω) satisfies t𝑔(𝜉, 𝜂) = 𝑔(t𝜉, 𝜂) for all orthogonal transformations t ∈ 𝑆𝑂 𝜂 (3) and all 𝜉 ∈ Ω. Then, for all 𝜁 ∈ Ω and 𝑖 ∈ {1, 2, 3}, we have ∫   −1 𝑣 (𝑖,𝑖) p𝑛 (𝜁, 𝜉)𝑔(𝜉, 𝜂) 𝑑𝜔(𝜉) = 𝜇 𝑛(𝑖) (𝑂 (𝑖) 𝑔) ∧ (𝑛)𝑜 𝜁(𝑖) 𝑃𝑛 (𝜁 · 𝜂), (10.9) Ω

where (𝑂

(𝑖)

∧

∫

1

𝑔) (𝑛) = 2𝜋

𝐺 𝑖 (𝑡)𝑃𝑛 (𝑡) 𝑑𝑡,

(10.10)

−1

and

𝐺 𝑖 (𝜉 · 𝜂) = 𝑂 (𝑖) 𝜉 𝑔(𝜉, 𝜂).

(10.11)

Within the concept of (vectorial) zonal kernel functions, this theorem leads us to the following statement. Theorem 10.4 Any l2(𝑖,𝑖) (Ω)-zonal rank-2 tensor kernel function 𝑣 k (𝑖,𝑖) can be represented as a Legendre series of the form

10.2 Tensor Zonal Kernel Functions of Rank Two in Vectorial Context 𝑣 (𝑖,𝑖)

k

∞ ∑︁ 2𝑛 + 1 (𝑖,𝑖) ∧ (k ) (𝑛) 𝑣 p𝑛(𝑖,𝑖) (𝜉, ·), 4𝜋 𝑛=0

(𝜉, ·) =

527

(10.12)

𝑖

where ( 𝑣 k (𝑖,𝑖) ) ∧ (𝑛) = 2𝜋𝜇 𝑛(𝑖)

∫

1

𝐾 (𝑖) (𝑡)𝑃𝑛 (𝑡) 𝑑𝑡 = 𝜇 𝑛(𝑖) (𝐾 (𝑖) ) ∧ (𝑛).

(10.13)

−1

Proof We deal with the case 𝑖 = 1. As 𝑣 k (1,1) (𝜉, ·) is a member of the space l2 (Ω), we have ∫ (1) (1) (1) (1) 𝑜 (1) (𝜉 · 𝜂) · 𝑜 (1) (𝜉 · 𝜂) 𝑑𝜔(𝜂) < ∞. (10.14) 𝜉 𝑜𝜂 𝐾 𝜉 𝑜𝜂 𝐾 Ω

Furthermore, for f ∈ l2 (Ω) and 𝑔 ∈ l2 (Ω) we obtain ∫ ∫ f (𝜂) · 𝑜 (1) 𝑔(𝜂) 𝑑𝜔(𝜂) = 𝑂 (1) 𝜂 𝜂 f (𝜂) · 𝑔(𝜂) 𝑑𝜔(𝜂). Ω

(10.15)

Ω

Hence, it follows that ∫ (1) (1) 𝑜 (1) (𝜉 · 𝜂) · 𝑜 (1) (𝜉 · 𝜂) 𝑑𝑤(𝜂) 𝜂 𝐾 𝜂 𝐾 Ω ∫ (1) (1) (1) (1) = 𝑂 (1) (𝜉 · 𝜂) · 𝑜 (1) (𝜉 · 𝜂) 𝑑𝜔(𝜂) 𝜂 𝐾 𝜉 𝑜𝜉 𝑜𝜂 𝐾 Ω ∫ (1) (1) (1) (1) = 𝑜 (1) (𝜉 · 𝜂) · 𝑜 (1) (𝜉 · 𝜂) 𝑑𝜔(𝜂). 𝜉 𝑜𝜂 𝐾 𝜉 𝑜𝜂 𝐾

(10.16)

Ω (1) (𝜉 ·𝜂), 𝜂 ∈ Ω, we can express 𝑔 as a Fourier (orthogonal) Defining 𝑔 𝜉 (𝜂) = 𝑜 (𝑖) 𝜉 𝜂 𝐾 series. More explicitly, using the addition theorem and the vectorial Funk-Hecke formula, we readily find (1) 𝑜 (𝑖) (𝜉 · 𝜂) = 𝑔 𝜉 (𝜂) 𝜂 𝐾

=

∞ 2𝑛+1 ∑︁ ∑︁ 

(10.17) (𝑔 𝜉 ) (𝑖)

∧

(𝑖) (𝑛, 𝑚)𝑦 𝑛,𝑚 (𝜂)

𝑛=0𝑖 𝑚=1

=

=

=

∞ 2𝑛+1 ∑︁ ∑︁ ∫ 𝑛=0𝑖 𝑚=1 ∞ ∑︁ 𝑛=0𝑖 ∞ ∑︁ 𝑛=0𝑖

(𝑖) (𝑖) 𝑔 𝜉 (𝜁) · 𝑦 𝑛,𝑚 (𝜁) 𝑑𝜔(𝜁) 𝑦 𝑛,𝑚 (𝜂)

Ω

2𝑛 + 1 4𝜋

∫

𝑣 (𝑖,𝑖) p𝑛 (𝜂, 𝜁)𝑔 𝜉 (𝜁) 𝑑𝜔(𝜁)

Ω

2𝑛 + 1  (𝑖)  −1  (𝑖)  ∧ 𝜇𝑛 𝑂 𝑔 𝜉 (𝑛) 𝑜 (𝑖) 𝜂 𝑃 𝑛 (𝜂 · 𝜉), 4𝜋

528

10 Vector Zonal Kernel Functions

where (𝑂 (𝑖) 𝑔 𝜉 ) ∧ (𝑛) = 2𝜋

∫

1

𝐺 𝑖 (𝑡)𝑃𝑛 (𝑡) 𝑑𝑡,

(10.18)

−1

with

𝐺 𝑖 (𝜉 · 𝜁) = 𝑂 𝜁(𝑖) 𝑔 𝜉 (𝜁).

Therefore, for 𝑖 = 1, we have ∫ 1 ∫ (𝑂 (1) 𝑔 𝜉 ) ∧ (𝑛) = 2𝜋 𝐾ˆ (𝑡)𝑃𝑛 (𝑡) 𝑑𝑡 = 2𝜋 −1

1

(10.19)

𝐾 (1) (𝑡)𝑃𝑛 (𝑡) 𝑑𝑡,

(10.20)

−1

where (1) (1) 𝐾ˆ (𝜉 · 𝜂) = 𝑂 (1) (𝜉 · 𝜂) = (𝜉 · 𝜉) 𝐾 (1) (𝜉 · 𝜂) = 𝐾 (1) (𝜉 · 𝜂), 𝜉 𝑜𝜂 𝐾

(10.21)

thereby identifying, as usual, 𝐾ˆ : Ω × Ω → R3 with 𝐾ˆ : [−1, 1] → R3 , (i.e., 𝐾ˆ (𝑡) = 𝐾ˆ (𝜉 · 𝜂)). Summarizing our results for 𝑖 = 1, we therefore obtain 𝑣 (1,1)

k

(1) (1) (𝜉 · 𝜂) = 𝑜 (1) (𝜉 · 𝜂) 𝜉 𝑜𝜂 𝐾

= 𝑜 (1) 𝜉 𝑔 𝜉 (𝜂) = 𝑜 (1) 𝜉

∞ ∑︁ 2𝑛 + 1 𝑛=0

=

4𝜋

∞ ∑︁ 2𝑛 + 1 𝑛=0

4𝜋

(𝜇 𝑛(1) ) −1 (𝑂 (1) 𝑔 𝜉 ) ∧ (𝑛)𝑜 (1) 𝜂 𝑃 𝑛 (𝜉 · 𝜂)

(𝑂 (1) 𝑔 𝜉 ) ∧ (𝑛) 𝑣 p𝑛(1,1) (𝜉, 𝜂).

(10.22)

Observing the fact that 𝜇 𝑛(1) = 1, we finally get the wanted assertion. Next, we come to the cases 𝑖 = 2, 3. Now, 𝐾 (𝑖) (𝜉·) is differentiable and, thus, in L2 (Ω). The kernel 𝐾 (𝑖) admits the Legendre series expansion 𝐾 (𝑖) =

∞ ∑︁ 2𝑛 + 1 𝑛=0

4𝜋

where (𝐾 (𝑖) ) ∧ (𝑛) = 2𝜋

∫

(𝐾 (𝑖) ) ∧ (𝑛)𝑃𝑛 ,

1

𝐾 (𝑖) (𝑡)𝑃𝑛 (𝑡) 𝑑𝑡.

(10.23)

(10.24)

−1

This leads us to the desired identity (𝑖) (𝑖) k (𝑖) (𝜉, 𝜂) = 𝑜 (𝑖) (𝜉 · 𝜂) 𝜉 𝑜𝜂 𝐾

(10.25)

10.2 Tensor Zonal Kernel Functions of Rank Two in Vectorial Context

529

∞ ∑︁ 2𝑛 + 1 (𝑖) ∧ (𝐾 ) (𝑛)𝜇 𝑛(𝑖) 𝑣 p𝑛(𝑖,𝑖) (𝜉, 𝜂). 4𝜋 𝑛=0

=

𝑖

□

Altogether, Theorem 10.4 is verified.

A key property of an l2 (Ω)-tensor zonal kernel function 𝑣 k is its invariance under orthogonal transformations t, i.e., 𝑣 k(t𝜉, t𝜂) = t𝑣 k(𝜉, 𝜂)t𝑇 , 𝜉, 𝜂 ∈ Ω. In addition, it is not difficult to show the following result. Theorem 10.5 A (vectorial) zonal rank-2 tensor kernel function of type 𝑖 𝑣 k (𝑖,𝑖) : Ω × Ω → R3 ⊗ R3 is an l2(𝑖,𝑖) (Ω)-zonal rank-2 tensor kernel function, if and only if ∞ ∑︁ 2𝑛 + 1  𝑣 (𝑖,𝑖) ∧  2 ( k ) (𝑛) < ∞, 4𝜋 𝑛=0

(10.26)

( 𝑣 k (𝑖,𝑖) ) ∧ (𝑛) = 𝜇 𝑛(𝑖) (𝐾 (𝑖) ) ∧ (𝑛).

(10.27)

𝑖

where

Of course, this theorem follows directly from the addition theorem 2𝑛+1 ∑︁

(𝑖) (𝑦 𝑛,𝑚 (𝜉)) 2 =

𝑚=1

2𝑛 + 1 , 4𝜋

𝜉 ∈ Ω.

(10.28)

The introduction of convolutions involving (vectorial) zonal rank-2 tensor kernel functions is quite similar to the scalar case. Definition 10.6 Let 𝑣 k, 𝑣 h be l2 (Ω)-zonal rank-2 tensor kernel functions. Suppose that 𝑓 is a vector field of class l2 (Ω). Then, 𝑣 k ∗ 𝑓 defined by ∫ 𝑣 𝑣 ( k ∗ 𝑓 ) (𝜉) = k(𝜉, 𝜂) 𝑓 (𝜂) 𝑑𝜔(𝜂), 𝜉 ∈ Ω, (10.29) Ω

is called the convolution of

𝑣k

against 𝑓 . Furthermore, 𝑣 h ∗ 𝑣 k defined by

∫ 𝑣

𝑣

𝑣

( h ∗ k) (𝜉, 𝜂) =

h(𝜉, 𝜁) 𝑣 k(𝜁, 𝜂) 𝑑𝜔(𝜁),

𝜉, 𝜂 ∈ Ω,

(10.30)

Ω

is said to be the convolution of 𝑣 h against 𝑣 k. Note that the symbol “∗” is again used simultaneously for different types of convolutions.

530

10 Vector Zonal Kernel Functions

Obviously, 𝑣 k ∗ 𝑓 is a member of class l2 (Ω). In spectral formulation, we have 𝑣 (𝑖,𝑖)

k

∗𝑓 =

∞ ∑︁

( 𝑣 k (𝑖,𝑖) ) ∧ (𝑛)

𝑛=0𝑖

and 𝑣 (𝑖)

h

2𝑛+1 ∑︁

(𝑖) ( 𝑓 (𝑖) ) ∧ (𝑛, 𝑚)𝑦 𝑛,𝑚 ,

∞ ∑︁ 2𝑛 + 1 𝑣 (𝑖,𝑖) ∧ ( h ) (𝑛)( 𝑣 k (𝑖,𝑖) ) ∧ (𝑛) 𝑣 p𝑛(𝑖,𝑖) , 4𝜋 𝑛=0

∗ 𝑣k =

(10.31)

𝑚=1

(10.32)

𝑖

𝑖 = 1, 2, 3. By virtue of the orthogonal expansion in terms of Legendre tensors (10.32), it is not hard to verify that, for every point 𝜉 ∈ Ω, ( 𝑣 h ∗ 𝑣 k) (𝜉, ·) is continuous on the sphere Ω. Lemma 10.7 Let 𝑣 k = Then

Í3

𝑣 (𝑖,𝑖)

k

𝑖=1

𝑣 k (𝑖,𝑖)

be an l2 (Ω)-zonal rank-2 tensor kernel function.

∞ ∑︁ 2𝑛 + 1 𝑣 (𝑖,𝑖) p𝑛 (𝜉, ·) ∗ 𝑣 k(·, 𝜂), 4𝜋 𝑛=0

(𝜉, 𝜂) =

(10.33)

𝑖

𝜉, 𝜂 ∈ Ω. Finally, we mention the representation of an l2 (Ω)-vector field 𝑓 in terms of Legendre tensors 𝑣 p𝑛(𝑖,𝑖) 3 ∑︁ ∞ ∑︁ 2𝑛 + 1 𝑣 (𝑖,𝑖) 𝑓 = p𝑛 ∗ 𝑓 , (10.34) 4𝜋 𝑖=1 𝑛=0 𝑖

where the equality in (10.34) is understood in ∥ · ∥ l2 (Ω) -sense. Remark 10.8 As we have shown, the Legendre tensor fields 𝑣 p˜ 𝑛(𝑖,𝑖) can be expressed in terms of the tensor fields 𝑣 p𝑛(𝑖,𝑖) . This is the reason why it also makes sense to introduce (vectorial) zonal rank-2 tensor kernel functions with respect to the { 𝑣 p˜ 𝑛(𝑖,𝑖) }-system by letting 𝑣 ˜ (𝑖,𝑖)

k

(𝜉, 𝜂) =

∞ ∑︁ 2𝑛 + 1 𝑣 ˜ (𝑖,𝑖)  ∧ k (𝑛) 𝑣 p˜ 𝑛(𝑖,𝑖) (𝜉, 𝜂), 4𝜋 𝑛=0

(10.35)

𝑖

(𝜉, 𝜂) ∈ Ω × Ω, where ( 𝑣 k˜ (𝑖,𝑖) ) ∧ (𝑛) 𝑣 p𝑛(𝑖,𝑖) (𝜉, 𝜂) =

∫

𝑣 ˜ (𝑖,𝑖)

k

Ω

(𝜉, 𝜁)

2𝑛 + 1 𝑣 (𝑖,𝑖) p˜ 𝑛 (𝜁, 𝜂) 𝑑𝜔(𝜂). 4𝜋

(10.36)

Clearly, all results being valid for the { 𝑣 p˜ 𝑛(𝑖,𝑖) }-system can be formulated in parallel.

10.3 Vector Zonal Kernel Functions in Vectorial Context

531

10.3 Vector Zonal Kernel Functions in Vectorial Context

Remembering the second vectorial variant of the addition theorem (Theorem 6.46) in vector theory, we now turn to the definition of vector zonal kernel functions. As already announced, they are created by a single application of the operators 𝑜 (𝑖) to scalar zonal kernel functions. Definition 10.9 Assume that 𝐾 (𝑖) ∈ C (0𝑖 ) [−1, 1], 𝑖 ∈ {1, 2, 3}, are scalar zonal kernel functions, respectively. A function 𝑘 (𝑖) : Ω × Ω → R3 (more precisely, 𝑣 𝑘 (𝑖) ), given by (𝑖) 𝑘 (𝑖) (𝜉, 𝜂) = 𝑜 (𝑖) (𝜉 · 𝜂), 𝜉, 𝜂 ∈ Ω, (10.37) 𝜉 𝐾 is called a (vectorial) zonal vector kernel function of type 𝑖 (with respect to {𝑝 𝑛(𝑖) }), and 3 ∑︁ 𝑘= 𝑘 (𝑖) (10.38) 𝑖=1

is called a (vectorial) zonal vector kernel function (with respect to {𝑝 𝑛 }). Definition 10.10 A zonal vector kernel function 𝑘 (𝑖) : Ω × Ω → R3 of type 𝑖 is called an l2(𝑖) (Ω)-zonal vector kernel function, if 𝑘 (𝑖) (𝜉, ·) is in l2 (Ω) for each 𝜉 ∈ Ω. Í Furthermore, 𝑘 = 3𝑖=1 𝑘 (𝑖) is called an l2 (Ω)-zonal vector kernel function. The following result can be directly derived from the identities (10.4), (10.5), and (10.6). Theorem 10.11 An l2(𝑖) (Ω)-zonal vector kernel function 𝑘 (𝑖) : Ω × Ω → R3 of type 𝑖 can be expressed as a Legendre series in the form 𝑘 (𝑖) (𝜉, ·) =

∞ ∑︁ 2𝑛 + 1 (𝑖) ∧ (𝑘 ) (𝑛) 𝑝 𝑛 (𝑖) (𝜉, ·), 4𝜋 𝑛=0

(10.39)

𝑖

where

  1/2 (𝑘 (𝑖) ) ∧ (𝑛) = 𝜇 𝑛(𝑖) (𝐾 (𝑖) ) ∧ (𝑛).

(10.40)

Proof Again, we first deal with the case 𝑖 = 1. As 𝑘 (1) (𝜉, ·) is in l2 (Ω), it is easy to see that ∫ ∫ (1) (1) 𝐾 (1) (𝜉 · 𝜂)𝐾 (1) (𝜉 · 𝜂) 𝑑𝜔(𝜂) = 𝐾 (1) (𝜉 · 𝜂)𝑂 (1) (𝜉 · 𝜂) 𝑑𝜔(𝜂) 𝜉 𝑜𝜉 𝐾 Ω Ω ∫ (1) (1) = 𝑜 (1) (𝜉 · 𝜂) · 𝑜 (1) (𝜉·) 𝑑𝜔(𝜂) 𝜉 𝐾 𝜉 𝐾 Ω

532

10 Vector Zonal Kernel Functions

∫ =

𝑘 (1) (𝜉 · 𝜂) · 𝑘 (1) (𝜉 · 𝜂) 𝑑𝜔(𝜂) < ∞.

Ω

Thus we have 𝐾 (1) (𝜉·) ∈ L2 (Ω), such that 𝐾 (1) (𝜉·) can be written as a Legendre series. This leads us to the identities 𝑘 (1) (𝜉, ·) = 𝑜 (1)

∞ ∑︁ 2𝑛 + 1 𝑛=0

=

∞  ∑︁

𝜇 𝑛(1)

𝑛=0

=

4𝜋

(𝐾 (1) ) ∧ (𝑛)𝑃𝑛 (𝜉, ·)

(10.41)

 1/2 2𝑛 + 1 (𝐾 (1) ) ∧ (𝑛) 𝑝 𝑛(1) (𝜉, ·) 4𝜋

∞ ∑︁ 2𝑛 + 1 𝑛=0

where

4𝜋

(𝑘 (1) ) ∧ (𝑛) 𝑝 (1) (𝜉, ·),

  1/2 (𝑘 (1) ) ∧ (𝑛) = 𝜇 𝑛(1) (𝐾 (1) ) ∧ (𝑛).

(10.42)

This is the required result for 𝑖 = 1. For the cases 𝑖 = 2, 3, we observe that 𝐾 (𝑖) (𝜉·) is assumed to be differentiable and, therefore, in L2 (Ω). The assertion of our theorem follows by the same arguments as shown for the case 𝑖 = 1. □ Theorem 10.12 A vector zonal kernel function 𝑘 (𝑖) : Ω × Ω → R3 of type 𝑖 is an l2(𝑖) (Ω)-vector zonal kernel function, if and only if ∞ ∑︁ 2𝑛 + 1  (𝑖) ∧  2 (𝑘 ) (𝑛) < ∞, 4𝜋 𝑛=0

(10.43)

  1/2 (𝑘 (𝑖) ) ∧ (𝑛) = 𝜇 𝑛(𝑖) (𝐾 (𝑖) ) ∧ (𝑛).

(10.44)

𝑖

where

Proof Observing that 𝑥 · 𝑦 = tr 𝑥 ⊗ 𝑦, 𝑥, 𝑦 ∈ R3 , we obtain (𝑘 (𝜉, ·), 𝑘 (𝜉, ·)) l2 (Ω) ! ∫ ∑︁ 3 ∑︁ ∞ 2𝑛 + 1 (𝑖) ∧ (𝑖) = (𝑘 ) (𝑛) 𝑝 𝑛 (𝜉, 𝜂) 4𝜋 Ω 𝑖=1 𝑛=0

(10.45)

𝑖

3 ∞ ©∑︁ ∑︁ 2𝑙 + 1 ( 𝑗 ) ∧ ª ( 𝑗) ·­ (𝑘 ) (𝑙) 𝑝 𝑙 (𝜉, 𝜂) ® 𝑑𝜔(𝜂) 4𝜋 « 𝑗=1 𝑙=0 𝑗 ¬ 3 ∑︁ 3 ∞ 2𝑛+1 ∑︁ ∑︁ ∑︁ ( 𝑗) (𝑖) (𝑘 (𝑖) ) ∧ (𝑛)(𝑘 ( 𝑗 ) ) ∧ (𝑛)𝑦 𝑛,𝑚 (𝜉) · 𝑦 𝑛,𝑚 (𝜉) 𝑖=1 𝑗=1 𝑛=max (0𝑖 ,0 𝑗 ) 𝑚=1

10.4 Convolutions Involving Vector Zonal Kernel Functions

=

3 ∑︁ 3 ∑︁

∞ ∑︁

(𝑘 (𝑖) ) ∧ (𝑛)(𝑘 ( 𝑗 ) ) ∧ (𝑛)

𝑖=1 𝑗=1 𝑛=max (0𝑖 ,0 𝑗 )

533

2𝑛 + 1 𝑣 (𝑖, 𝑗 ) tr ( p𝑛 (𝜉, 𝜉)). 4𝜋

The desired assertion follows by taking into account that (𝑖, 𝑗 )

tr( 𝑣 p𝑛

(𝜉, 𝜉)) = 𝛿𝑖 𝑗 .

(10.46) □

This yields the proof of Theorem 10.12.

Remark 10.13 From Lemma 6.63, we know that the vector Legendre kernels 𝑝˜ 𝑛(𝑖) can be written in terms of 𝑝 𝑛(𝑖) . Therefore, it also makes sense to introduce, in parallel, (vectorial) zonal vector kernel functions with respect to { 𝑝˜ 𝑛(𝑖) } by letting 𝑘˜ (𝑖) (𝜉, 𝜂) =

∞ ∑︁ 2𝑛 + 1 ˜ (𝑖) ∧ ( 𝑘 ) (𝑛) 𝑝˜ 𝑛(𝑖) (𝜉, 𝜂), 4𝜋 𝑛=0

(10.47)

𝑖

(𝜉, 𝜂) ∈ Ω × Ω, where ( 𝑘˜ (𝑖) ) ∧ (𝑛) 𝑝˜ 𝑛(𝑖) (𝜉, 𝜂) =

∫

2𝑛 + 1 (𝑖) 𝑘˜ (𝑖) (𝜉, 𝜁) 𝑝˜ 𝑛 (𝜂, 𝜁) 𝑑𝜔. 4𝜋 Ω

(10.48)

10.4 Convolutions Involving Vector Zonal Kernel Functions

Next, we introduce convolutions in the vectorial context. Definition 10.14 Let 𝑘 be an l2 (Ω)-zonal vector kernel function, 𝑓 ∈ l2 (Ω), 𝐹 ∈ L2 (Ω). Then 𝑘 ∗ 𝑓 defined by ∫ 𝑘 ∗ 𝑓 (𝜉) = 𝑘 (𝜂, 𝜉) · 𝑓 (𝜂) 𝑑𝜔(𝜂) (10.49) Ω

is called the convolution of 𝑘 against 𝑓 . Moreover, 𝑘 (𝑖) ★ 𝐹, 𝑖 = 1, 2, 3, given by ∫ (𝑖) 𝑘 ★ 𝐹 (𝜉) = 𝑘 (𝑖) (𝜉, 𝜂)𝐹 (𝜂) 𝑑𝜔(𝜂) (10.50) Ω

is called the convolution of 𝑘 (𝑖) against 𝐹. Note that we use different symbols for the convolutions to point out their different nature.

534

10 Vector Zonal Kernel Functions

For 𝑘, 𝑘ˆ being l2 (Ω)-zonal vector kernel functions we let 𝑘ˆ ★ (𝑘 ∗ 𝑓 ) =

3 ∑︁

  𝑘ˆ (𝑖) ★ 𝑘 (𝑖) ∗ 𝑓 .

(10.51)

𝑖=1

Note that   ∫ ∫ 𝑘ˆ (𝑖) ★ 𝑘 (𝑖) ∗ 𝑓 = 𝑘ˆ (𝑖) (·, 𝜉) ⊗ 𝑘 (𝑖) (𝜂, 𝜉) 𝑓 (𝜂) 𝑑𝜔(𝜂)𝑑𝜔(𝜉). Ω

(10.52)

Ω

This motivates the following rank-2 tensorial setting. Definition 10.15 Let 𝑘ˆ (𝑖) , 𝑘 (𝑖) be two l2(𝑖) (Ω)-zonal vector kernel functions. Then, we define the convolution of 𝑘ˆ (𝑖) against 𝑘 (𝑖) by 

∫  ˆ𝑘 (𝑖) ★ 𝑘 (𝑖) (𝜉, 𝜂) = 𝑘ˆ (𝑖) (𝜉, 𝜁) ⊗ 𝑘 (𝑖) (𝜂, 𝜁) 𝑑𝜔(𝜁).

(10.53)

Ω

Furthermore, 𝑘ˆ ★ 𝑘 is understood to be 𝑘ˆ ★ 𝑘 =

3 ∑︁

𝑘ˆ (𝑖) ★ 𝑘 (𝑖) .

(10.54)

𝑖=1

The following theorem can be verified easily by standard arguments. Theorem 10.16 Let 𝑘ˆ (𝑖) , 𝑘 (𝑖) be two l2(𝑖) (Ω)-zonal vector kernel functions. Then the convolution 𝑘ˆ (𝑖) ★ 𝑘 (𝑖) is an l2 (Ω)-zonal tensor kernel function, such that (𝑖)

𝑘ˆ (𝑖) ★ 𝑘 (𝑖) (𝜉, ·) =

∞ 2𝑛+1 ∑︁ ∑︁ 𝑛=0𝑖 𝑚=1

( 𝑘ˆ (𝑖) ) ∧ (𝑛)(𝑘 (𝑖) ) ∧ (𝑛)

2𝑛 + 1 𝑣 (𝑖,𝑖) p𝑛 (𝜉, ·). 4𝜋

(10.55)

From the expansion (10.55) in terms of Legendre rank-2 tensors, it can be derived that 𝑘ˆ (𝑖) ★ 𝑘 (𝑖) (𝜉, ·) is continuous on the sphere Ω for every 𝜉 ∈ Ω. The Parseval identity for vector spherical harmonics enables us to verify the following theorem (see M. Bayer et al. (1998), S. Beth (2000)). ˆ 𝑘 are l2 (Ω)-zonal vector Theorem 10.17 Let 𝑓 be of class l2 (Ω). Assume that 𝑘, 2 ˆ kernel functions, whereas k, k are l (Ω)-zonal rank-2 tensor kernel functions with ( 𝑣 kˆ (𝑖,𝑖) ) ∧ (𝑛) = ( 𝑘ˆ (𝑖) ) ∧ (𝑛),

(10.56)

10.5 Dirac Families of Zonal Vector Kernel Functions

535

and ( 𝑣 k (𝑖,𝑖) ) ∧ (𝑛) = (𝑘 (𝑖) ) ∧ (𝑛),

(10.57)

k ∗ 𝑣 k ∗ 𝑓 = 𝑘ˆ ★ (𝑘 ∗ 𝑓 ).

(10.58)

for all 𝑖 = 1, 2, 3. Then 𝑣ˆ

In other words, the different ways of forming convolutions (10.58) against vector fields either by tensor zonal kernel function or vector zonal kernels are equivalent. This is, in fact, a remarkable result. In consequence, due to Theorem 10.17, we can substitute rank-2 tensor zonal kernel functions by vector zonal kernel basis functions which is of importance not only for numerical purposes: once again, vector zonal functions require first order derivatives of the Legendre polynomials, whereas tensor zonal kernel functions make it necessary to compute the second order derivatives. Furthermore, the operational effort is reduced as we do not have to calculate tensor products when we turn over to vector zonal kernel functions. The structural price that must be paid in comparison to the tensor approach, however, is a bilinear framework for the vectorial kernels involved in the convolutions.

10.5 Dirac Families of Zonal Vector Kernel Functions

Starting from a Dirac family {Φ𝜌 } 𝜌∈ (0,∞) of scalar zonal kernel functions, we are able to construct a Dirac family of zonal rank-2 tensor kernel functions {𝝋𝜌 } 𝜌∈ (0,∞) as follows (note that we restrict ourselves to the system of dual operators 𝑜 (𝑖) , 𝑂 (𝑖) , 𝑖 ∈ {1, 2, 3}): 3 ∑︁ 𝑣 (𝑖,𝑖) 𝝋𝜌 (𝜉, 𝜂) = 𝝋𝜌 (𝜉, 𝜂) (10.59) 𝑖=1

with 𝑣

𝝋𝜌(𝑖,𝑖) (𝜉, 𝜂) =

∞ ∑︁

(Φ𝜌 ) ∧ (𝑛)

𝑛=𝑂𝑖

2𝑛+1 ∑︁

(𝑖) (𝑖) 𝑦 𝑛, 𝑗 (𝜉) ⊗ 𝑦 𝑛, 𝑗 (𝜂),

(10.60)

𝑗=1

𝜉, 𝜂 ∈ Ω. Correspondingly, a Dirac family {𝜑𝜌 } 𝜌∈ (0,∞) of vector zonal kernel functions 𝜑𝜌 reads as follows: 𝜑𝜌 (𝜉, 𝜂) =

3 ∑︁

𝜑𝜌(𝑖) (𝜉, 𝜂)

(10.61)

𝑖=1

with 𝜑𝜌(𝑖) (𝜉, 𝜂) =

∞ ∑︁ 𝑛=𝑂𝑖

(𝑖) (Φ𝜌 ) ∧ (𝑛)𝑌𝑛, 𝑗 (𝜉)𝑦 𝑛, 𝑗 (𝜂),

(10.62)

536

10 Vector Zonal Kernel Functions

𝜉, 𝜂 ∈ Ω. As an immediate consequence, we obtain the following linear and bilinear approach for rank-2 tensor Dirac families and the bilinear approach for vector Dirac families. Theorem 10.18 Let {Φ𝜌 } 𝜌∈ (0,∞) be a scalar Dirac family. Then lim ∥ 𝑓 − 𝝋𝜌 ∗ 𝑓 ∥ l2 (Ω) = 0,

(10.63)

lim ∥ 𝑓 − 𝝋𝜌 ∗ 𝝋𝜌 ∗ 𝑓 ∥ l2 (Ω) = 0

(10.64)

lim ∥ 𝑓 − 𝜑𝜌 ★ 𝜑𝜌 ∗ 𝑓 ∥ l2 (Ω) = 0

(10.65)

𝜌→0

𝜌→0

and 𝜌→0

for all 𝑓 ∈ l2 (Ω). This means that we have extended the notion of an approximate identity in a canonical way to spherical vector field thereby using two different, but (in bilinear sense) equivalent approaches to Dirac families. Seen from the point of spherical functions on the sphere, we should have a closer look to the Dirac families involved in the approximation. It is clear that 𝑣

(1) 𝝋𝜌(1,1) (𝜉, 𝜂) = 𝑜 (1) 𝜉 𝑜 𝜂 Φ𝜌 (𝜉 · 𝜂)

and 𝑣

(𝑖) 𝝋𝜌(𝑖,𝑖) (𝜉, 𝜂) = −𝑜 (𝑖) 𝜉 𝑜𝜂

∫

𝐺 (Δ∗ ; 𝜉 · 𝜁)Φ𝜌 (𝜁 · 𝜂) 𝑑𝜔(𝜉),

(10.66)

(10.67)

Ω

𝜉, 𝜂 ∈ Ω, 𝑖 = 2, 3. Equivalently, we have (𝑖) 𝝋𝜌(𝑖,𝑖) (𝜉, 𝜂) = 𝑜 (𝑖) 𝜉 𝑜𝜂

∞ ∑︁ 2𝑛 + 1 𝑛=1

4𝜋

1 (Φ𝜌 ) ∧ (𝑛)𝑃𝑛 (𝜉 · 𝜂), 𝑛(𝑛 + 1)

(10.68)

𝜉, 𝜂 ∈ Ω, 𝑖 = 2, 3. In an analogous way, we find 𝜑𝜌(𝑖) (𝜉, 𝜂) = 𝑜 (𝑖) 𝜂

∞ ∑︁ 2𝑛 + 1 𝑛=1

4𝜋

1 √︁

𝑛(𝑛 + 1)

(Φ𝜌 ) ∧ (𝑛)𝑃𝑛 (𝜉 · 𝜂),

(10.69)

𝑖 = 2, 3. Remark 10.19 Our approach has shown that an isotropic operator mapping a vector field onto a vector field refuses the representation in terms of a vector zonal kernel function. In that context, in fact, zonal tensor functions have to be taken into account.

10.6 Bibliographical Notes

537

Zonal tensor functions, indeed, fall back upon an addition theorem involving the tensor product of vector spherical harmonics. Although they do not allow us to describe isotropic vector fields, zonal rank-2 tensor fields are of advantage for the approximation of vector fields in form of splines (W. Freeden, T. Gervens (1991)), W. Freeden et al. (1994) and wavelets (W. Freeden et al. (1998)) (by matrix-vector multiplications with constant vectors). The numerical disadvantage of a representation of vector fields based on tensor zonal kernel fields is easily understood. We have to deal with matrix-vector multiplications. In comparison with vector zonal functions, a further drawback comes up. While vector zonal functions require only the first derivative of a scalar radial zonal function, zonal tensor functions even need second derivatives. This makes them more difficult to handle in vector field modeling, particularly when the scalar zonal function is not known elementary in a closed representation, but only as series expansion in terms of Legendre polynomials. Nevertheless, tensor zonal functions are an important tool in the characterization of vector fields (comparable to the scalar case). Of course, tensor zonal functions are natural structures to observe rotational symmetry within a tensor framework, and in this case, second derivatives for the occurring Legendre polynomials are canonical.

10.6 Bibliographical Notes

Zonal kernel functions in the vector context have been introduced in a double sense in twofold way (i) as vector fields generated by applying the operators 𝑜 (𝑖) , 𝑜˜ (𝑖) , respectively, on scalar zonal kernel functions (see M. Bayer et al. (1998), S. Beth (2000), H. Nutz (2002), C. Mayer (2003), W. Freeden, T. Maier (2003)) (ii) as tensor fields generated by a double application of the operators 𝑜 (𝑖) , 𝑜˜ (𝑖) , respectively, on scalar zonal functions (see W. Freeden et al. (1998), S. Beth (2000), H. Nutz (2002), M.K. Abeyratne (2003)). In the first case, we are led to zonal vector kernel functions, whereas the second case leads to zonal tensor kernel functions (to be used within the vector context). Both types of isotropic functions are basic tools for approximation techniques like spherical splines and wavelets (see, e.g., G. Wahba (1982), W. Freeden, T. Gervens (1991), W. Freeden, U. Windheuser (1996), U. Windheuser (1995), W. Freeden et al. (1998), W. Freeden, M. Schreiner (1997), H. Nutz (2002), M.J. Fengler (2005), W. Freeden, M. Schreiner (2006a), S. Gramsch (2006), T. Fehlinger et al. (2008), W. Freeden et al. (2018a)).

Chapter 11

Tensorial Zonal Kernel Functions

Next, we come to zonal kernel functions in the tensor context. In analogy to the vectorial case, we are able to derive two variants based on the known addition theorems (Theorem 7.21 and Theorem 7.34). In more detail, we obtain zonal kernel functions in the tensor context by applying the operators o (𝑖,𝑘 ) , 𝑖, 𝑘 ∈ {1, 2, 3}, once and twice to scalar zonal kernel functions (see Table 11.1). Table 11.1 Overview on (tensorial) zonal rank-4/rank-2 tensor kernel functions in relation to Legendre kernel functions.

Zonal kernel (generating system)

Linear approach

Bilinear approach

scalar field ({𝑌𝑛, 𝑗 }system)

scalar zonal kernel function ( { 𝑃𝑛 }-system)

tensor field (𝑖,𝑘) ({y𝑛, 𝑗 }system)

(tensorial) zonal rank-4 tensor kernel function ({P𝑛(𝑖,𝑘,𝑖,𝑘) }-system)

K (𝑖,𝑘)

(tensorial) zonal rank-2 tensor kernel function ({ 𝑡 p𝑛(𝑖,𝑘) }-system)

𝑡 k (𝑖,𝑘)

tensor field (𝑖,𝑘) ({ y˜ 𝑛, 𝑗 }system)

(tensorial) zonal rank-4 tensor kernel function ({ P˜ 𝑛(𝑖,𝑘,𝑖,𝑘) }-system)

˜ (𝑖,𝑘) K

(tensorial) zonal rank-2 tensor kernel function ({ 𝑡 p˜ 𝑛(𝑖,𝑘) }-system)

𝑡k ˜ (𝑖,𝑘)

𝐾

© The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2022 W. Freeden and M. Schreiner, Spherical Functions of Mathematical Geosciences, Geosystems Mathematics, https://doi.org/10.1007/978-3-662-65692-1_11

539

540

11 Tensorial Zonal Kernel Functions

11.1 Preparatory Material

Our work is based on the following already known lemma, which demonstrates that the operators o (𝑖,𝑘 ) are easily applicable to scalar zonal kernel functions. Lemma 11.1 Let 𝐾 be of class C (0𝑖𝑘 ) [−1, 1]. Suppose that 𝜂 ∈ Ω is fixed. Then, for all 𝜉 ∈ Ω, o (1,1) 𝐾 (𝜉 · 𝜂) = 𝐾 (𝜉 · 𝜂)𝜉 ⊗ 𝜂, 𝜉

(11.1)

o (1,2) 𝐾 (𝜉 𝜉 o (1,3) 𝐾 (𝜉 𝜉 o (2,1) 𝐾 (𝜉 𝜉 o (2,2) 𝐾 (𝜉 𝜉 (2,3) o 𝜉 𝐾 (𝜉

· 𝜂) = 𝐾 (𝜉 · 𝜂)𝜉 ⊗ (𝜂 − (𝜉 · 𝜂)𝜉),

(11.2)

· 𝜂) = 𝐾 ′ (𝜉 · 𝜂)𝜉 ⊗ (𝜉 ∧ 𝜂),

(11.3)

· 𝜂) = 𝐾 ′ (𝜉 · 𝜂)(𝜂 − (𝜉 · 𝜂)𝜉) ⊗ 𝜉,

(11.4)

· 𝜂) = 𝐾 (𝜉 · 𝜂)itan (𝜉),

(11.5)

· 𝜂) = 𝐾 ′′ (𝜉 · 𝜂) ((𝜂 − (𝜉 · 𝜂)𝜉) ⊗ (𝜂 − (𝜉 · 𝜂)𝜉)

(11.6)

′

− (𝜉 ∧ 𝜂) ⊗ (𝜉 ∧ 𝜂)) , o (3,1) 𝐾 (𝜉 𝜉 o (3,2) 𝐾 (𝜉 𝜉

· 𝜂) = 𝐾 ′ (𝜉 · 𝜂)(𝜉 ∧ 𝜂) ⊗ 𝜉,

(11.7)

· 𝜂) = 𝐾 ′′ (𝜉 · 𝜂) ((𝜂 − (𝜉 · 𝜂)𝜉) ⊗ (𝜉 ∧ 𝜂)

(11.8)

+ (𝜉 ∧ 𝜂) ⊗ (𝜂 − (𝜉 · 𝜂)𝜉)) , o (3,3) 𝐾 (𝜉 · 𝜂) = 𝐾 (𝜉 · 𝜂)jtan (𝜉). 𝜉

(11.9)

For simplicity, we omit the explicit representations of a double application of the operators 𝑜 (𝑖,𝑘 ) to the scalar kernel 𝐾.

11.2 Tensor Zonal Kernel Functions of Rank Four in Tensorial Context

First, we are interested in defining (tensorial) rank-4 tensor zonal kernel functions by a double application of operators o (𝑖,𝑘 ) on (sufficiently often differentiable) scalar zonal kernel functions.

11.2 Tensor Zonal Kernel Functions of Rank Four in Tensorial Context

541

Definition 11.2 Assume that 𝐾 (𝑖,𝑘 ) ∈ C (2·0𝑖𝑘 ) [−1, 1], 𝑖, 𝑘 ∈ {1, 2, 3}, are scalar zonal kernel functions. A function K (𝑖,𝑘 ) : Ω × Ω → R3 ⊗ R3 ⊗ R3 ⊗ R3 , (more precisely, 𝑡 K (𝑖,𝑘 ) ) given by ) (𝑖,𝑘 ) (𝑖,𝑘 ) K (𝑖,𝑘 ) (𝜉, 𝜂) = o (𝑖,𝑘 o𝜂 𝐾 (𝜉 · 𝜂), 𝜉

𝜉, 𝜂 ∈ Ω,

(11.10)

is called a (tensorial) zonal rank-4 tensor kernel function of type (𝑖, 𝑘) (with respect to {P𝑛(𝑖,𝑘,𝑖,𝑘 ) }). Furthermore, we let K=

3 ∑︁ 3 ∑︁

K (𝑖,𝑘 )

(11.11)

𝑖=1 𝑘=1

K is called a (tensorial) zonal rank-4 tensor kernel function (with respect to {P𝑛 }). In close analogy to the vector case, we introduce the following setting. Definition 11.3 A (tensorial) zonal rank-4 tensor kernel function of type (𝑖, 𝑘), K (𝑖,𝑘 ) : Ω × Ω → R3 ⊗ R3 ⊗ R3 ⊗ R3 , 𝑖, 𝑘 ∈ {1, 2, 3}, is called an l2(𝑖,𝑘 ) (Ω)(tensorial) zonal rank-4 tensor kernel function, if K (𝑖,𝑘 ) (𝜉, ·) is square-integrable on Ω for every 𝜉 ∈ Ω. Furthermore, K=

3 ∑︁ 3 ∑︁

K (𝑖,𝑘 )

(11.12)

𝑖=1 𝑘=1

is called an l2 (Ω)-tensorial zonal rank-4 tensor kernel function, if K (𝑖,𝑘 ) are l2(𝑖,𝑘 ) (Ω)-tensorial zonal rank-4 tensor kernel functions. Clearly, (tensorial) zonal rank-4 tensor kernel functions can be expanded in terms of the Legendre functions. For that purpose, we need the corresponding tensorial variant of the Funk-Hecke formula. As in the vector case, for the convenience of the reader, it will be recapitulated briefly. Theorem 11.4 (Funke-Hecke Formula in Tensorial Context) Let 𝜂 ∈ Ω be fixed. Assume that h(·, 𝜂) ∈ c (2) (Ω) satisfies h(t𝜉, 𝜂) = th(𝜉, 𝜂)t𝑇 for all orthogonal transformations t ∈ 𝑆𝑂 𝜂 (3) and all 𝜉 ∈ Ω. Then, for all 𝜁 ∈ Ω and for all 𝑖, 𝑘 ∈ {1, 2, 3}, ∫   −1  ∧ P𝑛(𝑖,𝑘,𝑖,𝑘 ) (𝜁, 𝜉)h(𝜁, 𝜂) 𝑑𝜔(𝜉) = 𝜇 𝑛(𝑖,𝑘 ) 𝑂 (𝑖,𝑘 ) h (𝑛)o 𝜁(𝑖,𝑘 ) 𝑃𝑛 (𝜁 · 𝜂), Ω

(11.13)

where  𝑂

(𝑖,𝑘 )

h

∧

∫

1

(𝑛) = 2𝜋

𝐻𝑖𝑘 (𝑡)𝑃𝑛 (𝑡) 𝑑𝑡, −1

(11.14)

542

and

11 Tensorial Zonal Kernel Functions

) 𝐻𝑖𝑘 (𝜉 · 𝜂) = 𝑂 (𝑖,𝑘 h(𝜉, 𝜂). 𝜉

(11.15)

In parallel to the vector case, we formulate the following result. Theorem 11.5 An l2(𝑖,𝑘 ) (Ω)-tensorial zonal rank-4 tensor kernel function K (𝑖,𝑘 ) can be represented as a Legendre series in the form ∞ ∑︁ 2𝑛 + 1 (𝑖,𝑘 ) ∧ (K ) (𝑛)P𝑛(𝑖,𝑘,𝑖,𝑘 ) (𝜉, ·), 4𝜋 𝑛=0

K (𝑖,𝑘 ) (𝜉, ·) =

(11.16)

𝑖𝑘

where

(K (𝑖,𝑘 ) ) ∧ (𝑛) = 𝜇 𝑛(𝑖,𝑘 ) (𝐾 (𝑖,𝑘 ) ) ∧ (𝑛).

(11.17)

The proof of Theorem 11.5 follows in close analogy to its vectorial counterpart (Theorem 8.4). Thus, it is omitted here. Theorem 11.6 A (tensorial) zonal rank-4 tensor kernel function of type (𝑖, 𝑘), K (𝑖,𝑘 ) : Ω × Ω → R3 ⊗ R3 ⊗ R3 ⊗ R3 is an l2(𝑖,𝑘 ) (Ω) (tensorial) zonal rank-4 tensor kernel function, if and only if ∞ ∑︁ 2𝑛 + 1  (𝑖,𝑘 ) )∧  2 (K (𝑛) < ∞, 4𝜋 𝑛=0

(11.18)

(K (𝑖,𝑘 ) ) ∧ (𝑛) = 𝜇 𝑛(𝑖,𝑘 ) (𝐾 (𝑖,𝑘 ) ) ∧ (𝑛).

(11.19)

𝑖𝑘

where

Theorem 11.6 follows directly from the identity 2𝑛+1 ∑︁ 𝑚=1

(𝑖,𝑘 ) |y𝑛,𝑚 (𝜉)| 2 =

2𝑛 + 1 . 4𝜋

(11.20)

11.3 Convolutions Involving Zonal Tensor Kernel Functions

We are now going to introduce convolutions in the tensor context. Definition 11.7 Let H, K be l2 (Ω)-tensorial zonal rank-4 tensor kernel functions. Suppose that f is of class l2 (Ω). Then K ∗ f defined by

11.3 Convolutions Involving Zonal Tensor Kernel Functions

543

∫ (K ∗ f)(𝜉) =

K(𝜉, 𝜂)f (𝜂) 𝑑𝜔(𝜂),

(11.21)

Ω

𝜉 ∈ Ω, is called the convolution of K against f. Furthermore, H ∗ K defined by ∫ (H ∗ K)(𝜉, 𝜂) = H(𝜉, 𝜁)K(𝜁, 𝜂) 𝑑𝜔(𝜁), (11.22) Ω

𝜉, 𝜂 ∈ Ω, is called the convolution of H against K. Different variants of convolutions are definable. From the addition theorem, the orthogonality of the tensor spherical harmonics, together with the identity f (g · h) = (f ⊗ g)h, we are able to show that K

(𝑖,𝑘 )

∗f =

∫ ∑︁ ∞

(K (𝑖,𝑘 ) ) ∧ (𝑛)

Ω 𝑛=0 𝑖𝑘

×

2𝑛+1 ∑︁

(𝑖,𝑘 ) (𝑖,𝑘 ) y𝑛,𝑚 (𝜉) ⊗ y𝑛,𝑚 (𝜂)

𝑚=1

𝑝+1 ∞ 2∑︁ ∑︁

) (f (𝑖,𝑘 ) ) ∧ ( 𝑝, 𝑞)y (𝑖,𝑘 𝑝,𝑞 (𝜂) 𝑑𝜔(𝜂)

𝑝=0𝑖𝑘 𝑞=1

=

∞ ∑︁

(K (𝑖,𝑘 ) ) ∧ (𝑛)

𝑛=0𝑖𝑘

2𝑛+1 ∑︁

(𝑖,𝑘 ) (f (𝑖,𝑘 ) ) ∧ (𝑛, 𝑚) y𝑛,𝑚 .

(11.23)

𝑚=1

Further on, because of (F ⊗ G)(H ⊗ I) = (G · H)F ⊗ I, we find H (𝑖,𝑘 ) ∗ K (𝑖,𝑘 ) =

∞ ∑︁ 2𝑛 + 1 (𝑖,𝑘 ) ∧ (H ) (𝑛)(K (𝑖,𝑘 ) ) ∧ (𝑛)P𝑛(𝑖,𝑘,𝑖,𝑘 ) . 4𝜋 𝑛=0

(11.24)

𝑖𝑘

Using the Legendre series expansion (11.24), we easily see that the convolution H (𝑖,𝑘 ) ∗ K(𝜉, ·) is continuous on Ω for each point 𝜉 ∈ Ω. Í Lemma 11.8 Let K = 3𝑖,𝑘=1 K (𝑖,𝑘 ) be an l2 (Ω)-tensorial zonal rank-4 tensor kernel function. Then K (𝑖,𝑘 ) is expressible in the form K (𝑖,𝑘 ) (𝜉, 𝜂) =

∞ ∑︁ 2𝑛 + 1 (𝑖,𝑘,𝑖,𝑘 ) P𝑛 (𝜉, ·) ∗ K(·, 𝜂), 4𝜋 𝑛=0

(11.25)

𝑖𝑘

𝜉, 𝜂 ∈ Ω. Finally, we are able to deduce from (7.330) that every f ∈ l2 (Ω) can be represented in the form 3 ∞ ∑︁ ∑︁ 2𝑛 + 1 (𝑖,𝑘,𝑖,𝑘 ) f= P𝑛 ∗f (11.26) 4𝜋 𝑖,𝑘=1 𝑛=0 𝑖𝑘

544

11 Tensorial Zonal Kernel Functions

(in ∥ · ∥ l2 (Ω) -sense). Remark 11.9 The (tensorial) Legendre rank-4 tensor kernel functions P˜ 𝑛(𝑖,𝑘,𝑖,𝑘 ) are expressible in terms of P𝑛(𝑖,𝑘,𝑖,𝑘 ) . Therefore, (tensorial) zonal rank-4 tensor kernel functions with respect to the {P˜ 𝑛(𝑖,𝑘,𝑖,𝑘 ) }-system read as follows: ˜ (𝑖,𝑘 ) (𝜉, 𝜂) = K

∞ ∑︁ 2𝑛 + 1  ˜ (𝑖,𝑘 )  ∧ K (𝑛) P˜ 𝑛(𝑖,𝑘,𝑖,𝑘 ) , 4𝜋 𝑛=0

(11.27)

𝑖𝑘

(𝜉, 𝜂) ∈ Ω × Ω, where ˜ (𝑖,𝑘 ) ) ∧ (𝑛) P˜ 𝑛(𝑖,𝑘,𝑖,𝑘 ) (𝜉, 𝜂) = (K

∫

˜ (𝑖,𝑘 ) (𝜉, 𝜁) 2𝑛 + 1 P˜ 𝑛(𝑖,𝑘,𝑖,𝑘 ) (𝜉, 𝜂) 𝑑𝜔(𝜁). K 4𝜋 Ω (11.28)

11.4 Tensor Zonal Kernel Functions of Rank Two in Tensorial Context

Tensorial rank-2 tensor zonal kernel functions are defined by a single application of the operators o (𝑖,𝑘 ) to (sufficiently smooth) scalar zonal kernel functions. Seen from operational point of view, they are of importance for two reasons: First, the computational effort is reduced because we do not have to calculate the tensor product of a tensor of rank four and a tensor of rank two. Furthermore, according to Lemma 11.1, we only need the second order derivatives of the generating scalar zonal kernel functions, whereas in the first approach involving tensorial rank-4 tensor zonal kernel functions, we need fourth order derivatives. Definition 11.10 Assume that 𝐾 (𝑖,𝑘 ) : [−1, 1] → R are (sufficiently often differentiable) scalar zonal kernel functions, i.e., 𝐾 (𝑖,𝑘 ) ∈ C (0𝑖𝑘 ) [−1, 1], 𝑖, 𝑘 ∈ {1, 2, 3}. ) A function 𝑡 k (𝑖,𝑘 : Ω × Ω → R3 ⊗ R3 , given by 𝜉 𝑡 (𝑖,𝑘 ) k 𝜉 (𝜉, 𝜂)

) (𝑖,𝑘 ) = o (𝑖,𝑘 𝐾 (𝜉 · 𝜂), 𝜉

𝜉, 𝜂 ∈ Ω,

(11.29)

is called a (tensorial) zonal rank-2 tensor kernel function of type (𝑖, 𝑘) (with respect to {𝑡 p𝑛(𝑖,𝑘 ) }), while 3 ∑︁ 3 ∑︁ 𝑡 (𝑖,𝑘 ) 𝑡 k= k (11.30) 𝑖=1 𝑘=1

is called a (tensorial) zonal rank-2 tensor kernel function (with respect to {𝑡 p𝑛(𝑖,𝑘 ) }).

11.4 Tensor Zonal Kernel Functions of Rank Two in Tensorial Context

545

In analogy to our above considerations, we introduce the following definition. Definition 11.11 A (tensorial) zonal rank-2 tensor kernel function of kind (𝑖, 𝑘), k (𝑖,𝑘 ) : Ω × Ω → R3 ⊗ R3 is called an l2(𝑖,𝑘 ) (Ω)-tensorial zonal rank-2 tensor kernel function, if 𝑡 k (𝑖,𝑘 ) (𝜉, ·) is in l2 (Ω) for each 𝜉 ∈ Ω. Furthermore, 𝑡 k = Í3 Í3 𝑡 (𝑖,𝑘 ) is called an l2 (Ω)-(tensorial) zonal rank-2 tensor kernel function, 𝑖=1 𝑘=1 k (𝑖,𝑘 ) 2 if k are l (𝑖,𝑘 ) (Ω)-tensorial zonal rank-2 tensor kernel functions. In accordance with our approach, we are immediately able to prove the following property of an l2(𝑖,𝑘 ) (Ω)-(tensorial) zonal rank-2 tensor kernel function. Theorem 11.12 An l2(𝑖,𝑘 ) (Ω)-zonal rank-2 tensor zonal kernel function 𝑡 k (𝑖,𝑘 ) can be represented as a Legendre series in the form 𝑡 (𝑖,𝑘 )

k

(𝜉, ·) =

∞ ∑︁ 2𝑛 + 1 𝑡 (𝑖,𝑘 ) ∧ 𝑡 (𝑖,𝑘 ) (k ) (𝑛) p𝑛 (𝜉, ·), 4𝜋 𝑛=0

(11.31)

𝑖𝑘

where

  1/2 ( 𝑡 k (𝑖,𝑘 ) ) ∧ (𝑛) = 𝜇 𝑛(𝑖,𝑘 ) (𝐾 (𝑖,𝑘 ) ) ∧ (𝑛).

(11.32)

The proof of Theorem 11.12 parallels that one known from the vectorial case, hence, we do not formulate it. Theorem 11.13 A (tensorial) zonal rank-2 tensor function of type (𝑖, 𝑘) 𝑡 k (𝑖,𝑘 ) : Ω × Ω → R3 ⊗ R3 is an l2(𝑖,𝑘 ) (Ω)-zonal rank-2 tensor kernel function, if and only if ∞ ∑︁ 2𝑛 + 1  𝑡 (𝑖,𝑘 ) ∧  2 (k ) (𝑛) < ∞, 4𝜋 𝑛=0

(11.33)

  1/2 ( 𝑡 k (𝑖,𝑘 ) ) ∧ (𝑛) = 𝜇 𝑛(𝑖,𝑘 ) (𝐾 (𝑖,𝑘 ) ) ∧ (𝑛).

(11.34)

𝑖𝑘

where

Remark 11.14 From Lemma 7.43, we know that the (tensorial) Legendre rank-2 tensor kernel functions 𝑡 p˜ 𝑛(𝑖,𝑘 ) are expressible in terms of p𝑛(𝑖,𝑘 ) . Therefore, we are able to introduce zonal rank-2 tensor kernel functions with respect to the {𝑡 p˜ 𝑛(𝑖,𝑘 ) }system ∞ ∑︁ 2𝑛 + 1 𝑡 ˜ (𝑖,𝑘 )  ∧ 𝑡 ˜ (𝑖,𝑘 ) k (𝜉, 𝜂) = k (𝑛) 𝑡 p˜ 𝑛(𝑖,𝑘 ) (𝜉, 𝜂), (11.35) 4𝜋 𝑛=0 𝑖

(𝜉, 𝜂) ∈ Ω × Ω, where

546

11 Tensorial Zonal Kernel Functions

( 𝑡 k˜ (𝑖,𝑘 ) ) ∧ (𝑛) 𝑡 p𝑛(𝑖,𝑘 ) (𝜉, 𝜂) =

∫

𝑡 ˜ (𝑖,𝑘 )

k

(𝜉, 𝜁)

Ω

2𝑛 + 1 𝑡 (𝑖,𝑘 ) p˜ 𝑛 (𝜁, 𝜂) 𝑑𝜔(𝜂). 4𝜋

(11.36)

Next, we want to define the convolution in a (tensorial) rank-2 tensor context . Definition 11.15 Let 𝑡 k be a l2 (Ω)-tensorial zonal rank-2 tensor kernel functions. Furthermore, assume that f ∈ l2 (Ω), 𝐹 ∈ L2 (Ω). Then 𝑡 k ∗ f defined by ∫ 𝑡 𝑡 k ∗ f (𝜉) = k(𝜂, 𝜉) · f (𝜂) 𝑑𝜔(𝜂) (11.37) Ω

is called the convolution of 𝑡 k against f. Moreover, 𝑡 k (𝑖,𝑘 ) ★ 𝐹, 𝑖, 𝑘 ∈ {1, 2, 3}, given by ∫ 𝑡 (𝑖,𝑘 )

k

𝑡 (𝑖,𝑘 )

★ 𝐹 (𝜉) =

k

(𝜉, 𝜂)𝐹 (𝜂) 𝑑𝜔(𝜂)

(11.38)

  ★ 𝑡 k (𝑖,𝑘 ) ∗ f .

(11.39)

Ω

is called the convolution of k (𝑖,𝑘 ) against 𝐹. For brevity, we write 𝑡

kˆ ★ ( 𝑡 k ∗ f) =

3 ∑︁ 3 ∑︁

𝑡 ˆ (𝑖,𝑘 )

k

𝑖=1 𝑘=1

Since it is not difficult to see that   ∫ ∫ 𝑡 (𝑖,𝑘 ) 𝑡 (𝑖,𝑘 ) h ★ 𝑡 k (𝑖,𝑘 ) ∗ f = h (·, 𝜉) ⊗ 𝑡 k (𝑖,𝑘 ) (𝜂, 𝜉)f (𝜂)𝑑𝜔(𝜂) 𝑑𝜔(𝜉), (11.40) Ω

Ω

we are finally led to the following setting. Definition 11.16 Let 𝑡 h (𝑖,𝑘 ) , 𝑡 k (𝑖,𝑘 ) be two l2(𝑖,𝑘 ) (Ω)-(tensorial) zonal rank-2 tensor kernel functions. Then, we define the convolution of 𝑡 h (𝑖,𝑘 ) against 𝑡 k (𝑖,𝑘 ) by ∫ 𝑡 (𝑖,𝑘 ) 𝑡 (𝑖,𝑘 ) 𝑡 (𝑖,𝑘 ) h ★ k (𝜉, 𝜂) = h (𝜉, 𝜁) ⊗ 𝑡 k (𝑖,𝑘 ) (𝜂, 𝜁) 𝑑𝜔(𝜁). (11.41) Ω

Moreover, 𝑡 h ★ 𝑡 k is given by 𝑡

h ★ 𝑡k =

3 ∑︁ 3 ∑︁

𝑡 (𝑖,𝑘 )

h

★ 𝑡 k (𝑖,𝑘 ) .

(11.42)

𝑖=1 𝑘=1

Collecting our material on the rank-2 tensor context, we are led to formulate the following result.

11.4 Tensor Zonal Kernel Functions of Rank Two in Tensorial Context

547

Theorem 11.17 Let 𝑡 h (𝑖,𝑘 ) , 𝑡 k (𝑖,𝑘 ) be two l2(𝑖,𝑘 ) (Ω)-(tensorial) zonal rank-2 tensor kernel functions. Then the convolution 𝑡 h (𝑖,𝑘 ) ★ 𝑡 k (𝑖,𝑘 ) is an l2(𝑖) (Ω)-tensorial zonal rank-4 tensor kernel function, and we have 𝑡 (𝑖,𝑘 )

h

★ 𝑡 k (𝑖,𝑘 ) (𝜉, ·) =

∞ 2𝑛+1 ∑︁ ∑︁

( 𝑡 h (𝑖,𝑘 ) ) ∧ (𝑛)( 𝑡 k (𝑖,𝑘 ) ) ∧ (𝑛)

𝑛=0𝑖𝑘 𝑚=1

2𝑛 + 1 (𝑖,𝑘,𝑖,𝑘 ) P𝑛 (𝜉, ·). 4𝜋 (11.43)

By observing the property (11.43) we are able to deduce that, for every 𝜉 ∈ Ω, h (𝑖,𝑘 ) ★ k (𝑖,𝑘 ) (𝜉, ·) is continuous on Ω. Theorem 11.18 Let f of class l2 (Ω). Suppose that 𝑡 h and 𝑡 k are l2(𝑖,𝑘 ) (Ω)-zonal rank˜ K are l2 (Ω)-zonal rank-4 tensor kernel 2 tensor kernel functions, whereas K, (𝑖,𝑘 ) functions satisfying (H (𝑖,𝑘 ) ) ∧ (𝑛) = ( 𝑡 h (𝑖,𝑘 ) ) ∧ (𝑛), (11.44) and (K (𝑖,𝑘 ) ) ∧ (𝑛) = (k (𝑖,𝑘 ) ) ∧ (𝑛),

(11.45)

for all 𝑖, 𝑘 ∈ {1, 2, 3}, 𝑛 ≥ 0𝑖𝑘 . Then H ∗ K ∗ f = 𝑡 h ★ 𝑡 k ∗ f.

(11.46)

Proof Observing the Legendre series expansion of f and of the zonal kernel functions, the addition theorem, and the orthogonality of the spherical harmonics, we obtain for the left hand side H∗K∗f =

(11.47)

3 ∑︁

∞ ∑︁

(H (𝑖,𝑘 ) ) ∧ (𝑛)(K (𝑖,𝑘 ) ) ∧ (𝑛)

𝑖,𝑘=1 𝑛=0𝑖𝑘

2𝑛+1 ∑︁

(𝑖,𝑘 ) (f (𝑖,𝑘 ) ) ∧ (𝑛, 𝑚) y𝑛,𝑚 .

𝑚=1

For the right hand side, we find 𝑡

h ★ 𝑡k ∗ f =

3 ∑︁

𝑡 (𝑖,𝑘 )

h

★ 𝑡 k (𝑖,𝑘 ) ∗ f

(11.48)

𝑖,𝑘=1

=

3 ∫ ∫ ∑︁ 𝑖,𝑘=1

=

Ω

3 ∞ ∑︁ ∑︁ 𝑖,𝑘=1 𝑛=0𝑖𝑘

𝑡 (𝑖,𝑘 )

h

(·, 𝜁) 𝑡 k (𝑖,𝑘 ) (𝜂, 𝜁) · f (𝜂) 𝑑𝜔(𝜂) 𝑑𝜔(𝜁)

Ω 𝑡 (𝑖,𝑘 )∧

h

(𝑛) 𝑡 k (𝑖,𝑘 )∧ (𝑛)

2𝑛+1 ∑︁

(𝑖,𝑘 ) f (𝑖,𝑘 )∧ (𝑛, 𝑚) y𝑛,𝑚 .

𝑚=1

In connection with (11.44) and (11.45), we obtain the desired result.

□

548

11 Tensorial Zonal Kernel Functions

11.5 Dirac Families of Zonal Tensor Kernel Functions

Starting once more from a scalar Dirac family {Φ𝜌 } 𝜌∈ (0,∞) , we are able to construct a Dirac family {𝚽𝜌 } 𝜌∈ (0,∞) of zonal rank-4 tensorial kernel functions as follows. 𝚽𝜌 (𝜉, 𝜂) =

3 ∑︁ 3 ∑︁

𝚽𝜌(𝑖,𝑘,𝑖,𝑘 ) (𝜉, 𝜂),

𝜉, 𝜂 ∈ Ω,

(11.49)

𝑖=1 𝑘=1

with 𝚽𝜌(𝑖,𝑘,𝑖,𝑘 ) (𝜉, 𝜂) =

∞ ∑︁

(Φ𝜌 ) ∧ (𝑛)

𝑛=0𝑖𝑘

2𝑛+1 ∑︁

(𝑖,𝑘 ) (𝑖,𝑘 ) y𝑛,𝑚 (𝜉) ⊗ y𝑛,𝑚 (𝜂),

(11.50)

𝑚=1

𝜉, 𝜂 ∈ Ω. Correspondingly, a Dirac family {𝝋𝜌 } 𝜌∈ (0,∞) of zonal rank-2 tensor kernel functions reads as follows 𝑡

𝝋𝜌 (𝜉, 𝜂) =

3 ∑︁ 3 ∑︁

𝑡

𝝋𝜌(𝑖,𝑘 ) (𝜉, 𝜂)

(11.51)

𝑖=1 𝑘=1

with 𝑡

𝝋𝜌(𝑖,𝑘 ) (𝜉, 𝜂) =

∞ ∑︁ 𝑛=0𝑖,𝑘

(Φ𝜌 ) ∧ (𝑛)

2𝑛+1 ∑︁

(𝑖,𝑘 ) 𝑌𝑛,𝑚 (𝜉)y𝑛,𝑚 (𝜂),

(11.52)

𝑚=1

𝜉, 𝜂 ∈ Ω. From our consideration, it is clear that the following theorem holds true. Theorem 11.19 Let {Φ𝜌 } 𝜌∈ (0,∞) be a scalar function as defined by (11.49). Then lim ∥f − 𝚽𝜌 ∗ f ∥ l2 (Ω) = 0,

(11.53)

lim ∥f − 𝚽𝜌 ∗ 𝚽𝜌 ∗ f ∥ l2 (Ω) = 0

(11.54)

lim ∥f − 𝝋𝜌 ★ 𝝋𝜌 ∗ f ∥ l2 (Ω) = 0.

(11.55)

𝜌→0

𝜌→0

and 𝜌→0

Theorem 11.19 extends the notion of an approximate identity to tensor spherical fields. Obviously, 𝚽𝜌(1,1,1,1) (𝜉, 𝜂) = 𝑜 (1,1) 𝑜 (1,1) Φ𝜌 (𝜉 · 𝜂). 𝜂 𝜉 𝜉, 𝜂 ∈ Ω. For 𝑖 ∈ {2, 3} we have

(11.56)

11.6 Bibliographical Notes

549

𝚽𝜌(𝑖,𝑖,𝑖,𝑖) (𝜉, 𝜂) =

1 (𝑖,𝑖) (𝑖,𝑖) 𝑜 𝑜 𝜂 Φ𝜌 (𝜉 · 𝜂), 2 𝜉

𝜉, 𝜂 ∈ Ω, while, for (𝑖, 𝑘) ∈ {(1, 2), (1, 3), (2, 1), (3, 1)}, ∫ ) (𝑖,𝑘 ) 𝚽𝜌(𝑖,𝑘,𝑖,𝑘 ) (𝜉, 𝜂) = −𝑜 (𝑖,𝑘 𝑜 𝐺 (Δ∗ ; 𝜉 · 𝜁)Φ𝜌 (𝜁 · 𝜂) 𝑑𝜔(𝜁), 𝜂 𝜉

(11.57)

(11.58)

Ω

𝜉, 𝜂 ∈ Ω. Finally, for (𝑖, 𝑘) ∈ {(2, 3), (3, 2)}, 1 𝚽𝜌(𝑖,𝑘,𝑖,𝑘 ) (𝜉, 𝜂) = 𝑛(𝑛 + 1)(𝑛(𝑛 + 1) − 2) ∫ (𝑖,𝑘 ) (𝑖,𝑘 ) 𝑜𝜉 𝑜𝜂 𝐺 (Δ∗ (Δ∗ − 2); 𝜉 · 𝜁)Φ𝜌 (𝜁 · 𝜂) 𝑑𝜔(𝜁),

(11.59)

Ω

𝜉, 𝜂 ∈ Ω.

11.6 Bibliographical Notes

The generalization of zonal kernel functions to the tensor context as proposed here is essentially based on M. Schreiner (1994), W. Freeden et al. (1994), W. Freeden et al. (1998), H. Nutz (2002), and W. Freeden, M. Schreiner (2009).

Part V

Geopotentials

Chapter 12

Potentials in Euclidean Space

In what follows, we discuss function systems occurring in Cartesian and spherical potential theory to handle problems, e.g., in geosciences, prospection, and exploration.

12.1 Newton’s Law of Gravitation

Newton’s law of gravitation, first published in his Principia in 1687, asserts that the force exerted on a point mass 𝑄 at 𝑥 ∈ R3 by a system of finitely many point masses 𝑞 𝑖 at 𝑦 𝑖 ∈ R3 , 𝑖 = 1, . . . , 𝑁, is equal to 𝑥 ↦→ 𝑣(𝑥) = 𝐶

𝑁 ∑︁ 𝑖=1

𝑞𝑖 𝑄 𝑥 − 𝑦𝑖 , |𝑥 − 𝑦 𝑖 | 2 |𝑥 − 𝑦 𝑖 |

𝑥 ≠ 𝑦 𝑖 , 𝑖 = 1, . . . , 𝑁,

(12.1)

with a constant 𝐶 < 0 (like masses attract). The same law of interaction between point charges was discovered experimentally by C. A. de Coulomb (1736-1806) and announced in 1785, now with 𝐶 > 0 (like charges repel). Note that the numerical value of the constant 𝐶 depends on the unit system one is using to measure force, mass (or charge), and distance. After the introduction of the function 𝑥 ↦→ 𝑉 (𝑥) = 𝐶

𝑁 ∑︁

𝑞𝑖 , |𝑥 − 𝑦 𝑖 |

𝑥 ≠ 𝑦 𝑖 , 𝑖 = 1, . . . , 𝑁,

(12.2)

𝑖=1

into the theory of gravitation by D. Bernoulli in 1748, J.-L. Lagrange noticed in 1773 that 𝑣(𝑥) = 𝑄 ∇𝑉 (𝑥), 𝑥 ≠ 𝑦 𝑖 , 𝑖 = 1, . . . , 𝑁. (12.3) © The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2022 W. Freeden and M. Schreiner, Spherical Functions of Mathematical Geosciences, Geosystems Mathematics, https://doi.org/10.1007/978-3-662-65692-1_12

553

554

12 Potentials in Euclidean Space

Hence, the function 𝐹 completely describes the gravitational (or electrostatic) field. For a continuous distribution of charges with density 𝐹, vanishing outside G, the potential becomes ∫ 𝐶 1 𝑉 (𝑥) = 𝐹 (𝑦) 𝑑𝑦, 𝑥 ∈ R3 , (12.4) 4𝜋 G |𝑥 − 𝑦| where, as usual, 𝑑𝑦 is the volume element and G 𝑐 = R3 \G is the outer space of G. As observed by P.S. Laplace in 1782, the function 𝐺 (Δ; | · −𝑦|), 𝑦 ∈ R3 , given by 𝐺 (Δ; |𝑥 − 𝑦|) =

1 1 , 4𝜋 |𝑥 − 𝑦|

𝑥 ≠ 𝑦,

(12.5)

in today’s jargon called the fundamental solution of the Laplace equation (12.6) in R3 \{𝑦}, i.e., Δ 𝑥 𝐺 (Δ; |𝑥 − 𝑦|) = 0, 𝑥 ∈ R3 \{𝑦}. (12.6) Later, the solutions of the homogeneous Laplace equation came to be known as harmonic functions. It should, however, be remarked that the Laplace equation had been also considered by Lagrange in 1760 in connection with his study of fluid flow problems. Laplace’s result was completed by his student S. D. Poisson (1781-1840) in 1813, when he showed that Δ𝑉 = −𝐶𝐹 for smooth enough densities 𝐹.

12.2 Volume Potential

In what follows, we collect some basic mathematical material well-known from classical potential theory in Euclidean space R3 . First we have a closer look at the Newtonian volume integral (see, e.g., O.D. Kellogg (1929), J. Schauder (1931), N.M. G¨unter (1957), R. Leis (1967), E. Martensen (1968), C. M¨uller (1969), W. Walter (1971), S.G. Michlin (1975), W. Freeden, M. Schreiner (2009), W. Freeden, C. Gerhards (2013), and the references therein). Theorem 12.1 Suppose that G is a regular region in R3 . (i) Let 𝐹 : G → R be an integrable bounded function. Then ∫ 𝑉 (𝑥) = 𝐶 𝐹 (𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝑦, 𝑥 ∈ G 𝑐 , G

satisfies

(12.7)

12.2 Volume Potential

555

∫ 𝐹 (𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝑦 = 0

Δ𝐶

(12.8)

G

for all 𝑥 ∈ G 𝑐 , i.e., 𝑉 is harmonic in G 𝑐 . (ii) Let 𝐹 : G → R be of class C (0) (G). Then 𝑉 as defined by the volume integral in (12.7) is of class C (0) (G). Furthermore, we have ∫ ∇𝑉 (𝑥) = 𝐶 𝐹 (𝑦) ∇ 𝑥 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝑦, 𝑥 ∈ G. (12.9) G

Proof We only verify (ii). By one-dimensional Taylor linearization we obtain 1 1 1 1 3 1 (𝑢 − 𝑢 0 ) + (𝑢 − 𝑢 0 ) 2 √ ≈√ − 8 (𝑢 0 + 𝜗(𝑢 − 𝑢 0 )) 52 𝑢 0 2 32 𝑢 𝑢0 for some 𝜗 ∈ (0, 1). Setting 𝑢 = 𝑟 2 and 𝑢 0 = 𝜌 2 we therefore find   1 1 𝑟2 3 1 ≈ 3− 2 + (𝑟 2 − 𝜌 2 ) 2 . 𝑟 2𝜌 8 (𝜌 2 + 𝜗(𝑟 2 − 𝜌 2 )) 52 𝜌

(12.10)

(12.11)

In other words, by letting 𝑟 = |𝑥 − 𝑦| we are able to “mollifier” the fundamental solution of the Laplace operator 𝐺 (Δ; 𝑟) =

1 , 4𝜋𝑟

𝑟 > 0,

(12.12)

by 𝐺 𝜌 (Δ; 𝑟) =

   1 1 2   3 − 2 𝑟 , 𝑟 ≤ 𝜌,   𝜌  8𝜋𝜌 

(12.13)   1    , 𝑟 > 𝜌.  4𝜋𝑟 such that 𝐺 𝜌 (Δ; ·) is continuously differentiable for all 𝑟 ≥ 0 . Obviously, 𝐺 (Δ; 𝑟) = 𝐺 𝜌 (Δ; 𝑟) for all 𝑟 > 𝜌. Consequently, we have 𝐺 (Δ; |𝑥 − 𝑦|) =

1 1 , 4𝜋 |𝑥 − 𝑦|

|𝑥 − 𝑦| ≠ 0,

admits a “mollifier regularization” of the form   1 1  2   3 − |𝑥 − 𝑦| , |𝑥 − 𝑦| ≤ 𝜌,  8𝜋𝜌  𝜌2   𝜌 𝐺 (Δ; |𝑥 − 𝑦|) =   1   , 𝜌 < |𝑥 − 𝑦|.   4𝜋|𝑥 − 𝑦|

(12.14)

(12.15)

556

12 Potentials in Euclidean Space

For brevity, we set ∫ 𝑈 (𝑥) =

𝐹 (𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝑦

(12.16)

𝐹 (𝑦) 𝐺 𝜌 (Δ; |𝑥 − 𝑦|) 𝑑𝑦.

(12.17)

G

and

∫ 𝜌

𝑈 (𝑥) = G

The integrands of 𝑈 and 𝑈 𝜌 differ only in the ball Ωint 𝜌 (𝑥) around the point 𝑥 with radius 𝜌. Moreover, the function 𝐹 : G → R is supposed to be continuous on G. Hence, it is uniformly bounded on G. This shows us that ! ∫ |𝑈 (𝑥) − 𝑈 𝜌 (𝑥)| = 𝑂

|𝐺 (Δ; |𝑥 − 𝑦|) − 𝐺 𝜌 (Δ; |𝑥 − 𝑦|)| 𝑑𝑦 Ωint 𝜌 (𝑥)

= 𝑂 (𝜌 2 ).

(12.18)

Therefore, 𝑈 is of class C (0) (G) as the limit of a uniformly convergent sequence of continuous functions on G. Furthermore, we let ∫ 𝑢(𝑥) = 𝐹 (𝑦) ∇ 𝑥 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝑦 (12.19) G

and

∫ 𝜌

𝐹 (𝑦) ∇ 𝑥 𝐺 𝜌 (Δ; |𝑥 − 𝑦|) 𝑑𝑦.

𝑢 (𝑥) =

(12.20)

G

Because of |∇ 𝑥 𝐺 (Δ; |𝑥 − 𝑦|)| = 𝑂 (|𝑥 − 𝑦| −2 ), the integrals 𝑢 and 𝑢 𝜌 exist for all 𝑥 ∈ G. It is not hard to see that sup |𝑢(𝑥) − 𝑢 𝜌 (𝑥)| = sup |∇ 𝑥 𝑈 (𝑥) − ∇ 𝑥 𝑈 𝜌 (𝑥)| = 𝑂 (𝜌). 𝑥∈ G

(12.21)

𝑥∈ G

Consequently, 𝑢 is a continuous vector field on G. Moreover, as the relation (12.21) holds uniformly on G, we obtain ∫ 𝑢(𝑥) = ∇𝑈 (𝑥) = 𝐹 (𝑦) ∇ 𝑥 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝑦. (12.22) G

□

This is the desired result.

Moreover, the aforementioned Poisson equation under the assumption of 𝜇-H¨older continuity, 𝜇 ∈ (0, 1], can be formulated by the following mollifier method: Theorem 12.2 If 𝐹 is of class C (0, 𝜇) (G), 𝜇 ∈ (0, 1], then the Poisson differential equation ∫ −Δ𝐶 𝐹 (𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝑦 = 𝐶𝐹 (𝑥) (12.23) G

12.2 Volume Potential

557

holds true for all 𝑥 ∈ G. 3

3

−5

Proof The Taylor linearization of 𝑠 − 2 is 𝑠02 − 32 𝑠0 2 (𝑠 − 𝑠0 ). Hence, by letting 𝑠 = 𝑟 2 and 𝑠0 = 𝜌 2 , we are able to replace 𝑟 −3 by 2𝜌1 3 (5 − 𝜌32 𝑟 2 ). Hence, as mollifier regularization of 𝑍 (Δ; |𝑥 − 𝑦|) =

1 , 4𝜋|𝑥 − 𝑦| 3

|𝑥 − 𝑦| ≠ 0,

(12.24)

we introduce

𝑍 𝜌 (Δ; |𝑥 − 𝑦|) =

  1 3  2   5 − |𝑥 − 𝑦| , |𝑥 − 𝑦| ≤ 𝜌,   𝜌2  8𝜋𝜌 3    1   ,   4𝜋|𝑥 − 𝑦| 3

(12.25)

𝜌 < |𝑥 − 𝑦|.

𝑍 𝜌 (Δ; ·) is continuously differentiable for all 𝑟 ≥ 0. Moreover, by already known arguments, it can be shown that the vector field ∫ 𝑧 𝜌 (𝑥) = − 𝐹 (𝑦) 𝑍 𝜌 (Δ; |𝑥 − 𝑦|)(𝑥 − 𝑦) 𝑑𝑦 (12.26) G

converges uniformly on G to the limit field ∫ 𝑢(𝑥) = ∇𝑈 (𝑥) = 𝐹 (𝑦) ∇ 𝑥 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝑦.

(12.27)

G

For all 𝑦 ∈ Ωint 𝜌 (𝑥) we obtain by a simple calculation ∇ 𝑥 · ((𝑥 − 𝑦) 𝑍 𝜌 (Δ; |𝑥 − 𝑦|)) =

  15 1 |𝑥 − 𝑦| 2 − . 8𝜋 𝜌 3 𝜌5

(12.28)

Furthermore, we find ∫ ∇ 𝑥 · ((𝑥 − 𝑦) 𝑍 𝜌 (Δ; |𝑥 − 𝑦|)) 𝑑𝑦 = 1.

(12.29)

Ωint 𝜌 (𝑥)

Hence, we obtain ∫ 𝜌

∇ · 𝑧 (𝑥) = − ∇ · ∫ =−

𝐹 (𝑦) 𝑍 𝜌 (Δ; |𝑥 − 𝑦|)(𝑥 − 𝑦) 𝑑𝑦 G

Ωint 𝜌 (𝑥)

= − 𝐹 (𝑥)

𝐹 (𝑦)∇ 𝑥 · (𝑍 𝜌 (Δ; |𝑥 − 𝑦|)(𝑥 − 𝑦)) 𝑑𝑦

(12.30)

558

12 Potentials in Euclidean Space

∫ (𝐹 (𝑥) − 𝐹 (𝑦))∇ 𝑥 · (𝑍 𝜌 (Δ|𝑥 − 𝑦|)(𝑥 − 𝑦)) 𝑑𝑦.

+ Ωint 𝜌 (𝑥)

The 𝜇-H¨older continuity of 𝐹 guarantees the estimate sup |∇ · 𝑧 𝜌 (𝑥) + 𝐹 (𝑥)| = 𝑂 (𝜌 𝜇 )

(12.31)

𝑥∈ G

uniformly with respect to 𝑥 ∈ G. In an analogous way, we are able to show that the first partial derivatives of (12.26) converge uniformly to continuous limit fields. Moreover, we have ∫ ∇ · 𝑢(𝑥) = Δ𝑈 (𝑥) = Δ 𝐹 (𝑦)𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝑦 = −𝐹 (𝑥), 𝑥 ∈ G, (12.32) G

as required.

□

In Theorem 12.2 the assumption of 𝜇-H¨older continuity of 𝐹, 𝜇 ∈ (0, 1], is needed for its proof. As a matter of fact, H. Petrini (1900) showed that the 𝜇-H¨older continuity of 𝐹, 𝜇 ∈ (0, 1], is necessary to imply the second continuous differentiability of the Newton volume potential. Finally, it should be mentioned that the mollifier regularization techniques as proposed here are used in W. Freeden (2021), C. Blick et al. (2021) (see also W. Freeden, M.Z. Nashed (2018c), W. Freeden, M.Z. Nashed (2020a)) to decorrelate mass density distribution and magnetization distribution, respectively, by mollifier wavelets for geologic purposes of inverse gravimetry and magnetometry.

12.3 Interior/Exterior Green’s Formula

Observing the specific properties of the fundamental solution of the Laplace equation we are able to formulate the third interior Green’s formula. Mean value theorems and maximum/minimum principle are the canonical consequences. Harmonic functions are recognized to be analytic in their harmonicity domain. The Kelvin transform enables us to study harmonic functions which are regular at infinity. Keeping the regularity at infinity in mind we are finally led to exterior Green’s formulas. The third exterior Green’s formula is formulated in analogy to its interior counterpart (for the proofs the reader is referred to , e.g., W. Freeden, C. Gerhards (2013)).

Theorem 12.3 (Interior Third Green’s Theorem)

12.3 Interior/Exterior Green’s Formula

559

Let G ⊂ R3 be a regular region with continuously differentiable boundary 𝜕G. Suppose that 𝐹 : G → R is of class C (1) (G) ∩C (2) (G) with Δ𝐹 Lebesgue- integrable on G. Then  ∫  𝜕 𝜕 𝐺 (Δ; |𝑥 − 𝑦|) 𝐹 (𝑦) − 𝐹 (𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝜔(𝑦) 𝜕𝜈(𝑦) 𝜕𝜈(𝑦) 𝜕G ∫ − 𝐺 (Δ; |𝑥 − 𝑦|)Δ𝐹 (𝑦) 𝑑𝑦) = 𝛼(𝑥)𝐹 (𝑥), (12.33) G

where 𝛼(𝑥) id solid angle 𝛼(𝑥) subtended by the boundary 𝜕G at the point 𝑥 ∈ R3 As special case we obtain for a regular region with continuously differentiable boundary 𝜕G Corollary 12.4 Let G ⊂ R3 be a regular region with continuously differentiable boundary 𝜕G. Suppose that 𝐹 : G → R is of class C (1) (G) ∩ C (2) (G) with Δ𝐹 = 0 on G. Then  ∫  𝜕𝐹 𝜕 𝐺 (Δ; |𝑥 − 𝑦|) (𝑦) − 𝐹 (𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝜔(𝑦) (12.34) 𝜕𝜈(𝑦) 𝜕𝜈(𝑦) 𝜕G  𝐹 (𝑥),    1  = 𝐹 (𝑥),  2    0, 

𝑥 ∈ G, 𝑥 ∈ 𝜕G, 𝑥 ∈ G𝑐 .

These Green’s formulas turn out to be the point of departure for the limit and jump relations in potential theory (see, e.g., O.D. Kellogg (1929), J. Schauder (1931), W. Freeden, C. Gerhards (2013)). Letting 𝐹 = 1 in G, we obviously find in connection with (12.5) and Corollary 12.4 Definition 12.5 (Solid Angle) Let G ⊂ R3 be a regular region. Then the solid angle 𝛼(𝑥) subtended by the boundary 𝜕G at the point 𝑥 ∈ R3 is given by ∫ 𝜕 𝛼(𝑥) = − 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝜔(𝑦). (12.35) 𝜕G 𝜕𝜈(𝑦)

Note that we have

560

12 Potentials in Euclidean Space

 𝑥 ∈ G,   1,  1 , 𝑥 𝛼(𝑥) = 2 ∈ 𝜕G,   0, 𝑥 ∉ G, 

(12.36)

provided that G ⊂ R𝑞 is a regular region with continuously differentiable boundary 𝜕G (cf. Fig. 12.1). In the case of the cube G = (−1, 1) 3 ⊂ R3 we especilly have (i) 𝛼(𝑥) = 1 if 𝑥 is located in the open cube G, (ii) 𝛼(𝑥) = 1/2 if 𝑥 is located on one of the six faces of the boundary 𝜕G of the cube G but not on an edge or in a vertex, (iii) 𝛼(𝑥) = 1/4 if 𝑥 is located on one of the eight edges of 𝜕G but not in a vertex, (iv) 𝛼(𝑥) = 1/8 if 𝑥 is located in one of the eight vertices of 𝜕G (cf. Fig. 12.2). 𝛼( 𝑥 ) = 1/2 𝛼( 𝑥 ) = 0

𝛼( 𝑥 ) = 1

Fig. 12.1: Solid angle subtended at 𝑥 ∈ R3 by the surface 𝜕G of a regular region G with “smooth boundary”.

𝛼( 𝑥 ) = 1/2

𝛼( 𝑥 ) = 0

𝛼( 𝑥 ) = 1 𝛼( 𝑥 ) = 1/8

𝛼( 𝑥 ) = 1/4

Fig. 12.2: Solid angle subtended at 𝑥 ∈ R3 by the surface 𝜕G of the “non-smooth” cube G = (−1, 1) 3 .

It should be remarked that the divergence theorem first appeared in Lagrange’s 1860 posthumous work, and it was proved in a special case already by Gauss in 1813. The general three-dimensional case was treated by M. V. Ostrogradsky in

12.3 Interior/Exterior Green’s Formula

561

1826. In a preliminary section of his groundbreaking 1828 essay, George Green’s proved several reductions of three-dimensional volume integrals to surface integrals, similar in spirit to the divergence theorem, and independently of M. V. Ostrogradsky. Nowadays, those are called Green’s identities and best viewed as consequences of the Gauss integral theorem. Next we are concerned with the mean value theorem (in today’s mathematical language) that dates back to C.F. Gauss (1840). Theorem 12.6 Let G ⊂ R3 be a regular region. Then the following statements are equivalent: (i) 𝑈 : G → R is harmonic in G, i.e., 𝑈 ∈ C (2) (G) and Δ𝑈 = 0 in G, (ii) 𝑈 : G → R possesses the Mean Value Property on G, i.e., 𝑈 is of class C (0) (G) and for all 𝑥 ∈ G and all 𝑟 > 0 with Ω𝑟int (𝑥) ⋐ G ∫ 1 𝑈 (𝑥) = 𝑈 (𝑦) 𝑑𝜔(𝑦), (12.37) 4𝜋𝑟 2 | 𝑥 |=𝑟 (iii) 𝑈 is of class C (0) (G) and for all 𝑟 > 0 with Ω𝑟int (𝑥) ⋐ G ∫ (𝑈 (𝑥) − 𝑈 (𝑦)) 𝑑𝑦 = 0.

(12.38)

| 𝑥 | ≤𝑟

A central result in the theory of harmonic functions is the Maximum/Minimum Principle. Essential tool for its proof is the Mean Value Property. Theorem 12.7 (Maximum/Minimum Principle) Let G ⊂ R3 be a regular region. Suppose that 𝑈 is harmonic in G and non-constant. Then 𝑈 does not reach its minimum or maximum in G. If, in addition, 𝑈 is of class C (0) (G), then 𝑈 reaches its minimum and maximum in G, and the extremal points are lying on 𝜕G. More precisely, sup |𝑈 (𝑥)| ≤ sup |𝑈 (𝑥)|. 𝑥∈ G

(12.39)

𝑥 ∈𝜕G

A direct consequence of the Maximum/Minimum Principle is the following stability theorem.

562

12 Potentials in Euclidean Space

Theorem 12.8 Let G ⊂ R3 be a regular region. Suppose that 𝑈 and 𝑉 are of class C (0) (G) ∩ C (2) (G), and harmonic in G. Let 𝜀 be an arbitrary positive number. If sup |𝑈 (𝑥) − 𝑉 (𝑥)| ≤ 𝜀,

(12.40)

𝑥 ∈𝜕G

then sup |𝑈 (𝑥) − 𝑉 (𝑥)| ≤ 𝜀.

(12.41)

𝑥∈ G

Now we are prepared to establish the (real) analyticity of harmonic functions. Theorem 12.9 (Analyticity) Let G ⊂ R3 be a regular region. Suppose that 𝑈 is harmonic on G. Then 𝑈 is (real) analytic, i.e., for 𝑥 0 ∈ G there exists 𝜌 > 0 such that 𝑈 (𝑥0 + ℎ) =

∞ ∑︁ 1 ((ℎ · ∇) 𝑗 𝑈)(𝑥0 ) 𝑗! 𝑗=0

(12.42)

for all ℎ ∈ R3 with |ℎ| < 𝜌.

𝑅 0

𝑥ˇ 𝑅

𝑥

𝑦

Fig. 12.3: The inversion 𝑥 ↦→ 𝑥ˇ 𝑅 with respect to the sphere Ω𝑅 . Kelvin Transform. The mapping (cf. Fig. 12.3) 

𝑅 𝑥 ↦→ 𝑥ˇ 𝑅 = |𝑥|

2 𝑥,

𝑥 ∈ Ωint \{0},

(12.43)

int transforms Ω𝑅 int\{0} into R3 \Ωint 𝑅 and Ω 𝑅 = 𝜕Ω 𝑅 onto itself. Referring to Fig. 12.3, we observe that the two triangles with edges ( 𝑥ˇ 𝑅 , 𝑦, 0) and (𝑥, 𝑦, 0) are similar whenever 𝑦 ∈ Ω𝑅 . Furthermore, the ratios |𝑥|/|𝑦| and |𝑦|/| 𝑥ˇ 𝑅 | are equal, provided that 𝑦 ∈ Ω𝑅 .

On the one hand, for 𝑥 = |𝑥|𝜉, 𝜉 ∈ Ω, and 𝑦 = |𝑦|𝜂, 𝑅 = |𝑦|, 𝜂 ∈ Ω, we have

12.3 Interior/Exterior Green’s Formula

563

|𝑥 − 𝑦| 2 = 𝑥 2 + 𝑦 2 − 2𝑥 · 𝑦 = |𝑥| 2 + 𝑅 2 − 2|𝑥|𝑅 𝜉 · 𝜂. On the other hand we see that 2  2 2  4  2 |𝑥| 𝑅 𝑅 𝑅2 2 2 = |𝑥| 𝑥 − 𝑦 |𝑥| + 𝑅 − 2 𝑥 · 𝑦 𝑅 |𝑥| 2 𝑅 2 |𝑥| 4 |𝑥| 2

(12.44)

(12.45)

= |𝑥| 2 + 𝑅 2 − 2|𝑥|𝑅 𝜉 · 𝜂. Lemma 12.10 For all 𝑦 ∈ Ω𝑅 and 𝑥 ∈ Ωint 𝑅, |𝑥 − 𝑦| =

|𝑥| | 𝑥ˇ 𝑅 − 𝑦|. 𝑅

(12.46)

After these preparations about the inversion of points with respect to a sphere Ω𝑅 , 𝑅 > 0, we discuss the Kelvin transform. For simplicity, we first choose 𝑅 = 1, i.e., we restrict the inversion to the unit sphere Ω. Theorem 12.11 Assume that 𝑈 is of class C (2) (G), G ⊂ R3 \{0} open. Let Gˇ be the image of G under the inversion 𝑥 ↦→ 𝑥ˇ = |𝑥| −2 𝑥, and denote by 𝑈ˇ = 𝐾 [𝑈] : Gˇ → R, with   1 𝑥 𝑈ˇ (𝑥) = 𝐾 [𝑈] (𝑥) = 𝑈 , (12.47) |𝑥| |𝑥| 2 the Kelvin transform of 𝑈. Then   1 𝑥 ˇ Δ𝑈 (𝑥) = 5 Δ𝑈 . |𝑥| 2 |𝑥|

(12.48)

For the proof the reader is referred to any textbook of potential theory, e.g., W. Freeden, C. Gerhards (2013). Corollary 12.12 Assume that 𝑈 is of class C (2) (G), G ⊂ R3 \{0} open. Let Gˇ𝑅 be the image of G under the inversion 𝑥 ↦→ 𝑥ˇ 𝑅 = 𝑅 2 |𝑥| −2 𝑥, and denote by 𝑈ˇ = 𝐾 𝑅 [𝑈]: Gˇ𝑅 → R, with  2 ! 𝑅 𝑅 𝑅 𝑈ˇ (𝑥) = 𝐾 [𝑈] (𝑥) = 𝑈 𝑥 , (12.49) |𝑥| |𝑥| the Kelvin transform of 𝑈 with respect to Ω𝑅 = 𝜕Ωint 𝑅 . Then 

𝑅 Δ𝑈ˇ (𝑥) = |𝑥|

5

 Δ𝑈

𝑅 |𝑥|

2 ! 𝑥 .

(12.50)

564

12 Potentials in Euclidean Space

The Newton (volume) potential extended over G is harmonic in the exterior G 𝑐 = R3 \G. This is the reason why potential theory under geoscientifically relevant aspects essentially aims at concepts in the outer space of a regular region. The treatment of the outer space in the Euclidean space R3 , however, includes the discussion at infinity. As a consequence, Green’s integral theorems must be formulated under geophysically relevant conditions imposed on harmonic functions at infinity. Mathematically (see, e.g., W. Freeden, C. Gerhards (2013)), the “regularity at infinity” can be deduced via the Kelvin transform by a transition from functions harmonic in the inner space to their counterparts in outer space, and vice versa. Theorem 12.13 If 𝑈 is harmonic in G 𝑐 = R3 \G and 𝑈 converges to zero for |𝑥| → ∞ uniformly with respect to all directions, then |𝑥||𝑈 (𝑥)| and |𝑥| 2 |∇𝑈 (𝑥)| are bounded for |𝑥| → ∞. Theorem 12.13 leads to the definition of the “regularity at infinity”. Definition 12.14 A function 𝑈 : G 𝑐 → R is called regular at infinity, if 𝑈 satisfies the asymptotic relation |𝑈 (𝑥)| = 𝑂 (|𝑥| −1 ) and |∇𝑈 (𝑥)| = 𝑂 (|𝑥| −2 ), |𝑥| → ∞, uniformly with respect to all directions 𝑥/|𝑥|. Now we are prepared to discuss exterior versions of Green’s identities involving harmonic functions regular at infinity. All these identities can be obtained by first considering the auxiliary set G𝑅𝑐 (0) = G 𝑐 ∩ Ωint 𝑅 (with 𝑅 sufficiently large such that int int int G ⋐ Ω𝑅 , i.e., G ⊂ Ω𝑅 and dist(𝜕G, 𝜕Ω𝑅 ) > 0) and afterwards letting 𝑅 tend to infinity (note that G𝑅𝑐 (0) as the difference of the two regular regions Ωint 𝑅 and G allows the application of the interior Green’s formulas). Theorem 12.15 (Exterior First Green’s Theorem) Let 𝐹 be a function of class C (2) (G 𝑐 ) ∩ C (1) (G 𝑐 ) such that 𝐹 is harmonic in G 𝑐 and regular at infinity. Suppose that the function 𝐻 ∈ C (1) (G 𝑐 ) satisfies the asymptotic relations |𝑦| 2 |𝐹 (𝑦)∇𝐻 (𝑦)| = 𝑂 (1) (12.51) and

 |∇𝐹 (𝑦) · ∇𝐻 (𝑦)| = 𝑂

Then

∫

 1 , 𝜀 > 0. |𝑦| 3+𝜀

∫ ∇𝐹 (𝑦) · ∇𝐻 (𝑦) 𝑑𝑦 = G𝑐

𝐹 (𝑦) 𝜕G

𝜕𝐻 (𝑦) 𝑑𝑆(𝑦), 𝜕𝜈

(12.52)

(12.53)

where 𝜈 is the outer unit normal field to G 𝑐 , i.e., the inner unit normal field to G.

12.4 Volume Potential and Gravitational Field

565

Theorem 12.16 (Exterior Second Green’s Theorem) Let the function 𝐹, 𝐺 ∈ C (1) (G 𝑐 ) ∩ C (2) (G 𝑐 ) be harmonic in G 𝑐 and regular at infinity. Then  ∫  𝜕 𝜕 𝐹 (𝑦) 𝐻 (𝑦) − 𝐻 (𝑦) 𝐹 (𝑦) 𝑑𝜔(𝑦) = 0. (12.54) 𝜕𝜈 𝜕𝜈 𝜕G Theorem 12.17 (Exterior Third Green’s Theorem) Suppose that G is a regular region with continuously differentiable boundary 𝜕G. Let 𝑈 be of class C (1) (G 𝑐 ) ∩ C (2) (G 𝑐 ) such that 𝑈 is harmonic in G 𝑐 and regular at infinity. Then  ∫  𝜕 𝜕 𝐺 (Δ; |𝑥 − 𝑦|) 𝑈 (𝑦) − 𝑈 (𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝜔(𝑦) 𝜕𝜈(𝑦) 𝜕𝜈(𝑦) 𝜕G 𝑐    𝑈 (𝑥), 𝑥 ∈ G ,  = 12 𝑈 (𝑥), 𝑥 ∈ 𝜕G,   0, 𝑥 ∈ G,  where 𝜈 is the outer unit normal field to G 𝑐 , i.e., the inner unit normal field to G, and 𝛼(𝑥) is the solid angle subtended by the boundary 𝜕G at 𝑥 ∈ R3 .

12.4 Volume Potential and Gravitational Field

According to the classical Newton’s Law of Gravitation (1687), knowing the density distribution 𝐹 of a body, the gravitational potential can be computed everywhere in R3 . More explicitly, the gravitational potential 𝑉 for the exterior of a regular region G is given by ∫ 𝐹 (𝑦) 𝑉 (𝑥) = 𝑑𝑉 (𝑦), 𝑥 ∈ R3 \G. (12.55) |𝑥 − 𝑦| G The properties of the gravitational potential (12.55) in the exterior of G are easily described as follows: Δ𝑉 (𝑥) = 0, 𝑥 ∈ R3 \G. (12.56)

566

12 Potentials in Euclidean Space 𝑦

𝑥 0

|𝑥| 2

Fig. 12.4: Regularity at infinity.

Moreover, the gravitational potential 𝑉 is regular at infinity, i.e.,   1 |𝑉 (𝑥)| = 𝑂 , |𝑥| → ∞, |𝑥|   1 |∇𝑉 (𝑥)| = 𝑂 , |𝑥| → ∞. |𝑥| 2 Note that, for suitably large values |𝑥| (see Fig. 12.4), we have |𝑦| ≤ |𝑥 − 𝑦| ≥ ||𝑥| − |𝑦|| ≥ 12 |𝑥|.

(12.57) (12.58) 1 2 |𝑥|,

hence,

Clearly, the gravitational field 𝑣 = ∇𝑉 fulfills the following identities: 𝐿 · ∇𝑉 (𝑥) = 0, 𝑥 ∈ R3 \G

(12.59) 3

∇ · ∇𝑉 (𝑥) = Δ𝑉 (𝑥) = 0, 𝑥 ∈ R \G.

(12.60)

However, the problem is that in georeality the density distribution is very irregular and known only for parts of the upper crust of the Earth. It is actually so that geoscientists would like to know it from measuring the gravitational field. Even if the Earth is supposed to be spherical, the determination of the gravitational potential by integrating Newton’s potential is not achievable. This is the reason why, in spherical nomenclature, geosciences expand the gravitational potential of the spherical Earth Ωint 𝑅 into a series of spherical harmonics. In doing so, we observe that the so-called reciprocal distance (“Newton kernel”) can be expressed as a Legendre series as follows: 𝑛   ∞  1 1 ∑︁ |𝑦| 𝑥 𝑦 = 𝑃𝑛 · , (12.61) |𝑥 − 𝑦| |𝑥| 𝑛=0 |𝑥| |𝑥| |𝑥| ext 𝑦 ∈ Ωint 𝑅 , 𝑥 ∈ Ω 𝑅 , i.e., |𝑦| ≤ 𝑅 < |𝑥|. Relating (12.61) to the radius 𝑅, we obtain

12.4 Volume Potential and Gravitational Field

567

∞ 2𝑛+1

∑︁ ∑︁ 4𝜋𝑅 1 𝑅 = 𝐻𝑅 (𝑥)𝐻𝑛,𝑘 (𝑦), |𝑥 − 𝑦| 𝑛=0 𝑗=1 2𝑛 + 1 −𝑛−1,𝑘

(12.62)

𝑅 is an inner harmonic of degree 𝑛 and order 𝑘 given by where 𝐻𝑛,𝑘

𝑅 𝐻𝑛,𝑘 (𝑥)

 𝑛 1 |𝑥| = 𝑌𝑛,𝑘 (𝜉), 𝑅 𝑅

𝑥 = |𝑥|𝜉, 𝜉 ∈ Ω,

(12.63)

𝑅 and 𝐻 −𝑛−1,𝑘 is an outer harmonic of degree 𝑛 and order 𝑘 given by

𝑅 𝐻 −𝑛−1,𝑘 (𝑥) =

{𝑌𝑛,𝑘 }

𝑛=0,1,... 𝑘=1,...,2𝑛+1

  𝑛+1 1 𝑅 𝑌𝑛,𝑘 (𝜉), 𝑅 |𝑥|

𝑥 = |𝑥|𝜉, 𝜉 ∈ Ω.

(12.64)

is an L2 (Ω)-orthonormal system of scalar spherical harmonics.

Insertion of the series expansion (12.62) into the Newton formula for the gravitational potential yields for 𝑥 ∈ Ωext 𝑅 : ∞ 2𝑛+1 ∑︁ ∑︁ 4𝜋𝑅 ∫ 𝑅 𝑅 𝑉 (𝑥) = 𝐹 (𝑦) 𝐻𝑛,𝑘 (𝑦) 𝑑𝑉 (𝑦) 𝐻 −𝑛−1,𝑘 (𝑥). int 2𝑛 + 1 Ω 𝑅 𝑛=0 𝑘=1

(12.65)

The integrals involving inner/outer harmonics are regular, and a closer look at the individual terms reveals their geophysical relevance: The zero term gives the potential with mass equal to that of the gravitating mass distribution of the spherical Earth’s body Ωint 𝑅 . The first order term relates to dipole mass moments. Hence, looking at a potential with the same zero and first order coefficient as 𝑉 we are led to a “disturbing potential” as difference of both potentials, such that the zero and first order coefficients of the disturbing potential vanish. The quadrupole moments obtained by the second order term reflect the oblateness of the mass distribution. Unfortunately, the expansion coefficients of the series (12.65) are not computable in geopractice, since their determination requires the knowledge of the density function 𝜌 in the Earth’s interior Ωint 𝑅 . In fact, it turns out that there are infinitely many mass distributions, which have the given gravitational potential of the Earth as exterior potential. To overcome the difficulties, the solution of the (Dirichlet) boundary-value problem Δ𝑉 (𝑥) = 0, 𝑥 ∈ Ωint 𝑅 , corresponding to the boundary condition 𝑉 |Ω 𝑅 ∈ C(Ω𝑅 ) would suffice for purposes of determining the exterior gravitational potential, in principle, from the geophysical point of view, the expansion coefficients can be expressed using the “boundary function” 𝑉 |Ω𝑅 . However, the comparison of such spherical harmonic coefficients for 𝑉 |Ω𝑅 leads to an infinite number of equations

568

12 Potentials in Euclidean Space

relating 𝑉 |Ω𝑅 on the (spherical Earth’s) surface Ω𝑅 to the density distribution 𝐹 inside the (spherical) Earth Ωint 𝑅 . In other words, the knowledge of the density function inside the Earth allows the Fourier (orthogonal) expansion in terms of the potential coefficients. Inversely, given the potential coefficients as derived from the terrestrial potential, 𝑉 |Ω𝑅 does not suffice to determine the density distribution. In geological exploration, this ambiguity is known as the inverse gravimetry problem of determining the density distribution (see W. Freeden (2021) and the references therein) from certain potential values. Collecting the results on the Earth’s gravitational field 𝑣 in the context of the outer space Ωext 𝑅 , we are confronted with the following (mathematical) characterization: 𝑣 is an infinitely often differentiable vector field in the exterior of the Earth such that (v1) 𝑣 satisfies the differential equations div 𝑣 = ∇ · 𝑣 = 0,

curl 𝑣 = 𝐿 · 𝑣 = 0

(12.66)

in the Earth’s exterior, (v2) 𝑣 is regular at infinity:  |𝑣(𝑥)| = 𝑂

 1 , |𝑥| 2

|𝑥| → ∞.

(12.67)

Seen from mathematical point of view, the properties (v1) and (v2) imply that the Earth’s gravitational field 𝑣 in Ωext 𝑅 (see, e.g., O.D. Kellogg (1929), M.E. Gurtin (1971), A. Wangerin (1921)) is a gradient field 𝑣 = ∇𝑉,

(12.68)

where the gravitational potential 𝑉 fulfills the properties: 𝑉 is an infinitely often differentiable scalar field in Ωext 𝑅 such that (V1) 𝑉 is harmonic in Ωext 𝑅 , i.e., Δ𝑉 = 0, (V2) 𝑉 is regular at infinity, i.e., 

 1 , |𝑥|   1 |∇𝑉 (𝑥)| = 𝑂 , |𝑥| 2 |𝑉 (𝑥)| = 𝑂

|𝑥| → ∞, |𝑥| → ∞,

(12.69) (12.70)

and vice versa. Moreover, the gradient field of the gravitational field (i.e., the Jacobi matrix field)

12.4 Volume Potential and Gravitational Field

569

v = ∇𝑣,

(12.71)

obeys the following properties: v is an infinitely often differentiable tensor field inΩext 𝑅 such that (v1) div v = ∇ · v = 0,

curl v = 𝐿 · v = 0

(12.72)

in the Earth’s exterior, (v2) v is regular at infinity: 

 1 |v(𝑥)| = 𝑂 , |𝑥| 3

|𝑥| → ∞,

(12.73)

and vice versa. Combining (12.72) with (12.71), we see that v can be represented as the Hesse tensor of the scalar field 𝑉, i.e., v = (∇ ⊗ ∇) 𝑉 = ∇ (2) 𝑉 .

(12.74)

Harmonic Functions and Harnack’s Convergence Theorem. As preparation for the theory of boundary-value problems in terms of outer harmonics, some results known from potential theory should be recapitulated briefly. More explicitly, we are interested in essential ingredients of potential theory in their specific formulation for the outer space Ωext 𝑅 of the sphere around the origin with radius 𝑅. ext 3 ext 3 3 Let 𝑉 : Ωext 𝑅 → R, 𝑣 : Ω 𝑅 → R , and v : Ω 𝑅 → R ⊗ R , respectively, be a scalar, ext vector, and tensor field on the set Ω𝑅 . We say that 𝑉, 𝑣, v, respectively, are harmonic ext on Ωext 𝑅 if 𝑉, 𝑣, v are twice continuously differentiable on Ω 𝑅 and Δ𝑉 = 0, Δ𝑣 = 0, Δv = 0 on Ωext . 𝑅

Finally, for the convenience of the reader, we recapitulate some well-known theorems concerning harmonic fields on Ωext 𝑅 (for the proofs see, for example, M.E. Gurtin (1971), O.D. Kellogg (1929)): ext (i) Every harmonic field in Ωext 𝑅 is analytic in Ω 𝑅 , i.e., every harmonic field is determined by its local properties. ext 3 (ii) Harnack’s convergence theorem: Let 𝑉 𝛿 : Ωext 𝑅 → R, 𝑣 𝛿 : Ω 𝑅 → R , and ext 3 3 ext v 𝛿 : Ω𝑅 → R ⊗ R , respectively, be harmonic on Ω𝑅 for each value 𝛿 (0 < 𝛿 < 𝛿0 ), and regular at infinity. Moreover, let

𝑉 𝛿 → 𝑉 , 𝛿 → 0, 𝛿 > 0,

570

12 Potentials in Euclidean Space

𝑣 𝛿 → 𝑣 , 𝛿 → 0, 𝛿 > 0, v 𝛿 → v , 𝛿 → 0, 𝛿 > 0, ext ext uniformly on each subset K of Ωext 𝑅 with dist(K, 𝜕Ω 𝑅 ) > 0. Then 𝑉 : Ω 𝑅 → R, ext 3 ext 3 3 𝑣 : Ω𝑅 → R , and v : Ω𝑅 → R ⊗ R , respectively, is harmonic on Ωext 𝑅 and regular at infinity. Furthermore, for each fixed integer 𝑛

∇ (𝑛) 𝑉 𝛿 → ∇ (𝑛) 𝑉 , 𝛿 → 0, 𝛿 > 0, ∇ (𝑛) 𝑣 𝛿 → ∇ (𝑛) 𝑣 , 𝛿 → 0, 𝛿 > 0, ∇ (𝑛) v 𝛿 → ∇ (𝑛) v , 𝛿 → 0, 𝛿 > 0, ext holds uniformly on each subset K of Ωext 𝑅 with dist(K, 𝜕Ω 𝑅 ) > 0.

12.5 Scalar Inner/Outer Harmonics Consider the sphere Ω𝑅 ⊂ R3 around the origin with radius 𝑅 > 0. As usual, Ωint 𝑅 is the inner space of Ω𝑅 , and Ωext 𝑅 is the outer space. By virtue of the isomorphism Ω ∋ 𝜉 ↦→ 𝑅𝜉 ∈ Ω𝑅 , we assume functions 𝐹 : Ω𝑅 → R to be defined on Ω. It is clear that the function spaces defined on Ω admit their natural generalizations as spaces of functions defined on Ω𝑅 . We have, for example, C (∞) (Ω𝑅 ), L 𝑝 (Ω𝑅 ), etc. Obviously, an L2 (Ω)-orthonormal system of spherical harmonics forms an orthogonal system on Ω𝑅 (with respect to (·, ·)L2 (Ω𝑅 ) ). More explicitly, we have 

∫ (𝑌𝑛,𝑘 , 𝑌 𝑝,𝑞 )L2 (Ω𝑅 ) =

𝑌𝑛,𝑘 Ω𝑅

   𝑥 𝑥 𝑌 𝑝,𝑞 𝑑𝜔(𝑥) = 𝑅 2 𝛿 𝑛 𝑝 𝛿 𝑘𝑞 . |𝑥| |𝑥|

(12.75)

With the relationship 𝜉 ↔ 𝑅𝜉, the surface gradient ∇∗;𝑅 and the Beltrami operator Δ∗;𝑅 on Ω𝑅 , respectively, have the representation ∇∗;𝑅 = (1/𝑅)∇∗;1 = (1/𝑅)∇∗ , Δ∗;𝑅 = (1/𝑅 2 )Δ∗;1 = (1/𝑅 2 )Δ∗ , where ∇∗ , Δ∗ are the surface gradient and the Beltrami operator of the unit sphere Ω. For 𝑌𝑛 ∈ Harm𝑛 (Ω), we have Δ∗;𝑅𝑌𝑛 = (1/𝑅 2 )(Δ∗ ) ∧ (𝑛)𝑌𝑛 . 𝑅 } We introduce the system {𝑌𝑛,𝑘

𝑛=0,1,... 𝑘=1,...,2𝑛+1

𝑅 𝑌𝑛,𝑘 (𝑥) =

𝑅 } Due to (12.75), the system {𝑌𝑛,𝑘

by letting

  1 𝑥 𝑌𝑛,𝑘 , 𝑅 |𝑥| 𝑛=0,1,... 𝑘=1,...,2𝑛+1

L2 (Ω𝑅 ) = span

𝑥 ∈ Ω𝑅 .

(12.76)

is an orthonormal basis in L2 (Ω𝑅 ):

𝑛=0,1,..., 𝑘=1,...,2𝑛+1

𝑅 ) (𝑌𝑛,𝑘

∥ · ∥ L2 (Ω

𝑅)

.

(12.77)

12.5 Scalar Inner/Outer Harmonics

571

𝑅 } 𝑅 of degree 𝑛 and order 𝑘 can The system {𝐻𝑛,𝑘 of inner harmonics 𝐻𝑛,𝑘 𝑛=0,1,... 𝑘=1,...,2𝑛+1 be written as  𝑛 |𝑥| 𝑅 𝑅 (𝑥) , 𝐻𝑛,𝑘 (𝑥) = 𝑌𝑛,𝑘 𝑥 ∈ R3 . (12.78) 𝑅

It satisfies the following properties: 𝑅 is of class C (∞) (R3 ) • 𝐻𝑛,𝑘 𝑅 satisfies Laplace’s equation in R3 : • 𝐻𝑛,𝑘 𝑅 Δ𝐻𝑛,𝑘 (𝑥) = 0, 𝑅 |Ω = 𝑌 𝑅 = • 𝐻𝑛,𝑘 𝑅 𝑛,𝑘

𝑥 ∈ R3

1 𝑅 𝑌𝑛,𝑘 ∫

𝑅 • (𝐻𝑛,𝑘 , 𝐻𝑅 𝑝,𝑞 )L2 (Ω 𝑅 ) =

𝑅 𝑅 (𝑥) 𝑌 𝑝,𝑞 (𝑥) 𝑑𝜔(𝑥) = 𝛿 𝑛 𝑝 𝛿 𝑘𝑞 𝑌𝑛,𝑘 Ω𝑅

𝑅 | 1 (note that in the case of Ω𝑅 = Ω, we have 𝐻𝑛,𝑘 𝑅=1 = 𝐻 𝑛,𝑘 = 𝑌𝑛,𝑘 for all 𝑛 = 0, 1, . . ., 𝑘 = 1, . . . , 2𝑛 + 1).

From the addition theorem of spherical harmonics, we obtain 2𝑛+1 ∑︁

𝑅 𝑅 𝐻𝑛,𝑘 (𝑥)𝐻𝑛,𝑘 (𝑦) =

𝑘=1

 𝑛   2𝑛 + 1 |𝑥||𝑦| 𝑥 𝑦 𝑃 · 𝑛 |𝑥| |𝑦| 4𝜋𝑅 2 𝑅2

(12.79)

int for all (𝑥, 𝑦) ∈ Ωint 𝑅 ×Ω 𝑅 , which is known as the addition theorem of inner harmonics (see (4.26)).

In accordance with our notation, Harm𝑛 (Ωint 𝑅 ) denotes the space of all inner harint monics of degree 𝑛 on Ωint 𝑅 , i.e., Harm𝑛 (Ω 𝑅 ) is equal to the space of all linear com𝑅 , . . . 𝐻𝑅 int binations of the 2𝑛 + 1 elements 𝐻𝑛,1 𝑛,2𝑛+1 . Consequently, 𝑑 (Harm𝑛 (Ω 𝑅 )) = 2𝑛 + 1. We let

Harm 𝑝,...,𝑞 (Ωint 𝑅) =

𝑞 Ê

Harm𝑛 (Ωint 𝑅 ),

0 ≤ 𝑝 ≤ 𝑞.

(12.80)

𝑛= 𝑝

The kernel 𝐾Harm

𝐾Harm

int 𝑝,...,𝑞 (Ω 𝑅 )

int 𝑝,...,𝑞 (Ω 𝑅 )

(𝑥, 𝑦) =

int (·, ·) : Ωint 𝑅 × Ω 𝑅 → R given by 𝑞 2𝑛+1 ∑︁ ∑︁ 𝑛= 𝑝 𝑘=1

𝑅 𝑅 int 𝐻𝑛,𝑘 (𝑥)𝐻𝑛,𝑘 (𝑦), (𝑥, 𝑦) ∈ Ωint 𝑅 × Ω𝑅 ,

(12.81)

572

12 Potentials in Euclidean Space

is the reproducing kernel of the space Harm 𝑝,...,𝑞 (Ωint 𝑅 ) with respect to ∥ · ∥ L2 (Ω𝑅 ) , i.e.: (i) For every 𝑦 𝐾Harm

int 𝑝,...,𝑞 (Ω 𝑅 )

∈

Ωint 𝑅 , the functions 𝐾Harm

int 𝑝,...,𝑞 (Ω 𝑅 )

(𝑦, ·) as well as

(·, 𝑦) belong to Harm 𝑝,...,𝑞 (Ωint 𝑅)

int (ii) For any 𝐻 ∈ Harm 𝑝,...,𝑞 (Ωint 𝑅 ) and any 𝑥 ∈ Ω 𝑅 , the reproducing property





𝐻 (𝑥) = 𝐾Harm

int 𝑝,...,𝑞 (Ω 𝑅 )

(·, 𝑥), 𝐻 L2 (Ω 𝑅 )





= 𝐾Harm

int 𝑝,...,𝑞 (Ω 𝑅 )

(𝑥, ·), 𝐻 L2 (Ω 𝑅 )

holds true. 𝑅 𝑅 The system {𝐻 −𝑛−1,𝑘 } 𝑛=0,1,... of outer harmonics 𝐻 −𝑛−1,𝑘 of degree 𝑛 and order 𝑘=1,...,2𝑛+1 𝑘 defined by

 𝑅 𝐻 −𝑛−1,𝑘 (𝑥) =

𝑅 |𝑥|

 𝑛+1 𝑅 (𝑥) , 𝑌𝑛,𝑘

𝑥 ∈ R3 \{0},

(12.82)

satisfies the following properties: 𝑅 • 𝐻 −𝑛−1,𝑘 is of class C (∞) (R3 \{0}) 𝑅 • 𝐻 −𝑛−1,𝑘 satisfies Laplace’s equation in R3 \{0}: 𝑅 Δ𝐻 −𝑛−1,𝑘 (𝑥) = 0,

𝑥 ∈ R3 \{0}

𝑅 • 𝐻 −𝑛−1 is regular at infinity, i.e.

𝑅 𝐻

−𝑛−1,𝑘

(𝑥) = 𝑂

and ∇𝐻 𝑅

−𝑛−1,𝑘 (𝑥) = 𝑂





 1 , |𝑥|  1 , |𝑥| 2

|𝑥| → ∞

|𝑥| → ∞

𝑅 𝑅 = 1𝑌 • 𝐻 −𝑛−1,𝑘 Ω𝑅 = 𝑌𝑛,𝑘 𝑛,𝑘   𝑅 𝑅 𝑅 • 𝐻 −𝑛−1,𝑘 , 𝐻 − 𝑝−1,𝑞 2 = 𝛿 𝑛 𝑝 𝛿 𝑘𝑞 . L (Ω 𝑅 )

The addition theorem of spherical harmonics now yields

(12.83)

12.5 Scalar Inner/Outer Harmonics 2𝑛+1 ∑︁

573

𝑅 𝑅 𝐻 −𝑛−1,𝑘 (𝑥)𝐻 −𝑛−1,𝑘 (𝑦)

𝑘=1

 2  𝑛+1   2𝑛 + 1 𝑅 𝑥 𝑦 = 𝑃𝑛 · |𝑥| |𝑦| 4𝜋𝑅 2 |𝑥| |𝑦|

(12.84)

ext for all (𝑥, 𝑦) ∈ Ωext 𝑅 ×Ω 𝑅 , which is known as the addition theorem of outer harmonics.

We let

  Harm𝑛 Ωext = 𝑅



span

𝑅 𝐻 −𝑛−1,𝑘 |Ωext 𝑅



(12.85)

𝑘=1,...,2𝑛+1

and

𝑞   Ê   ext . Harm 𝑝,...,𝑞 Ωext = Harm Ω 𝑛 𝑅 𝑅

(12.86)

𝑛= 𝑝

The kernel 𝐾Harm

𝐾Harm

ext 𝑝,...,𝑞 (Ω 𝑅 )

𝑝,...,𝑞 (Ωext ) 𝑅

ext (·, ·): Ωext 𝑅 × Ω 𝑅 → R given by

(𝑥, 𝑦) =

𝑞 2𝑛+1 ∑︁ ∑︁

𝑅 𝑅 𝐻 −𝑛−1,𝑘 (𝑥)𝐻 −𝑛−1,𝑘 (𝑦),

(12.87)

𝑛= 𝑝 𝑘=1

ext ext (𝑥, 𝑦) ∈ Ωext 𝑅 × Ω 𝑅 , is the reproducing kernel of the space Harm 𝑝,...,𝑞 (Ω 𝑅 ) with respect to ∥ · ∥ L2 (Ω𝑅 ) .

For brevity, we set   Harm 𝑝,...,𝑞 (𝐾) = Harm 𝑝,...,𝑞 Ωext 𝑅 |𝐾

(12.88)

for every subset 𝐾 of Ωext 𝑅 . 𝑅 is related to the corresponding outer It should be noted that an inner harmonic 𝐻𝑛,𝑘 𝑅 harmonic 𝐻 −𝑛−1,𝑘 in the following way:

 𝑅 𝐻 −𝑛−1,𝑘 (𝑥) =

𝑅 |𝑥|

 2𝑛+1 𝑅 𝐻𝑛,𝑘 (𝑥) =

 2  𝑅 𝑅 𝑅 𝐻𝑛,𝑘 𝑥 . |𝑥| |𝑥| 2

(12.89)

In other words, the outer harmonic is obtainable by the “Kelvin transform” 𝐾 𝑅 relative to the sphere Ω𝑅 from its inner counterpart as follows:   𝑅 𝑅 𝑅 𝑅 𝐻 −𝑛−1,𝑘 (𝑥) = 𝐾 𝑅 𝐻𝑛,𝑘 (𝑥) = 𝐻 (𝑥), |𝑥| 𝑛,𝑘

(12.90)

where the map 𝑥 ↦→ 𝑥 defined by 𝑥=

𝑅2 𝑥, |𝑥| 2

𝑥≠0

(12.91)

574

12 Potentials in Euclidean Space

is called the inversion of R3 relative to the sphere Ω𝑅 . Note that 𝑥 lies on the ray from the origin determined by 𝑥, with |𝑥| 𝑅 = . 𝑅 |𝑥|

(12.92)

It is well known (see, for example, W. Walter (1971)) that the inversion map of R3 relative to the sphere Ω𝑅 is continuous and its own inverse. Moreover, it is the identity on Ω𝑅 . Furthermore, it is easily seen that 𝑅 𝐻𝑛,𝑘 (𝑥) =

  𝑅 𝑅 𝑅 𝐻 −𝑛−1,𝑘 (𝑥) = 𝐾 𝑅 𝐻 −𝑛−1,𝑘 (𝑥), |𝑥|

(12.93)

which demonstrates that it is reasonable to introduce the Kelvin transform for the compactification R3 ∪ {∞} of R3 (by additionally letting 𝑥 = ∞ for 𝑥 = 0 and 𝑥 = 0 for 𝑥 = ∞). Next, we discuss the representations of outer harmonics on spheres of different altitudes. By convention, throughout this work, 𝑅 is the height of the ground level, while 𝑆 describes the satellite level such that 𝑆 > 𝑅 > 0. By virtue of (12.82), we are immediately able to deduce that  𝑛 𝑅 𝑅 𝑟 𝐻 −𝑛−1,𝑘 = 𝐻 −𝑛−1,𝑘 𝑟

(12.94)

for all 𝑟 ≥ 𝑅. Moreover, the radial derivative 𝜕𝑟 admits the following representations 𝑅 𝜕𝑟 𝐻 −𝑛−1,𝑘 =

𝑅 𝜕𝐻 −𝑛−1,𝑘

𝜕𝑟

𝑛+1 𝑅 𝐻 −𝑛−1,𝑘 𝑟  𝑛 𝑛+1 𝑅 𝑟 =− 𝐻 −𝑛−1,𝑘 𝑟 𝑟   𝑛+1 𝑛+1 𝑅 𝑟 =− 𝐻 −𝑛−1,𝑘 . 𝑅 𝑟 =−

Furthermore, for all 𝑟 ≥ 𝑅, we have     𝑛 𝑛+1 𝑛+2 𝑅 𝑅 𝑟 (𝜕𝑟 ) 2 𝐻 −𝑛−1,𝑘 = − − 𝐻 −𝑛−1,𝑘 𝑟 𝑟 𝑟   𝑛+2 (𝑛 + 1)(𝑛 + 2) 𝑅 𝑟 = 𝐻 −𝑛−1,𝑘 . 𝑟 𝑅2

(12.95) (12.96)

(12.97)

These results about “upward continuation” can be arranged in a scheme as shown in Table 12.1.

12.6 Surface Potentials

575

Table 12.1 Outer harmonics characterizing “upward continuation”.

→ − 𝑛+1 𝑆

𝑆 𝐻−𝑛−1,𝑘

Ω𝑆 -level:

↑ Ω𝑅 -level:


𝑅 𝑛 𝑆

·

→ − 𝑛+1 𝑅

𝑅 𝐻−𝑛−1,𝑘

𝑆 𝜕𝑟 𝐻−𝑛−1,𝑘

↑

·


𝑅 𝑛+1 𝑆

𝑅 𝜕𝑟 𝐻−𝑛−1,𝑘

The concise scheme in Table 12.1 connects the outer harmonics and their derivatives at the altitudes 𝑅 (ground level) and 𝑆 (satellite level), respectively. This scheme applies per degree and order. The vertical arrows characterize “upward continuation”, while the horizontal arrows describe transition from the function to its radial derivative.

12.6 Surface Potentials

This section is essentially based on the ideas and concepts developed in W. Freeden (1980a) and its extension in W. Freeden, C. Gerhards (2013). Let 𝑄 be a continuous function on the boundary 𝜕G of a regular region G. Then the functions 𝑈𝑛 : R3 \𝜕G → R, 𝑛 = 1, 2, . . ., defined by 

∫ 𝑈𝑛 (𝑥) = 𝑈𝑛 [𝑄] (𝑥) =

𝑄(𝑦) 𝜕G

𝜕 𝜕𝜈(𝑦)

 𝑛−1

1 𝑑𝜔(𝑦) |𝑥 − 𝑦|

(12.98)

are infinitely often differentiable and satisfy the Laplace equation in G and G 𝑐 = R3 \G (𝜈 is the (unit) normal field pointing into the outer space G 𝑐 ). In addition, the functions 𝑈𝑛 are regular at infinity. The function 𝑈1 given by ∫ 1 𝑈1 (𝑥) = 𝑈1 [𝑄] (𝑥) = 𝑄(𝑦) 𝑑𝜔(𝑦) (12.99) |𝑥 − 𝑦| 𝜕G is called the potential of the single layer on 𝜕G, while 𝑈2 given by ∫ 𝜕 1 𝑈2 (𝑥) = 𝑈2 [𝑄] (𝑥) = 𝑄(𝑦) 𝑑𝜔(𝑦) 𝜕𝜈(𝑦) |𝑥 − 𝑦| 𝜕G

(12.100)

576

12 Potentials in Euclidean Space

is called the potential of the double layer on 𝜕G. For 𝑄 ∈ C(𝜕G), the functions 𝑈𝑛 , 𝑛 = 1, 2, can be continued continuously to the surface 𝜕G, but the limits depend from which parallel surface (inner or outer) the points 𝑥 tend to 𝜕G. On the other hand, the functions 𝑈𝑛 , 𝑛 = 1, 2, also are defined on the surface 𝜕G, i.e., the integrals (12.99), (12.100) exist for 𝑥 ∈ 𝜕G. Furthermore, the integral ∫ 𝜕 1 𝑈1′ (𝑥) = 𝑈1′ [𝑄] (𝑥) = 𝑄(𝑦) 𝑑𝜔(𝑦) (12.101) 𝜕𝜈(𝑥) |𝑥 − 𝑦| 𝜕G exists for all 𝑥 ∈ 𝜕G and can be continued continuously to 𝜕G. However, the integrals do not coincide, in general, with the inner or outer limits of the potentials. From classical potential theory (see, for example, O.D. Kellogg (1929), J. Schauder (1931), R. Leis (1967), W. Walter (1971), S.G. Michlin (1975)) and the references therein), it is known that for all 𝑥 ∈ 𝜕G and 𝑄 ∈ C(𝜕G) lim 𝑈1 (𝑥 ± 𝜏𝜈(𝑥)) = 𝑈1 (𝑥),

(12.102)

𝜏→0 𝜏>0

lim 𝜏→0 𝜏>0

𝜕𝑈1 (𝑥 ± 𝜏𝜈(𝑥)) = ∓2𝜋𝑄(𝑥) + 𝑈1′ (𝑥), 𝜕𝜈(𝑥)

(12.103)

lim 𝑈2 (𝑥 ± 𝜏𝜈(𝑥)) = ±2𝜋𝑄(𝑥) + 𝑈2 (𝑥),

(12.104)

lim (𝑈1 (𝑥 + 𝜏𝜈(𝑥)) − 𝑈1 (𝑥 − 𝜏𝜈(𝑥))) = 0,

(12.105)

𝜏→0 𝜏>0

(limit relations) 𝜏→0 𝜏>0



 𝜕𝑈1 𝜕𝑈1 lim (𝑥 + 𝜏𝜈(𝑥)) − (𝑥 − 𝜏𝜈(𝑥)) = −4𝜋𝑄(𝑥), 𝜏→0 𝜕𝜈(𝑥) 𝜕𝜈(𝑥) 𝜏>0 lim (𝑈2 (𝑥 + 𝜏𝜈(𝑥) − 𝑈2 (𝑥 − 𝜏𝜈(𝑥)) = 4𝜋𝑄(𝑥),

(12.106) (12.107)

𝜏→0 𝜏>0



 𝜕𝑈2 𝜕𝑈2 lim (𝑥 + 𝜏𝜈(𝑥)) − (𝑥 − 𝜏𝜈(𝑥)) = 0 𝜏→0 𝜕𝜈(𝑥) 𝜕𝜈(𝑥) 𝜏>0

(12.108)

(jump relations). In addition, it was indicated by O.D. Kellogg (1929) that the preceding relations hold uniformly with respect to all 𝑥 ∈ 𝜕G. This means (see W. Freeden, C. Gerhards (2013) for detailed proofs) that lim sup |𝑈1 (𝑥 ± 𝜏𝜈(𝑥)) − 𝑈1 (𝑥)| = 0, 𝜏→0 𝜏>0

𝑥 ∈𝜕G

(12.109)

12.6 Surface Potentials

577

𝜕𝑈1 ′ lim sup (𝑥 ± 𝜏𝜈(𝑥)) ± 2𝜋𝑄(𝑥) − 𝑈1 (𝑥) = 0, 𝜏→0 𝜕𝜈(𝑥)

(12.110)

𝑥 ∈𝜕G

𝜏>0

lim sup 𝑈2 (𝑥 ± 𝜏𝜈(𝑥)) ∓ 2𝜋𝑄(𝑥) − 𝑈2 (𝑥) = 0 𝜏→0 𝜏>0

(12.111)

𝑥 ∈Ω 𝑅

and lim sup 𝑈1 (𝑥 + 𝜏𝜈(𝑥)) − 𝑈1 (𝑥 − 𝜏𝜈(𝑥)) = 0, 𝜏→0 𝜏>0

𝜕𝑈1 𝜕𝑈1 lim sup (𝑥 + 𝜏𝜈(𝑥)) − (𝑥 − 𝜏𝜈(𝑥)) + 4𝜋𝑄(𝑥) = 0, 𝜏→0 𝜕𝜈(𝑥) 𝜕𝜈(𝑥) 𝜏>0 𝑥 ∈𝜕G lim sup 𝑈2 (𝑥 + 𝜏𝜈(𝑥)) − 𝑈2 (𝑥 − 𝜏𝜈(𝑥)) − 4𝜋𝑄(𝑥) = 0, 𝜏→0 𝜏>0

(12.112)

𝑥 ∈𝜕G

(12.113) (12.114)

𝑥 ∈Ω 𝑅

𝜕𝑈2 𝜕𝑈2 lim sup (𝑥 + 𝜏𝜈(𝑥)) − (𝑥 − 𝜏𝜈(𝑥)) = 0 . 𝜏→0 𝜕𝜈(𝑥) 𝜕𝜈(𝑥) 𝜏>0 𝑥 ∈Ω 𝑅

(12.115)

Here we have written, as usual, 𝜕𝑈 𝑥 (𝑥 ± 𝜏𝜈(𝑥)) = · (∇𝑈)(𝑥 ± 𝜏𝜈(𝑥)) . 𝜕𝜈(𝑥) 𝑅

(12.116)

Furthermore, by means of functional analysis, W. Freeden (1980a) (see also W. Freeden, C. Mayer (2003)) was able to show that the limit and jump relations also hold true in L2 -topology. In more detail, ∫ lim 𝜏→0 𝜏>0

∫ lim 𝜏→0 𝜏>0

𝜕G

 1/2 = 0,

(12.117)

𝜕G

! 1/2 2 𝜕𝑈1 ′ = 0, 𝜕𝜈(𝑥) (𝑥 ± 𝜏𝜈(𝑥)) ± 2𝜋𝐹 (𝑥) − 𝑈1 (𝑥) 𝑑𝜔(𝑥)

∫

𝑈2 (𝑥 ± 𝜏𝜈(𝑥)) ∓ 2𝜋𝑄(𝑥) − 𝑈2 (𝑥) 2 𝑑𝜔(𝑥)

lim 𝜏→0 𝜏>0

|𝑈1 (𝑥 ± 𝜏𝜈(𝑥)) − 𝑈1 (𝑥)| 2 𝑑𝜔(𝑥)

(12.118)

 1/2 =0

(12.119)

𝜕G

and ∫ lim 𝜏→0 𝜏>0

𝑈1 (𝑥 + 𝜏𝜈(𝑥)) − 𝑈1 (𝑥 − 𝜏𝜈(𝑥)) 2 𝑑𝜔(𝑥)

 1/2 = 0,

(12.120)

𝜕G

! 1/2 2 𝜕𝑈1 𝜕𝑈 1 lim (𝑥 + 𝜏𝜈(𝑥)) − (𝑥 − 𝜏𝜈(𝑥)) + 4𝜋𝑄(𝑥) 𝑑𝜔(𝑥) = 0, 𝜏→0 𝜕𝜈(𝑥) 𝜕G 𝜕𝜈(𝑥) 𝜏>0 (12.121) ∫  1/2 2 𝑈2 (𝑥 + 𝜏𝜈(𝑥)) − 𝑈2 (𝑥 − 𝜏𝜈(𝑥)) − 4𝜋𝑄(𝑥) 𝑑𝜔(𝑥) lim = 0, (12.122) ∫

𝜏→0 𝜏>0

𝜕G

578

12 Potentials in Euclidean Space

! 1/2 2 𝜕𝑈2 𝜕𝑈 2 (𝑥 + 𝜏𝜈(𝑥)) − (𝑥 − 𝜏𝜈(𝑥)) 𝑑𝜔(𝑥) =0. 𝜕𝜈(𝑥) 𝜕G 𝜕𝜈(𝑥)

∫ lim 𝜏→0 𝜏>0

(12.123)

12.7 Boundary-Value Problems for the Laplace Operator

The task of solving a boundary-value problem for the Laplace equation Δ𝑈 = 0 from given data on the boundary 𝜕G of a regular region G arises in many applications (e.g., gravitation, magnetics, solid Earth mechanics, electromagnetism). Of particular importance is the Dirichlet (resp. Neumann) boundary-value problem, i.e., the determination of 𝑈 from given potential values (resp. normal derivatives) on the boundary. Finding the solution in the interior/exterior space of a geoscientifically relevant boundary (such as, e.g., sphere, ellipsoid, geoidal surface, (actual) Earth’s surface) is of significance in all geosciences. Formulation and Uniqueness. We begin with the formulation of the four classical boundary-value problems. Interior Dirichlet Problem (IDP): Suppose that a function 𝐹 of class C (0) (𝜕G) is given. We are looking for a function 𝑉 : G → R satisfying the following conditions: (i) 𝑉 is of class C (2) (G) ∩ C (0) (G), (ii) 𝑉 satisfies Laplace’s equation Δ𝑉 = 0 in G, (iii) 𝑉 − |𝜕G = 𝐹 (i.e., 𝑉 − (𝑥) = lim 𝑉 (𝑥 − 𝜏𝜈(𝑥)) = 𝐹 (𝑥), 𝑥 ∈ 𝜕G), where 𝜈 is the 𝜏→0+

normal field directed into the exterior of G). Interior Neumann Problem (INP): Suppose that 𝐹 ∈ C (0) (𝜕G), with

∫ 𝜕G

𝐹 (𝑥) 𝑑𝜔(𝑥) =

0, is given. We are looking for a function 𝑉 : G → R satisfying the following conditions: (i) 𝑉 is of class C (2) (G) ∩ C (1) (G), (ii) 𝑉 satisfies Laplace’s equation Δ𝑉 = 0 in G, (iii)



𝜕𝑉 − 𝜕𝜈 𝜕G

= 𝐹 (i.e.,

𝜕𝑉 − 𝜕𝜈

(𝑥) = lim 𝜈(𝑥) · (∇𝑉)(𝑥 − 𝜏𝜈(𝑥)) = 𝐹 (𝑥), 𝑥 ∈ 𝜕G, 𝜏→0+

where 𝜈 is the normal field directed into the exterior of G).

12.7 Boundary-Value Problems for the Laplace Operator

579

Exterior Dirichlet Problem (EDP): Suppose that 𝐹 ∈ C (0) (𝜕G) is given. We are looking for a function 𝑉 : G 𝑐 → R, G 𝑐 = R3 \G, satisfying the following conditions: (i) 𝑉 is of class C (2) (G 𝑐 ) ∩ C (0) (G 𝑐 ), (ii) 𝑉 satisfies Laplace’s equation Δ𝑉 = 0 in G 𝑐 , (iii) 𝑉 is regular at infinity, (iv) 𝑉 + |𝜕G = 𝐹 (i.e., 𝑉 + (𝑥) = lim 𝑉 (𝑥 + 𝜏𝜈(𝑥)) = 𝐹 (𝑥), 𝑥 ∈ 𝜕G, where 𝜈 is the 𝜏→0+

normal field directed into the exterior of G). Exterior Neumann Problem (ENP): Suppose that 𝐹 ∈ C (0) (𝜕G) is given. We are looking for a function 𝑉 : G 𝑐 → R satisfying the following conditions: (i) 𝑉 is of class C (2) (G 𝑐 ) ∩ C (1) (G 𝑐 ), (ii) 𝑉 satisfies Laplace’s equation Δ𝑉 = 0 in G 𝑐 , (iii) 𝑉 is regular at infinity, (iv)



𝜕𝑉 + 𝜕𝜈 𝜕G

= 𝐹 (i.e.,

𝜕𝑉 + 𝜕𝜈

(𝑥) = lim 𝜈(𝑥) · (∇𝑉)(𝑥 + 𝜏𝜈(𝑥)) = 𝐹 (𝑥), 𝑥 ∈ 𝜕G, 𝜏→0+

where 𝜈 is the normal field directed into the exterior of G). Remark 12.18 The indices “+” and “−” just emphasize if we approach the boundary 𝜕G frome the inside or the outside of G. When it is obvious if we are in the exterior or the interior case, we often just write 𝑉 |𝜕G or 𝜕𝑉 𝜕𝜈 |𝜕G. Then the normal 𝜈 is meant to be the actual outer normal, i.e., it is directed into the exterior of G in case of interior problems in G, and it is directed into the exterior of G 𝑐 (i.e., the interior of G) in case of exterior problems in G 𝑐 . In what follows the well-posedness of the boundary-value problems, i.e., uniqueness, existence, and continuous dependence of the boundary data should be discussed in more detail. We begin with the uniqueness. Lemma 12.19 A solution of (IDP) is uniquely determined. Proof Let 𝑉 and 𝑉˜ satisfy (IDP). Then the difference function 𝐷 : G → R, 𝐷 = ˜ is of class C (2) (G) ∩ C (0) (G), satisfies Laplace’s equation Δ𝐷 = 0 in G, 𝑉 − 𝑉, and 𝐷 − |𝜕G = 0. Under these assumptions the Maximum/Minimum Principle (see Theorem 12.7) tells us that sup 𝑥 ∈ G |𝐷 (𝑥)| ≤ sup 𝑥 ∈𝜕G |𝐷 (𝑥)| = 0. However, this means 𝐷 = 0 in G.

□

580

12 Potentials in Euclidean Space

Lemma 12.20 A solution of (INP) is uniquely determined (up to an additive constant). Proof Clearly, the difference 𝐷 : G → R of two solutions 𝑉, 𝑉˜ is of class C (2) (G) ∩ − C (1) (G), satisfies Laplace’s equation Δ𝐷 = 0 in G. Moreover, we have 𝜕𝐷 𝜕𝜈 |𝜕G = 0. Under these properties the Interior First Green’s Theorem (cf. Theorem 2.2) shows that ∫ ∫ 𝜕𝐷 |∇𝐷 (𝑥)| 2 𝑑𝑥 = 𝐷 (𝑥) (𝑥) 𝑑𝜔(𝑥) = 0. (12.124) 𝜕𝜈 G 𝜕G | {z } =0

Thus, ∇𝐷 = 0 in G such that 𝐷 = const on G. In other words, together with 𝑉 also the function 𝑉˜ = 𝑉 + 𝐶 is a solution of (INP). □ Lemma 12.21 A solution of (EDP) is uniquely determined. Proof We set G𝑅𝑐 = G 𝑐 ∩ Ωint 𝑅 (for 𝑅 sufficiently large such that sup 𝑥 ∈ G |𝑥| < 𝑅). Then, from the Maximum/Minimum Principle (see Theorem 12.7), we obtain for the difference 𝐷 of two solutions sup 𝑥 ∈ G 𝑐 |𝐷 (𝑥)| ≤ sup 𝑥 ∈𝜕G 𝑐 |𝐷 (𝑥)|. Now, we know 𝑅 𝑅 𝐷 + |𝜕G = 0 and, because of the regularity at infinity, we have sup 𝑥 ∈Ω𝑅 |𝐷 (𝑥)| → 0 for 𝑅 → ∞. This implies the uniqueness. □ Lemma 12.22 A solution of (ENP) is uniquely determined. Proof For the difference 𝐷 of two solutions it follows that ∫ ∫ ∫ 𝜕𝐷 𝜕𝐷 |∇𝐷 (𝑥)| 2 𝑑𝑥 = 𝐷 (𝑥) (𝑥) 𝑑𝜔(𝑥) + 𝐷 (𝑥) (𝑥) 𝑑𝜔(𝑥). (12.125) 𝑐 𝜕𝜈 𝜕𝜈 G𝑅 𝜕G Ω𝑅 | {z } =0

The regularity of 𝐷 at infinity yields ∫ 𝜕𝐷 𝐷 (𝑥) (𝑥) 𝑑𝜔(𝑥) = 𝑂 (1), 𝑅 → ∞. (12.126) 𝜕𝜈 Ω𝑅 ∫ Consequently, G 𝑐 |∇𝐷 (𝑥)| 2 𝑑𝑥 = 0 so that 𝐷 = const. in G 𝑐 . Thus, the regularity at infinity gives 𝐷 = 0 in G 𝑐 .

□

Already at this stage, the Maximum/Minimum Principle enables us to guarantee the continuous dependence of the solution on boundary data of Dirichlet’s type. Lemma 12.23 Let 𝑉, 𝑉˜ : G → R satisfy the inner Dirichlet problem (IDP) with ˜ respectively, such that 𝑉 − |𝜕G = 𝐹 and 𝑉˜ − |𝜕G = 𝐹, sup |𝐹 (𝑥) − 𝐹˜ (𝑥)| ≤ 𝜀 𝑥 ∈𝜕G

(12.127)

12.7 Boundary-Value Problems for the Laplace Operator

581

for some 𝜀 > 0. Then, sup |𝑉 (𝑥) − 𝑉˜ (𝑥)| ≤ sup |𝑉 (𝑥) − 𝑉˜ (𝑥)| = sup |𝐹 (𝑥) − 𝐹˜ (𝑥)| ≤ 𝜀. 𝑥∈ G

𝑥 ∈𝜕G

(12.128)

𝑥 ∈𝜕G

Lemma 12.24 Let 𝑉, 𝑉˜ : G 𝑐 → R satisfy the exterior Dirichlet problem (EDP) with ˜ respectively, such that 𝑉 |𝜕G = 𝐹 and 𝑉˜ |𝜕G = 𝐹, sup |𝐹 (𝑥) − 𝐹˜ (𝑥)| ≤ 𝜀

(12.129)

𝑥 ∈𝜕G

for some 𝜀 > 0. Then, because of the regularity at infinity, sup |𝑉 (𝑥) − 𝑉˜ (𝑥)| ≤ sup |𝑉 (𝑥) − 𝑉˜ (𝑥)| = sup |𝐹 (𝑥) − 𝐹˜ (𝑥)| ≤ 𝜀. 𝑥 ∈ G𝑐

𝑥 ∈𝜕G

(12.130)

𝑥 ∈𝜕G

Exterior Boundary-Value Problems for a Ball. Next, we come to boundary-value problems corresponding to spherical boundaries. The explicit formulas for exterior problems play a significant role in geophysical and geodetic reality. First, we deal with a method of solving the exterior Dirichlet problem (EDP) by use of a socalled Green’s function. In a subsequent step we apply the Kelvin transform to solve the interior Dirichlet problem (IDP). Second, we are concerned with the Neumann boundary-value problem corresponding to spherical boundaries. Definition 12.25 𝐺 (EDP) (Δ; ·, ·) is called Green’s function for R3 \Ωint 𝑅 with respect to the exterior Dirichlet problem (EDP), if the following conditions are valid: (i) for every 𝑥 ∈ R3 \Ωint 𝑅, 𝑦 ↦→ Φ(𝑥, 𝑦) = 𝐺 (EDP) (Δ; 𝑥, 𝑦) − 𝐺 (Δ; |𝑥 − 𝑦|)

(12.131)

is continuously differentiable on R3 \Ωint 𝑅 , twice continuously differentiable in 3 int R3 \Ωint 𝑅 , harmonic in R \Ω 𝑅 , and regular at infinity,

(ii) for every 𝑥 ∈ R3 \Ωint 𝑅, 𝐺 (EDP) (Δ; 𝑥, 𝑦) = 0, 𝑦 ∈ Ω𝑅 .

(12.132)

3 int Clearly, for every 𝑥 ∈ R3 \Ωint 𝑅 , the Green’s function for R \Ω 𝑅 with respect to the exterior Dirichlet problem (EDP) is uniquely determined. The importance of the Green’s function is based on the following fact: Suppose that a function 3 int 𝑈 : R3 \Ωint 𝑅 → R is continuously differentiable on R \Ω 𝑅 , twice continuously

582

12 Potentials in Euclidean Space

3 int differentiable in R3 \Ωint 𝑅 , harmonic in R \Ω 𝑅 , and regular at infinity. Then, from

Theorem 12.17, we obtain for every 𝑥 ∈ R3 \Ωint 𝑅, 

 𝜕𝑈 𝜕 (𝐺 (Δ; |𝑥 − 𝑦|)) 𝑑𝜔(𝑦), (𝑦)𝐺 (Δ; |𝑥 − 𝑦|) − 𝑈 (𝑦) 𝜕𝜈(𝑦) Ω 𝑅 𝜕𝜈 (12.133) where 𝜈 is the normal to Ω𝑅 pointing into Ωint . Moreover, from the Exterior Second 𝑅 Green’s Theorem, we are able to deduce that ∫

𝑈 (𝑥) =

∫

© ª ­ ® ­Δ 𝑦 𝑈 (𝑦) Φ(𝑥, 𝑦) − 𝑈 (𝑦) Δ 𝑦 Φ(𝑥, 𝑦) ® 𝑑𝑦 ­| {z } ® R3 \Ωint | {z } 𝑅 =0 =0 « ¬ ∫  𝜕𝑈 𝜕 = (𝑦)Φ(𝑥, 𝑦) − 𝑈 (𝑦) Φ(𝑥, 𝑦) 𝑑𝜔(𝑦) 𝜕𝜈(𝑦) Ω 𝑅 𝜕𝜈

0=

(12.134)

holds for every 𝑥 ∈ R3 \Ωint 𝑅 . Summing up (12.133) and (12.134), we therefore find ∫

© 𝜕𝑈 ª 𝜕 (EDP) ­ (Δ; 𝑥, 𝑦) −𝑈 (𝑦) 𝐺 (EDP) (Δ; 𝑥, 𝑦) ®® 𝑑𝜔(𝑦), ­ 𝜕𝜈 (𝑦) 𝐺 𝜕𝜈(𝑦) | {z } Ω𝑅 « ¬ =0 (12.135) where 𝜈 is the unit normal field on Ω𝑅 pointing into Ωint . This leads us to the 𝑅 3 int following statement: Let 𝑈 : R \Ω𝑅 → R satisfy the aforementioned assumptions with 𝑈|Ω𝑅 = 𝐹, 𝐹 ∈ C (0) (Ω𝑅 ). Then 𝑈 allows the integral representation ∫ 𝜕 𝑈 (𝑥) = − 𝐹 (𝑦) 𝐺 (EDP) (Δ; 𝑥, 𝑦) 𝑑𝜔(𝑦). (12.136) 𝜕𝜈(𝑦) Ω𝑅 𝑈 (𝑥) =

For the explicit construction of the Green’s function for R3 \Ωint 𝑅 with respect to the exterior Dirichlet problem (EDP) we have to find, for every 𝑥 ∈ R3 \Ωint 𝑅 , the function 𝑦 ↦→ Φ(𝑥, 𝑦) satisfying the following properties: (i) Φ(𝑥, ·) is continuously differentiable on R3 \Ωint 𝑅 , twice continuously differen3 int tiable in R3 \Ωint 𝑅 , harmonic in R \Ω 𝑅 , and regular at infinity,

(ii) On the sphere Ω𝑅 , Φ(𝑥, ·) coincides with −𝐺 (Δ; |𝑥 − ·|). For every 𝑥 ∈ R3 \Ωint 𝑅 , this problem is again an exterior Dirichlet problem (EDP). It can be solved by observing that, for all 𝑦 ∈ Ω𝑅 , the identity (12.46) holds true. Consequently it follows that

12.7 Boundary-Value Problems for the Laplace Operator

583

  2 𝑅 𝑅 (EDP) 𝐺 (Δ; 𝑥, 𝑦) = 𝐺 (Δ; |𝑥 − 𝑦|) − 𝐺 Δ; 2 𝑥 − 𝑦 |𝑥| |𝑥|

(12.137)

represents the Green function for R3 \Ωint 𝑅 with respect to the exterior Dirichlet problem (EDP). In connection with (12.136) our approach yields ∫ 𝑈 (𝑥) = −

𝐹 (𝑦) Ω𝑅

𝜕 𝜕𝜈(𝑦)

© 1 ­ ­ 4𝜋 «

© 1 ­ ­ |𝑥 − 𝑦| − «

|𝑥| 𝑅

ªª 1 ®®®® 𝑑𝜔(𝑦). 𝑅2 | 𝑥 | 2 𝑥 − 𝑦 ¬¬

(12.138)

We have to calculate the (inner) normal derivative 𝜕 𝑦 𝐺 (EDP) (Δ; 𝑥, 𝑦) = 𝜈(𝑦) · ∇ 𝑦 𝐺 (EDP) (Δ; 𝑥, 𝑦) = − · ∇ 𝑦 𝐺 (EDP) (Δ; 𝑥, 𝑦). 𝜕𝜈(𝑦) 𝑅 (12.139) 2 An easy manipulation yields, with 𝑥ˇ 𝑅 = |𝑅𝑥 | 2 𝑥, ∇𝑦

1 1 1 − 4𝜋 |𝑥 − 𝑦| 4𝜋

1 |𝑥| 𝑅

( 𝑥˜ 𝑅 − 𝑦)

! =

| 𝑥 |2 𝑦 𝑅2

−𝑦

4𝜋|𝑥 − 𝑦| 3

=

1 𝑅 2 − |𝑥| 2 𝑦. 4𝜋𝑅 2 |𝑥 − 𝑦| 3

(12.140)

Combining (12.139) and (12.140) we obtain the so-called Abel-Poisson formula ∫ 1 |𝑥| 2 − 𝑅 2 𝑈 (𝑥) = 𝐹 (𝑦) 𝑑𝜔(𝑦). (12.141) 4𝜋𝑅 Ω𝑅 |𝑥 − 𝑦| 3 All in all, the Abel-Poisson formula (12.141) solves the exterior Dirichlet problem for 3 int 𝑈 : R3 \Ωint 𝑅 → R that is continuously differentiable on R \Ω 𝑅 , twice continuously 3 int differentiable in R3 \Ωint 𝑅 , harmonic in R \Ω 𝑅 , and regular at infinity, such that (0) 𝑈|Ω𝑅 = 𝐹, 𝐹 ∈ C (Ω𝑅 ). From Theorem 4.46, we know that ∫ 1 𝑟 2 − 𝑅2 𝐹 (𝑅𝜉) = lim 𝐹 (𝑅𝜂) 𝑑𝜔(𝑅𝜂) (12.142) 3 𝑟→𝑅+ 4𝜋𝑅 Ω 𝑅 (𝑟 2 + 𝑅 2 − 2𝑅𝑟 (𝜉 · 𝜂)) 2

holds true for all 𝐹 ∈ C (0) (Ω𝑅 ). In consequence, even though (12.141) has been (1) (R3 \Ωint ), this formula gives derived for solutions 𝑈 of class C (2) (R3 \Ωint 𝑅 𝑅) ∩C the unique solution of the Dirichlet problem corresponding to continuous boundaryvalues on the sphere Ω𝑅 . Theorem 12.26 (Exterior Abel-Poisson Formula) Let 𝐹 be of class C (0) (Ω𝑅 ). Then the function 𝑈 : R3 \Ωint 𝑅 → R given by ∫ 𝑈 (𝑥) = 𝐹 (𝑦) 𝐷 (𝑥, 𝑦) 𝑑𝜔(𝑦), 𝑥 ∈ R3 \Ωint (12.143) 𝑅, Ω𝑅

584

12 Potentials in Euclidean Space

with the exterior Abel-Poisson kernel function 1 |𝑥| 2 − 𝑅 2 4𝜋𝑅 |𝑥 − 𝑦| 3

𝐷 (𝑥, 𝑦) =

(12.144)

is the unique solution of the exterior Dirichlet boundary-value problem (EDP): 3 int (i) 𝑈 is continuous in R3 \Ωint 𝑅 , twice continuously differentiable in R \Ω 𝑅 , har3 int monic on R3 \Ωint 𝑅 , i.e., Δ𝑈 = 0 in R \Ω 𝑅 , and regular at infinity,

(ii) 𝑈 + | Ω𝑅 = 𝐹. Remark 12.27 By use of the spherical harmonic expansion for the Abel-Poisson kernel (cf. Lemma 4.38), 𝑈 can be represented by a Fourier series expansion in terms of outer harmonics 𝑈=

∞ 2𝑛+1 ∑︁ ∑︁

𝐹

∧L2 (Ω

𝑅)

𝑅 (𝑛, 𝑘)𝐻 −𝑛−1,𝑘 ,

(12.145)

𝑛=0 𝑘=1

where the Fourier coefficients are given by ∫ ∧ 𝑅 𝐹 L2 (Ω𝑅 ) (𝑛, 𝑘) = 𝐹 (𝑦)𝐻 −𝑛−1,𝑘 (𝑦) 𝑑𝜔(𝑦),

(12.146)

Ω𝑅

𝑛 = 0, 1, . . ., 𝑘 = 1, . . . , 2𝑛 + 1, and the series expansion is absolutely and uniformly 3 int convergent on each set K with K ⋐ R3 \Ωint 𝑅 , i.e., K ⊂ R \Ω 𝑅 with dist(K, ΩR ) > 0. Next we turn over to the interior Dirichlet problem corresponding to continuous boundary values on the sphere Ω𝑅 . Theorem 12.28 (Interior Abel-Poisson Formula) Let 𝐹 be of class C (0) (Ω𝑅 ). Then the function 𝑉 : Ωint 𝑅 → R given by ∫ 𝑉 (𝑥) = 𝐹 (𝑦) 𝐷 (𝑥, 𝑦) 𝑑𝜔(𝑦), 𝑥 ∈ Ωint (12.147) 𝑅 Ω𝑅

with the interior Abel-Poisson kernel function 𝐷 (𝑥, 𝑦) =

1 𝑅 2 − |𝑥| 2 , 4𝜋𝑅 |𝑥 − 𝑦| 3

(12.148)

is the unique solution of the interior Dirichlet boundary-value problem (IDP):

12.7 Boundary-Value Problems for the Laplace Operator

585

int (i) 𝑉 is continuous in Ωint 𝑅 and twice continuously differentiable in Ω 𝑅 , harmonic int int in Ω𝑅 , i.e., Δ𝑉 = 0 in Ω𝑅 ,

(ii) 𝑉 − | Ω𝑅 = 𝐹. Furthermore, 𝑉 can be represented by a Fourier series expansion in terms of outer harmonics ∞ 2𝑛+1 ∑︁ ∑︁ ∧ 𝑅 𝑉= 𝐹 L2 (Ω𝑅 ) (𝑛, 𝑘) 𝐻𝑛,𝑘 , (12.149) 𝑛=0 𝑘=1

where the Fourier coefficients are given by ∫ ∧L2 (Ω ) 𝑅 𝑅 𝐹 (𝑛, 𝑘) = 𝐹 (𝑦)𝐻𝑛,𝑘 (𝑦) 𝑑𝜔(𝑦),

(12.150)

Ω𝑅

𝑛 ∈ N0 , 𝑘 = 1, . . . , 2𝑛 + 1, and the series expansion is absolutely and uniformly convergent on each set K with K ⋐ R3 \Ωint 𝑅. In order to verify the solution (12.147) of the Dirichlet problem for Ωint 𝑅 we use the Kelvin transform (see Theorem 12.11). We follow a two step strategy: First we apply the Kelvin transform 𝑥ˇ 𝑅 ↦→ 𝑥, 𝑥ˇ 𝑅 ∈ R3 \Ωint 𝑅 and introduce the new (unknown) func𝑅 tion 𝑈 : Ωint \{0} → R defined by 𝑈 (𝑥) = | 𝑥 | 𝑉 ( 𝑥ˇ 𝑅 ). We know that the singularity 𝑅 is removable. Second, we solve the corresponding exterior Dirichlet problem for 𝑈 corresponding to the boundary data 𝑈 (𝑥) = |𝑅𝑥 | 𝑉 ( 𝑥ˇ 𝑅 ), 𝑥 ∈ Ω𝑅 , by means of the known Abel-Poisson formula and return to the original coordinates in 𝑥 by inverting the Kelvin transform. Setting 𝐹 = 1 in (12.147) we find ∫ 1 𝑅 2 − |𝑥| 2 1= 𝑑𝜔(𝑦), 4𝜋𝑅 Ω𝑅 |𝑥 − 𝑦| 3

𝑥 ∈ Ωint 𝑅 .

(12.151)

Next we discuss the exterior Neumann problem (ENP). We start with Definition 12.29 𝐺 (ENP) (Δ; ·, ·) is called Green function for R3 \Ωint 𝑅 with respect to the exterior Neumann problem (ENP), if the following conditions are valid: (i) for every 𝑥 ∈ R3 \Ωint 𝑅, 𝑦 ↦→ Φ(𝑥, 𝑦) = 𝐺 (ENP) (Δ; 𝑥, 𝑦) − 𝐺 (Δ; |𝑥 − 𝑦|)

(12.152)

586

12 Potentials in Euclidean Space

is continuously differentiable on R3 \Ωint 𝑅 , twice continuously differentiable in 3 int R3 \Ωint 𝑅 , harmonic in R \Ω 𝑅 , and regular at infinity,

(ii) for every 𝑥 ∈ R3 \Ωint 𝑅, 𝜕 𝐺 (ENP) (Δ; 𝑥, 𝑦) = 0 𝜕𝜈(𝑦)

(12.153)

𝜕 𝜕 Φ(𝑥, 𝑦) = − 𝐺 (Δ; |𝑥 − 𝑦|) 𝜕𝜈(𝑦) 𝜕𝜈(𝑦)

(12.154)

i.e.,

on the sphere Ω𝑅 , where 𝜈 is the inner (unit) normal field to Ω𝑅 . 3 int For every 𝑥 ∈ R3 \Ωint 𝑅 , the Green’s function for R \Ω 𝑅 with respect to the exterior Neumann problem (EDP) is uniquely determined. Suppose that a function 3 int 𝑈 : R3 \Ωint 𝑅 → R is continuously differentiable on R \Ω 𝑅 , twice continuously 3 int differentiable in R3 \Ωint 𝑅 , harmonic in R \Ω 𝑅 , and regular at infinity. Then, from

analogous arguments as in the Dirichlet case, we obtain for every 𝑥 ∈ R3 \Ωint 𝑅 © ª ­ 𝜕𝑈 ® 𝜕 ­ ® (ENP) (ENP) 𝑈 (𝑥) = (𝑦)𝐺 (Δ; 𝑥, 𝑦) − 𝑈 (𝑦) 𝐺 (Δ; 𝑥, 𝑦) ® 𝑑𝜔(𝑦). ­ ­ ® 𝜕𝜈 𝜕𝜈(𝑦) Ω𝑅 ­ | {z }® =0 « ¬ (12.155) This leads us to the following statement: Let 𝑈 : R3 \Ωint → R satisfy the aforemen𝑅 (0) (Ω ). Then 𝑈 allows the integral tioned assumptions with 𝜕𝑈 = 𝐹, 𝐹 ∈ C Ω 𝑅 𝑅 𝜕𝜈 representation ∫ 𝑈 (𝑥) = 𝐹 (𝑦) 𝐺 (ENP) (Δ; 𝑥, 𝑦) 𝑑𝜔(𝑦), 𝑥 ∈ R3 \Ωint (12.156) 𝑅 . ∫

Ω𝑅

An elementary calculation shows that |𝑥||𝑦| + |𝑥| 𝑦 − 𝑅 1 + Φ(𝑥, 𝑦) = ln 2 4𝜋|𝑥| 𝑦 − |𝑅𝑥 | 2 𝑥 4𝜋𝑅 |𝑥||𝑦| + |𝑥| 𝑦 −



− 𝑅2



+ 𝑅2

𝑅2 𝑥 | 𝑥 |2 𝑅2 𝑥 | 𝑥 |2

(12.157)

satisfies the conditions (i) and (ii) of Definition 12.29. In polar coordinates 𝑥 = |𝑥|𝜉, 𝑦 = |𝑦|𝜂, 𝜉, 𝜂 ∈ Ω, (12.157) can be rewritten as follows: 𝑅

Φ(𝑥, 𝑦) = 4𝜋

√︁

𝑅4

+

|𝑥| 2 |𝑦| 2

− 2𝑅 2 |𝑥||𝑦|𝜉 · 𝜂

(12.158)

12.7 Boundary-Value Problems for the Laplace Operator

587

√︁ |𝑥||𝑦| + 𝑅 4 + |𝑥| 2 |𝑦| 2 − 2𝑅 2 |𝑥||𝑦|𝜉 · 𝜂 − 𝑅 2 1 + ln . √︁ 4𝜋𝑅 |𝑥||𝑦| + 𝑅 4 + |𝑥| 2 |𝑦| 2 − 2𝑅 2 |𝑥||𝑦|𝜉 · 𝜂 + 𝑅 2 This leads us to the following representation of the solution for the exterior Neumann problem (ENP). Theorem 12.30 (Exterior Neumann Problem) Let 𝐹 be of class C (0) (Ω𝑅 ). Then the function 𝑈 : R3 \Ωint 𝑅 → R given by ∫ 𝑅 𝑈 (𝑥) = 𝐹 (𝑦) 𝑁 (𝑥, 𝑦)𝑑𝜔(𝜂), 𝑥 ∈ R3 \Ωint (12.159) 𝑅, 4𝜋 Ω𝑅 with the Neumann kernel function   2𝑅 |𝑥| + |𝑥 − 𝑦| − 𝑅 𝑁 (𝑥, 𝑦) = + ln |𝑥 − 𝑦| |𝑥| + |𝑥 − 𝑦| + 𝑅

(12.160)

is the unique solution of the exterior Neumann boundary-value problem: (i) 𝑈 is continuously differentiable in R3 \Ωint 𝑅 and twice continuously differentiable 3 3 int int in R3 \Ωint 𝑅 , harmonic in R \Ω 𝑅 , i.e., Δ𝑈 = 0 in R \Ω 𝑅 , and regular at infinity,

(ii)



𝜕𝑈 + 𝜕𝜈 Ω 𝑅

= 𝐹.

Furthermore, 𝑈 can be represented by a Fourier series expansion in terms of outer harmonics ∞ 2𝑛+1 ∑︁ ∑︁ 𝑅 ∧ 𝑅 𝑈= 𝐹 L2 (Ω𝑅 ) (𝑛, 𝑘)𝐻 −𝑛−1,𝑘 , (12.161) 𝑛 + 1 𝑛=0 𝑘=1 where the Fourier coefficients are given by (12.146), and the series expansion is absolutely and uniformly convergent on each set K with K ⋐ Ωint 𝑅. It should be noted that the inner Neumann problem (INP) does not play a significant role in a geomathematically reflected potential theory. Nevertheless, the introduction of its Green’s function is of interest from mathematical point of view. To explain this approach in more detail we again start from the representation formula  ∫  𝜕𝑈 𝜕 (INP) (INP) 𝑈 (𝑥) = (𝑦)𝐺 (Δ; 𝑥, 𝑦) − 𝑈 (𝑦) 𝐺 (Δ; 𝑥, 𝑦) 𝑑𝜔(𝑦) 𝜕𝜈(𝑦) Ω 𝑅 𝜕𝜈 (12.162) for every 𝑥 ∈ Ωint , where 𝜈 is the outward directed (unit) normal field to Ω 𝑅 . If we 𝑅 made the attempt to assume that 𝜕𝜈𝜕( 𝑦) 𝐺 (INP) (Δ; 𝑥, 𝑦) = 0 for all 𝑦 ∈ Ω𝑅 , then we would have

588

12 Potentials in Euclidean Space

∫ Ω𝑅

𝜕 Φ(𝑥, 𝑦) 𝑑𝜔(𝑦) = − 𝜕𝜈(𝑦)

∫ Ω𝑅

𝜕 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝜔(𝑦) = 1 𝜕𝜈(𝑦)

for every 𝑥 ∈ Ωint 𝑅 . This contradicts the necessary condition ∫ 𝜕 Φ(𝑥, 𝑦) 𝑑𝜔(𝑦) = 0 Ω 𝑅 𝜕𝜈(𝑦)

(12.163)

(12.164)

for every 𝑥 ∈ Ωint 𝑅 . In other words, we may not expect to represent 𝑈 in the form ∫ 𝜕𝑈 𝑈 (𝑥) = (𝑦) 𝐺 (INP) (Δ; 𝑥, 𝑦) 𝑑𝜔(𝑦). (12.165) 𝜕𝜈 Ω𝑅 Consequently, we have to modify the usual procedure. Instead of 𝜕 𝜕 Φ(𝑥, 𝑦) = − 𝐺 (Δ; 𝑥, 𝑦) = 0 𝜕𝜈(𝑦) 𝜕𝜈(𝑦)

(12.166)

for all 𝑦 ∈ Ω𝑅 , we require 𝜕 𝜕 1 Φ(𝑥, 𝑦) = − 𝐺 (Δ; |𝑥 − 𝑦|) − , 𝑦 ∈ Ω𝑅 , 𝑥 ∈ Ωint 𝑅 . (12.167) 𝜕𝜈(𝑦) 𝜕𝜈(𝑦) 4𝜋𝑅 2 In this case the necessary condition is fulfilled, and we are led to Definition 12.31 𝐺 (INP) (Δ; ·, ·) is called Green’s function for Ωint 𝑅 with respect to the interior Neumann problem (INP), if the following conditions are valid: (i) for every 𝑥 ∈ R3 \Ωint 𝑅, 𝑦 ↦→ Φ(𝑥, 𝑦) = 𝐺 (INP) (Δ; , 𝑦) − 𝐺 (Δ; |𝑥 − 𝑦|)

(12.168)

int is continuously differentiable on Ωint 𝑅 , twice continuously differentiable in Ω 𝑅 , int and harmonic in Ω𝑅 ,

(ii) for every 𝑥 ∈ Ωint 𝑅, 𝜕 1 𝐺 (INP) (Δ; 𝑥, 𝑦) = − , 𝑦 ∈ Ω𝑅 , 𝜕𝜈(𝑦) 4𝜋𝑅 2 where 𝜈 is the outward directed (unit) normal field to Ω𝑅 . The representation formula now yields for 𝑥 ∈ Ωint 𝑅

(12.169)

12.7 Boundary-Value Problems for the Laplace Operator

∫ 𝑈 (𝑥) = Ω𝑅

589

𝜕𝑈 1 (𝑦) 𝐺 (INP) (Δ; 𝑥, 𝑦) 𝑑𝜔(𝑦) − 𝜕𝜈 4𝜋𝑅 2 |

∫ 𝑈 (𝑦) 𝑑𝜔(𝑦) . Ω𝑅

{z =𝐶

}

(12.170) Therefore, 𝑈 is determined only up to an additive constant 𝐶. We already know that this is characteristic for the inner Neumann problem (INP). Finally it should be remarked that 𝐺 (INP) (Δ; 𝑥, 𝑦) is explicitly given by 𝐺 (INP) (Δ; 𝑥, 𝑦) =

1 4𝜋

© 1 𝑅 ­ ­ |𝑥 − 𝑦| + 2 |𝑥| 𝑦 − |𝑅𝑥 | 2 𝑥 « 1 2𝑅 2 ln 𝑅 𝑅 2 − 𝑥 · 𝑦 + |𝑥| 𝑦 −

+

(12.171)

ª ® .

𝑅2 ® 𝑥 | 𝑥 |2 ¬

From a geomathematical point of view the solutions of the exterior Dirichlet and Neumann problem, respectively, can be used in the so-called Meissl scheme (cf. P.A. Meissl (1971b), R. Rummel (1997), H. Nutz (2002), W. Freeden, H. Nutz (2018)) to characterize “upward continuation” of gravitational data (in a “frequency scheme” based on the potential coefficients; see Table 12.2).

Table 12.2 Meissl scheme for the “upward continuation” from the (terrestrial) height 𝑅 to the (spaceborne) height 𝑆 (involving outer harmonics).

→ − 𝑛+1 𝑆

Ω𝑆 -level:

𝑈

∧L2 (Ω

↑ Ω𝑅 -level:

𝑈

𝑆)

(𝑛, 𝑘 )

·


𝑅 𝑛 𝑆

∧L2 (Ω

→



− 𝑛+1 𝑅

𝑅)

(𝑛, 𝑘 )



𝜕𝑈 𝜕𝜈

 ∧L2 (Ω

↑ 𝜕𝑈 𝜕𝜈

·

𝑆)

(𝑛, 𝑘 )


𝑅 𝑛+1 𝑆

 ∧L2 (Ω

𝑅)

(𝑛, 𝑘 )

Obviously, the vertical arrows
𝑛 characterizing the “upward continuation” amount to an attenuation by the factor 𝑅𝑆 . The opposite directions characterizing the “downward  𝑛 continuation” amount to an amplification by the factor 𝑅𝑆 .

590

12 Potentials in Euclidean Space

12.8 Existence and Regularity Method

Next, we change our focus from sphere oriented formulations to regular regions. Let G be a regular region. Pot(G) denotes the space of all functions 𝑈 ∈ C (2) (G) harmonic in G, while Pot(G 𝑐 ) denotes the space of all functions 𝑈 ∈ C (2) (G 𝑐 ) that are harmonic in G 𝑐 and regular at infinity. For 𝑘 = 0, 1, . . . we denote by Pot (𝑘 ) (G) the space of all 𝑈 ∈ C (𝑘 ) (G) such that 𝑈 is of class Pot(G). Analogously, Pot (𝑘 ) (G 𝑐 ), G 𝑐 = R3 \G, denotes the space of all 𝑈 ∈ C (𝑘 ) (G 𝑐 ) such that 𝑈 is of class Pot(G 𝑐 ). In shorthand notation, Pot (𝑘 ) (G) = Pot(G) ∩ C (𝑘 ) (G),

(12.172)

Pot (𝑘 ) (G 𝑐 ) = Pot(G 𝑐 ) ∩ C (𝑘 ) (G 𝑐 ).

(12.173)

In the nomenclature of these function spaces the boundary-value problems can be reformulated briefly as follows: Interior Dirichlet Problem (IDP): Given a function 𝐹 of class C (0) (𝜕G), find a function 𝑉 ∈ Pot (0) (G) such that 𝑉 − (𝑥) = lim 𝑉 (𝑥 − 𝜏𝜈(𝑥)) = 𝐹 (𝑥),

𝑥 ∈ 𝜕G.

(12.174)

𝜏→0+

Interior Neumann Problem (INP): Given a function 𝐺 of class C (0) (𝜕G) satisfying ∫ the property 𝜕G 𝐺 (𝑥) 𝑑𝜔(𝑥) = 0, find a function 𝑉 ∈ Pot (1) (G) such that 𝜕𝑉 − (𝑥) = lim 𝜈(𝑥) · (∇𝑉)(𝑥 − 𝜏𝜈(𝑥)) = 𝐺 (𝑥), 𝜏→0+ 𝜕𝜈

𝑥 ∈ 𝜕G.

(12.175)

Exterior Dirichlet Problem (EDP): Given a function 𝐺 of class C (0) (𝜕G), find a function 𝑉 ∈ Pot (0) (G 𝑐 ) such that 𝑉 + (𝑥) = lim 𝑉 (𝑥 + 𝜏𝜈(𝑥)) = 𝐺 (𝑥),

𝑥 ∈ 𝜕G.

(12.176)

𝜏→0+

Exterior Neumann Problem (ENP): Given a function 𝐹 of class C (0) (𝜕G), find 𝑉 ∈ Pot (1) (G 𝑐 ) such that 𝜕𝑉 + (𝑥) = lim 𝜈(𝑥) · (∇𝑉)(𝑥 + 𝜏𝜈(𝑥)) = 𝐹 (𝑥), 𝜏→0+ 𝜕𝜈

𝑥 ∈ 𝜕G.

(12.177)

12.8 Existence and Regularity Method

591

Following an idea of C. Neumann (1870) and I. Fredholm (1900) we use the doublelayer potential ∫ 𝜕 𝑉 (𝑥) = 𝑆(𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝜔(𝑦) (12.178) 𝜕𝜈(𝑦) 𝜕G to solve the Dirichlet problems (IDP) and (EDP), respectively. More explicitly, we search for a function 𝑆 ∈ C (0) (𝜕G) such that 𝑉 − |𝜕G = 𝐹 (IDP) and for a function 𝑆 ∈ C (0) (𝜕G) such that 𝑉 + |𝜕G = 𝐹 (EDP), respectively, where 𝐹 : 𝜕G → R is assumed to be of class C (0) (𝜕G). In fact, for every 𝑥 ∈ 𝜕G, the limit relations tell us that ∫ 𝜕 𝐹 (𝑥) = 𝑉 ± (𝑥) = lim 𝑆(𝑦) 𝐺 (Δ; |𝑥 ± 𝜏𝜈(𝑥) − 𝑦|) 𝑑𝜔(𝑦) 𝜏→0+ 𝜕G 𝜕𝜈(𝑥) ∫ 1 𝜕 = ± 𝑆(𝑥) + 𝑆(𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝜔(𝑦). (12.179) 2 𝜕𝜈(𝑦) 𝜕G In an analogous way we are interested in solving the Neumann problems (INP) and (ENP), respectively, by use of a single-layer-potential ∫ 𝑈 (𝑥) = 𝑆(𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝜔(𝑦). (12.180) 𝜕G −

Now we look for a function 𝑄 ∈ C (0) (𝜕G) such that 𝜕𝑈 𝜕𝜈 |𝜕G = 𝐺 (INP) and 𝜕𝑈 + |𝜕G = 𝐺 (ENP) respectively, where 𝐺 : 𝜕G → R is assumed to be of class 𝜕𝜈 (0) C (𝜕G). For 𝑥 ∈ 𝜕G, the pointwise limit relations lead to the integral equations ∫ 𝜕𝑈 ± 𝜕 𝐺 (𝑥) = (𝑥) = lim 𝑄(𝑦) 𝐺 (Δ; |𝑥 ± 𝜏𝜈(𝑥) − 𝑦|) 𝑑𝜔(𝑦) 𝜏→0+ 𝜕G 𝜕𝜈 𝜕𝜈(𝑥) ∫ 1 𝜕 = ∓ 𝑄(𝑥) + 𝑄(𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝜔(𝑦). (12.181) 2 𝜕𝜈(𝑥) 𝜕G Summarizing our results, we are confronted with the following integral equations: ∫ 1 𝜕 (ID) : 𝑆(𝑥) − 𝑆(𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝜔(𝑦) = −𝐹 (𝑥), 𝑥 ∈ 𝜕G, 2 𝜕𝜈(𝑦) 𝜕G ∫ 1 𝜕 (ED) : 𝑆(𝑥) + 𝑆(𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝜔(𝑦) = 𝐹 (𝑥), 𝑥 ∈ 𝜕G, 2 𝜕𝜈(𝑦) 𝜕G ∫ 1 𝜕 (IN) : 𝑄(𝑥) + 𝑄(𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝜔(𝑦) = 𝐺 (𝑥), 𝑥 ∈ 𝜕G, 2 𝜕𝜈(𝑥) 𝜕G ∫ 1 𝜕 (EN) : 𝑄(𝑥) − 𝑄(𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝜔(𝑦) = −𝐺 (𝑥), 𝑥 ∈ 𝜕G. 2 𝜕𝜈(𝑥) 𝜕G As a consequence, we obtain two pairs of adjoint integral equations, viz. (ID), (EN) and (ED), (IN), where the second integral equation is adjoint to the first integral

592

12 Potentials in Euclidean Space

equation. This is characteristic of so-called Fredholm integral equations. Indeed, there is a huge literature on Fredholm integral equations leading to the so-called Fredholm Alternative (see, for example, S.G. Michlin (1975)). As a matter of fact, in the mid of the last century, the specification of the Fredholm Alternative was a decisive step in creating a particular discipline of mathematics, namely Functional Analysis. In what follows, we do not provide all functional analytic facets of today’s Fredholm theory. We only give a summary of those results to be needed in our context without any proof (for the details see. e.g., W. Freeden, C. Gerhards (2013)): Theorem 12.32 For 𝐹 ∈ C (0) (𝜕G), the inner Dirchlet problem (IDP) is uniquely solvable, and the solution can be represented in the form ∫ 𝜕 𝑈 (𝑥) = 𝑆(𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝜔(𝑦), 𝑥 ∈ G, (12.182) 𝜕𝜈(𝑦) 𝜕G where 𝑆 ∈ C (0) (𝜕G) satisfies the integral equation (ID) ∫ 1 𝜕 𝑆(𝑥) − 𝑆(𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝜔(𝑦) = −𝐹 (𝑥), 𝑥 ∈ 𝜕G. 2 𝜕𝜈(𝑦) 𝜕G

(12.183)

For 𝐺 ∈ C (0) (𝜕G) the exterior Neumann problem (ENP) is uniquely solvable, and the solution can be represented in the form ∫ 𝑈 (𝑥) = 𝑄(𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝜔(𝑦), 𝑥 ∈ G 𝑐 , (12.184) 𝜕G

where 𝑄 ∈ C (0) (𝜕G) satisfies the integral equation (EN) ∫ 1 𝜕 𝑄(𝑥) − 𝑄(𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝜔(𝑦) = −𝐺 (𝑥), 𝑥 ∈ 𝜕G. 2 𝜕𝜈(𝑥) 𝜕G

(12.185)

Theorem 12.33 For 𝐹 ∈ C (0) (𝜕G), the exterior Dirichlet problem (EDP) is uniquely solvable, and the solution can be represented in the form   ∫ 𝜕 𝑈 (𝑥) = 𝑆(𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) + 𝐺 (Δ; |𝑥|) 𝑑𝜔(𝑦), 𝑥 ∈ G, (12.186) 𝜕𝜈(𝑦) 𝜕G where 𝑆 ∈ C (0) (𝜕G) satisfies the integral equation (MED)   ∫ 1 𝜕 𝑆(𝑥) + 𝑆(𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) + 𝐺 (Δ; |𝑥|) 𝑑𝜔(𝑦) = 𝐹 (𝑥), 𝑥 ∈ 𝜕G. 2 𝜕𝜈(𝑦) 𝜕G (12.187)

12.8 Existence and Regularity Method

593

Finally, we recall the role of layer potentials for the classical Dirichlet and Neumann boundary-value problems. We restrict ourselves to the geoscientifically more relevant exterior problems. (EDP): Let 𝐷 + denote the “boundary-value space” of Dirichlet’s type: n  o 𝐷 + = 𝑉 + |𝜕G : 𝑉 ∈ Pot (0) G 𝑐 .

(12.188)

The exterior Dirichlet problem (EDP) is always uniquely determined, hence, 𝐷 +|𝜕G = C (0) (𝜕G).

(12.189)

  The solution of the problem 𝑉 ∈ Pot (0) G 𝑐 , 𝑉 + |𝜕G = 𝐹, 𝐹 ∈ C (0) (𝜕G) can be formulated in terms of a potential of the form ∫ 𝜕 (𝐺 (Δ; |𝑥 − 𝑦|) + 𝐺 (Δ; |𝑥|)) 𝑑𝜔(𝑦), 𝑆 ∈ C (0) (𝜕G), 𝑉 (𝑥) = 𝑆(𝑦) 𝜕𝜈(𝑦) 𝜕G (12.190) where 𝑆 satisfies the integral equation   1 𝐹 = 𝑉 + |𝜕G = 𝐼 + 𝐴 + 𝑃 |2 [𝑆], 𝐹 ∈ C (0) (𝜕G) (12.191) 2 and

∫ 𝑃 |2 (0, 0) [𝑆] : 𝑥 ↦→

𝑆(𝑦) 𝜕G

𝜕 𝐺 (Δ; |𝑥 − 𝑦|)𝑑𝜔(𝑦), 𝜕𝜈(𝑦)

(12.192)

𝑆(𝑦)𝐺 (Δ; |𝑥|) 𝑑𝜔(𝑦).

(12.193)

∫ 𝐴[𝑆] : 𝑥 ↦→ 𝜕G

Furthermore, we know that ∗ 1 𝐼 + 𝐴 + 𝑃 |2 = {0}, 2  h i 1 𝐼 + 𝐴 + 𝑃 |2 C (0) (𝜕G) = 𝐷 + . 2 

kern

(12.194)

(12.195)

By completion, we therefore obtain L2 (𝜕G) = 𝐷 +

∥ · ∥ L2 (𝜕G)

= C (0) (𝜕G)

∥ · ∥ L2 (𝜕G)

.

(ENP): Let 𝑁 + denote the “boundary-value space” of Neumann’s type:  +  𝜕𝑉 + (1) 𝑐 𝑁 = 𝜕G : 𝑉 ∈ Pot (G ) . 𝜕𝜈

(12.196)

(12.197)

594

12 Potentials in Euclidean Space

The problem (ENP) is always uniquely determined, hence, 𝑁 + = C (0) (𝜕G). (12.198)   + (0) (𝜕G), can be The solution of the problem 𝑉 ∈ Pot (1) G 𝑐 , 𝜕𝑉 𝜕𝜈 |𝜕G = 𝐹, 𝐹 ∈ C formulated in terms of a single-layer potential ∫ 𝑉 (𝑥) = 𝑄(𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝜔(𝑦), 𝑄 ∈ C (0) (𝜕G), (12.199) 𝜕G

where 𝑄 satisfies the integral equations 𝐹=

  𝜕𝑉 + 1 = − 𝐼 + 𝑃 (0, 0) 𝑄 𝜕G |1 𝜕𝜈 2

with

(12.200)

∫ 𝑃 |1 [𝑄] : 𝑥 ↦→

𝑄(𝑦)𝐺 (Δ; |𝑥 − 𝑦|)𝑑𝜔(𝑦).

(12.201)

𝜕G

We have

 kern

so that



1 − 𝐼 + 𝑃 |1 2

∗ = {0},

(12.202)

h i 1 − 𝐼 + 𝑃 |1 C (0) (𝜕G) = 𝑁 + . 2

(12.203)

By completion, we therefore get L2 (𝜕G) = 𝑁 +

∥ · ∥ L2 (𝜕G)

= C (0) (𝜕G)

∥ · ∥ L2 (𝜕G)

.

(12.204)

Similar arguments, of course, hold true for the inner boundary-value problems. The details are left to the reader. Regularity Theorems. Let 𝑈 be of class Pot (0) (G 𝑐 ). Then the Maximum/Minimum Principle for the outer space G 𝑐 states sup |𝑈 (𝑥)| ≤ sup |𝑈 (𝑥)|. 𝑥 ∈ G𝑐

(12.205)

𝑥 ∈𝜕G

From the theory of integral equations we are able to verify the continuous dependence of boundary data of Neumann’s type (the proof can be found, e.g., in W. Freeden, C. Gerhards (2013)). Lemma 12.34 Let 𝑈 be of class Pot (1) (G 𝑐 ). Then there exist a constant 𝐶 (dependent on 𝜕G) such that

12.9 Locally and Globally Uniform Approximation

595

+ 𝜕𝑈 sup |𝑈 (𝑥)| ≤ 𝐶 sup (𝑥) . 𝑥 ∈𝜕G 𝜕𝜈 𝑥 ∈ G𝑐

(12.206)

Hence, the exterior Neumann problem (ENP) is well-posed in the sense that its solution exists, is unique, and depends continuously on the boundary data. In what follows we want to eplain analogous stability theorems for the Dirichlet as well as Neumann problem in the L2 (𝜕G)-context.   Theorem 12.35 Let 𝑈 be of class Pot (0) G 𝑐 . Then there exists a constant 𝐶 (= 𝐶 (𝑘; K, 𝜕G)) such that ∫   sup ∇ (𝑘 ) 𝑈 (𝑥) ≤ 𝐶

+ 2 𝑈 (𝑥) 𝑑𝜔(𝑥)

 12 (12.207)

𝜕G

𝑥∈K

for all K ⋐ G 𝑐 and all 𝑘 ∈ N0 . An analogous argument yields the following regularity theorem in the L2 (𝜕G)context. Theorem 12.36 Let 𝑈 be of class Pot (1) (G 𝑐 ). Then there exists a constant 𝐶 (= 𝐶 (𝑘; K, 𝜕G)) such that ! 12 + 2 𝜕𝑈 (𝑥) 𝑑𝜔(𝑥) 𝜕G 𝜕𝜈

∫   sup ∇ (𝑘 ) 𝑈 (𝑥) ≤ 𝐶

𝑥∈K

(12.208)

for all K ⋐ G 𝑐 and all 𝑘 ∈ N0 .

12.9 Locally and Globally Uniform Approximation

In boundary-value problems of potential theory a result first motivated by C. Runge (1885) in one-dimensional complex analysis and later generalized by J.L. Walsh (1929) to potential theory is of basic interest. For our geoscientifically relevant purpose, it may be formulated as follows: Any function satisfying Laplace’s equation in G 𝑐 = R3 \G and regular at infinity may be approximated by a function, harmonic outside an arbitrarily given Runge sphere inside G in the sense that for any given 𝜀 > 0, the absolute error between the two functions is smaller than 𝜀 for all points 𝑥 ∈ R3 outside and on any closed surface completely surrounding the surface 𝜕G in

596

12 Potentials in Euclidean Space

the outer space. The value 𝜀 may be arbitrarily small, and the surrounding surface may be arbitrarily close to the surface 𝜕G . Obviously, the Runge-Walsh theoremin its preceding formulation is a pure existence theorem. It guarantees only the existence of an approximating function and does not provide a method to find it. Nothing is said about the approximation procedure and the structure of the approximating function systems. The theorem describes merely the theoretical background for the approximation of a potential by another potential defined on a larger harmonicity domain. The situation, however, is completely different in a spherical model. Assuming that the boundary 𝜕G is a sphere around the origin, a constructive approximation of a potential in the outer space is available, e.g., by outer harmonic expansions. More concretely, in a spherical context, a constructive version of the Runge-Walsh theorem can be established by finite truncations of Fourier expansions in terms of outer harmonics. The only unknown information left when using an outer harmonic expansion is the a priori choice of the right truncation parameter. From a superficial point of view one could suggest that approximation by truncated series expansions in terms of outer harmonics is closely related to spherical boundaries. The purpose of our next work, however, is to show that the essential steps involved in the Fourier expansion method can be generalized to any regular region G. We restrict ourselves to the geoscientifically relevant exterior cases. The interior cases follow by obvious arguments. Closure in L2 -Topology. We begin our considerations with the recapitulation (see W. Freeden (1981c)) of the following results. Lemma 12.37 Let G ⊂ R2 be a regular surface such that 𝑅 < inf 𝑥 ∈𝜕G |𝑥| . Then the following statements are valid: o n 𝑅 (i) 𝐻 −𝑛−1, 𝑗 𝜕G (ii)

n



𝑛=0,1,... 𝑗=1,...,2𝑛+1

𝑅 𝜕 𝜕𝜈 𝐻 −𝑛−1, 𝑗 𝜕G

is linearly independent,

o 𝑛=0,1,... 𝑗=1,...,2𝑛+1

is linearly independent.

Next, our goal is to prove completeness and closure theorems (see W. Freeden (1980a)).

12.9 Locally and Globally Uniform Approximation

597

Theorem 12.38 Let G ⊂ R3 be a regular region such that 𝑅 < inf 𝑥 ∈𝜕G |𝑥|. Then the following statements are valid: o n ∥·∥ 2 𝑅 (i) 𝐻 −𝑛−1, 𝑗 𝜕G 𝑛=0,1,... is complete in L2 (𝜕G) = 𝐷 + L (𝜕G) , 𝑗=1,...,2𝑛+1 n o ∥·∥ 2 𝑅 𝜕 (ii) 𝜕𝜈 𝐻 −𝑛−1, 𝑗 𝜕G 𝑛=0,1,... is complete in L2 (𝜕G) = 𝑁 + L (𝜕G) . 𝑗=1,...,2𝑛+1

Proof We restrict our attention to statement(i). Suppose that 𝐹 ∈ L2 (𝜕G) satisfies ∫ 𝑅 𝑅 (𝐹, 𝐻 −𝑛−1, 𝑗 |𝜕G)L2 (𝜕G) = 𝐹 (𝑦)𝐻 −𝑛−1, 𝑗 (𝑦) 𝑑𝜔(𝑦) = 0 (12.209) 𝜕G

for all 𝑛 ∈ N0 , 𝑗 = 1, ..., 2𝑛 + 1. We have to show that 𝐹 = 0 in L2 (𝜕G). We remember that the series expansion 𝐺 (Δ; |𝑥 − 𝑦|) =

∞ ∑︁ 𝑛=0

2𝑛+1 1 |𝑥| 𝑛 ∑︁ 𝑌𝑛, 𝑗 (𝜉)𝑌𝑛, 𝑗 (𝜂), 2𝑛 + 1 |𝑦| 𝑛+1 𝑗=1

(12.210)

int 𝑥 = |𝑥|𝜉, 𝑦 = |𝑦|𝜂, is analytic in the variable 𝑥 on the ball Ωint 𝑅 . For all 𝑥 ∈ Ω 𝑅 we thus find by virtue of (12.209) ∫ 𝑈 (𝑥) = 𝐹 (𝑦)𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝜔(𝑦) (12.211) 𝜕G

=

∞ ∑︁ 𝑛=0

∑︁ 𝑅 𝑅 2𝑛+1 𝐻 (𝑥) 2𝑛 + 1 𝑗=1 𝑛, 𝑗

∫

𝑅

𝐹 (𝑦)𝐻 −𝑛−1, 𝑗 (𝑦) 𝑑𝜔(𝑦) 𝜕G

= 0. Analytic continuation shows that the single-layer potential 𝑈 vanishes at each point 𝑥 ∈ G. In other words, the equations 𝑈 (𝑥 − 𝜏𝜈(𝑥)) = 0, 𝜕𝑈 (𝑥 − 𝜏𝜈(𝑥)) = 0 𝜕𝜈

(12.212) (12.213)

hold for all 𝑥 ∈ 𝜕G and all sufficiently small 𝜏 > 0. Using the limit and jump relations in L2 (𝜕G)-nomenclature, we therefore obtain ∫ 2 lim (12.214) 𝑈 (𝑥 + 𝜏𝜈(𝑥)) 𝑑𝜔(𝑥) = 0, 𝜏→0+

𝜕G

2 𝜕𝑈 𝜕𝜈 (𝑥 + 𝜏𝜈(𝑥)) + 𝐹 (𝑥) 𝑑𝜔(𝑥) = 0, 𝜕G

∫ lim 𝜏→0+

and

(12.215)

598

12 Potentials in Euclidean Space

∫ lim 𝜏→0+

𝜕𝑈 2 1 (𝑥) + 𝐹 (𝑥) 𝑑𝜔(𝑥) = 0. 𝜕𝜈 2 𝜕G

(12.216)

Observing that the limit in the last equation can be omitted, we read that this equation in the explicit form ∫ 𝜕 1 − 𝐹 (𝑦) 𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝜔(𝑦) = − 𝐹 (𝑥) (12.217) 𝜕𝜈(𝑥) 2 𝜕G in the sense of L2 (𝜕G). The left hand side of (12.217) constitutes a continuous function. Thus, the function 𝐹 can be replaced by a function 𝐹˜ ∈ C (0) (𝜕G) satisfying ˜ however, the classical 𝐹 = 𝐹˜ in the sense of L2 (𝜕G). For the continuous function 𝐹, limit and jump relations are valid: lim 𝑈 (𝑥 + 𝜏𝜈(𝑥)) = 0,

𝑥 ∈ 𝜕G,

(12.218)

𝑥 ∈ 𝜕G.

(12.219)

𝜏→0+

lim 𝜏→0+

𝜕𝑈 (𝑥 + 𝜏𝜈(𝑥)) = −𝐹˜ (𝑥), 𝜕𝜈

Consequently, the uniqueness theorem of the exterior Dirichlet problem shows us that 𝑈 (𝑥) = 0 for all 𝑥 ∈ R3 \G 𝑐 . However, this means that 𝐹˜ = 0 on the surface 𝜕G, as required. The remaining statement (ii) follows by analogous arguments. □ From constructive approximation (see, e.g., P.J. Davis (1963)) we know that the properties of completeness and closure are equivalent in a Hilbert space such as L2 (𝜕G). This leads us to the following statement. Corollary 12.39 Under the assumptions of Theorem 12.38 the following statements are valid:  𝑅 (i) 𝐻 −𝑛−1, 𝑗 𝜕G

𝑛=0,1,..., 𝑗=1,...,2𝑛+1

is closed in L2 (𝜕G), i.e., for any given 𝐹 ∈ L2 (𝜕G) and

arbitrary 𝜀 > 0 there exists a linear combination 𝐻𝑚 =

𝑚 2𝑛+1 ∑︁ ∑︁

𝑅

𝑎 𝑛, 𝑗 𝐻 −𝑛−1, 𝑗

(12.220)

𝑛=0 𝑗=1

such that ∥𝐹 − 𝐻𝑚 ∥ L2 (𝜕G) ≤ 𝜀. (ii)



𝑅 𝜕 𝜕𝜈 𝐻 −𝑛−1, 𝑗 𝜕G

𝑛=0,1,..., 𝑗=1,...,2𝑛+1

(12.221)

is closed in L2 (𝜕G), i.e., for any given 𝐹 ∈ L2 (𝜕G)

and arbitrary 𝜀 > 0 there exists a linear combination 𝑆𝑚 =

𝑚 2𝑛+1 ∑︁ ∑︁ 𝑛=0 𝑗=1

𝑎 𝑛, 𝑗

𝜕 𝑅 𝐻 𝜕𝜈 −𝑛−1, 𝑗

(12.222)

12.9 Locally and Globally Uniform Approximation

599

such that ∥𝐹 − 𝑆 𝑚 ∥ L2 (𝜕G) ≤ 𝜀.

(12.223)

Fundamental Systems. Based on our results on outer harmonics, i.e., multipole expansions, a large number of “polynomially based” countable systems of potentials can be shown to possess the L2 -closure property on 𝜕G. Probably best known are “mass-pole representations” (i.e., fundamental solutions of the Laplace operator). Their L2 (𝜕G)-closure is adequately described by using the concept of “fundamental systems”, which should be recapitulated briefly (see W. Freeden (1980a), W. Freeden, V. Michel (2004)). Definition 12.40 Let K ⊂ R3 be a region. A point set Y = {𝑦 𝑛 } 𝑛=0,1,... ⊂ K (with 𝑦 𝑛 ≠ 𝑦 𝑙 for 𝑛 ≠ 𝑙) is called a fundamental system in K, if the following properties are satisfied: (i) dist(Y, 𝜕K) > 0, (ii) for each 𝑈 ∈ Pot(K) the conditions 𝑈 (𝑦 𝑛 ) = 0 for 𝑛 = 0, 1, . . . imply 𝑈 = 0 in K.

G

A

𝜕A

Fig. 12.5: Illustration of the positioning of a fundamental system on 𝜕A in G. Some examples of fundamental systems should be listed for a regular region G (note that analogous arguments hold for fundamental systems in G 𝑐 ): Y = {𝑦 𝑛 } 𝑛=0,1,... is a fundamental system in G if it is a dense set of points of one of the following subsets of G: (1) regular region A with A ⋐ G, (2) boundary 𝜕A of a regular region A with A ⋐ G ( cf. Fig. 12.5). Theorem 12.41 Let G be a regular region. Then the following statements are valid:

600

12 Potentials in Euclidean Space

(i) For every fundamental system Y = {𝑦 𝑛 } 𝑛=0,1,... in G the system {𝐺 (Δ; | · −𝑦 𝑛 |)} 𝑛=0,1,... is closed in L2 (𝜕G) = 𝐷 +

∥ · ∥ L2 (𝜕G)

.

(ii) For every fundamental system Y = {𝑦 𝑛 } 𝑛=0,1,... in G the system   𝜕 𝐺 (Δ; | · −𝑦 𝑛 |), 𝜕𝜈 𝑛=0,1,... is closed in L2 (𝜕G) = 𝑁 +

∥ · ∥ L2 (𝜕G)

(12.224)

(12.225)

.

Proof We restrict ourselves to the proof of the statement (i). Since 𝑦 𝑛 ≠ 𝑦 𝑚 for all 𝑛 ≠ 𝑚 it immediately follows that the system {𝐺 (Δ; | · −𝑦 𝑛 |)} 𝑛=0,1,... is linearly independent. Our purpose is to verify the completeness of the system (12.224) in L2 (𝜕G). To this end we consider a function 𝐹 ∈ L2 (𝜕G) with ∫ 𝐹 (𝑥)𝐺 (Δ; |𝑥 − 𝑦 𝑛 |) 𝑑𝜔(𝑥) = 0, 𝑛 ∈ N0 . (12.226) 𝜕G

We have to prove that 𝐹 = 0 in L2 (𝜕G). Clearly, the single-layer potential 𝑈 given by ∫ 𝑈 (𝑦) = 𝐹 (𝑥)𝐺 (Δ; |𝑥 − 𝑦|) 𝑑𝜔(𝑦) (12.227) 𝜕G

vanishes at all points 𝑦 ∈ G. As 𝑈 is continuous and even analytic on G, the required assumption imposed on the system {𝑦 𝑛 } 𝑛=0,1,... in G implies 𝑈 (𝑦) = 0 for all 𝑦 ∈ G. The same arguments as given in the proof of Theorem 12.38 then guarantee that 𝐹 = 0 in the sense of ∥ · ∥ L2 (𝜕G) , as required. The statement (ii) follows by analogous arguments. □ Besides the outer harmonics, i.e., multipoles (see Corollary 12.39) and the mass (single-)poles (see Theorem 12.41) there exist a variety of countable systems of potentials showing the properties of completeness and closure in L2 (𝜕G). Many systems, however, are much more difficult to handle numerically (for instance, the ellipsoidal systems of Lam´e or Mathieu functions). Although they are very interesting, e.g., in Physical Geodesy, they will not be discussed here (for more details see, e.g., E.W. Grafarend et al. (2010) and the references therein). Instead we study some further kernel systems generated by superposition (i.e., infinite clustering) of outer harmonics (such that a certain amount of space localization can be expected on inner Runge spheres. In addition, because of their explicit availability as elementary functions, these systems turn out to be particularly suitable for numerical purposes.

12.9 Locally and Globally Uniform Approximation

601

Theorem 12.42 Let G ⊂ R3 be a regular region such that 𝑅 < inf 𝑥 ∈𝜕G |𝑥|. We int consider the kernel function 𝐾 (·, ·) : R3 \Ωint 𝑅 × Ω 𝑅 → R given by 𝐾 (𝑥, 𝑦) =

∞ 2𝑘+1 ∑︁ ∑︁

𝐾 ∧ (𝑘)𝐻 −𝑘−1,𝑙 (𝑥)𝐻 𝑘,𝑙 (𝑦) 𝑅

𝑅

(12.228)

𝑘=0 𝑙=1 3 int int for 𝑦 ∈ Ωint 𝑅 , 𝑥 ∈ R \Ω 𝑅 . Let Y = {𝑦 𝑛 } 𝑛=0,1,... be a fundamental system in Ω 𝑅 with 𝑅 0 < 𝑅 < inf |𝑥|. Suppose that 𝑥 ∈𝜕G

∞ ∑︁

 𝑘 ∧ 𝑅0 (2𝑘 + 1) 𝐾 (𝑘) 0 and K ⋐ G 𝑐 there exist an integer 𝑚 (dependent on 𝜀) and a set of coefficients 𝑎 0,1 , ..., 𝑎 𝑚,1 , ..., 𝑎 𝑚,2𝑚+1 such that 1 2 2 ∫ 𝑚 2𝑛+1 ∑︁ ∑︁ © ª 𝑅 ­ 𝐹 (𝑥) − ® 𝑎 𝑛, 𝑗 𝐻 −𝑛−1, 𝑗 (𝑥) 𝑑𝜔(𝑥) ® ≤ 𝜀 ­ 𝜕G 𝑛=0 𝑗=1 « ¬

and

 𝑚 2𝑛+1   ∑︁ ∑︁ (𝑘 )  𝑅 sup ∇ 𝑈 (𝑥) − 𝑎 𝑛, 𝑗 ∇ (𝑘 ) 𝐻 −𝑛−1, 𝑗 (𝑥) ≤ 𝐶𝜀 𝑥∈K 𝑛=0 𝑗=1

hold for all 𝑘 ∈ N0 . (ENP): For given 𝐹 ∈ C (0) (𝜕G), let 𝑈 satisfy 𝑈 ∈ Pot (1) (G 𝑐 ) , 𝜕𝑈 𝜕𝜈 𝜕G = 𝐹. Then, 𝑐 for any given 𝜀 > 0 and K ⋐ G there exist an integer 𝑚 (dependent on 𝜀) and a set of coefficients 𝑎 0,1 , ..., 𝑎 𝑚,1 , ..., 𝑎 𝑚,2𝑚+1 such that

© ­ ­ « and

1 2 2 𝑅 𝑚 2𝑛+1 ∑︁ ∑︁ 𝜕𝐻 −𝑛−1, 𝑗 ª 𝐹 (𝑥) − ® 𝑎 𝑛, 𝑗 (𝑥) 𝑑𝜔(𝑥) ® ≤ 𝜀 𝜕𝜈 𝜕G 𝑛=0 𝑗=1 ¬

∫

 𝑚 2𝑛+1   ∑︁ ∑︁ (𝑘 )  𝑅 sup ∇ 𝑈 (𝑥) − 𝑎 𝑛, 𝑗 ∇ (𝑘 ) 𝐻 −𝑛−1, 𝑗 (𝑥) ≤ 𝐶𝜀 𝑥∈K 𝑛=0 𝑗=1

hold for all 𝑘 ∈ N0 . In other words, the L2 -approximation in terms of outer harmonics on 𝜕G implies the uniform approximation (in the ordinary sense) on each subset K with positive distance to 𝜕G (in particular, to each compact subset K with positive distance to 𝜕G such that an approximation is guaranteed in locally compact topology). Unfortunately, the theorems developed until now are non-constructive, since further information about the choice of 𝑚 and the coefficients of the approximating linear combination is needed. In order to derive a constructive approximation theorem the

12.9 Locally and Globally Uniform Approximation

605

system of potential values and normal derivatives, respectively, can be orthonormalized on 𝜕G. As a result we obtain a “(generalized) Fourier expansion” (orthogonal Fourier approximation) that shows locally uniform approximation. Theorem 12.46 Let G ⊂ R3 be a regular region such that 𝑅 < inf 𝑥 ∈𝜕G |𝑥|. (EDP): For given 𝐹 ∈ C (0) (𝜕G), let 𝑈 be the solution of 𝑈 ∈ Pot (0) (G 𝑐 ), 𝑈 + |𝜕G = 𝑅 𝐹. Corresponding to the countably infinite sequence {𝐻 −𝑛−1, 𝑗 } 𝑛∈N0 there exists a system {𝐻 −𝑛−1, 𝑗 (𝜕G; ·)} 𝑛∈N0 , 𝑗=1,...,2𝑛+1 ∈ Pot (0) (R3 \Ωint 𝑅 ) such that the normal derivative {𝐻 −𝑛−1, 𝑗 (𝜕G; ·)|𝜕G} 𝑛∈N0 , 𝑗=,...,2𝑛+1 is orthonormal in the sense that ∫ 𝐻 −𝑛−1, 𝑗 (𝜕G; 𝑦)𝐻 −𝑙−1,𝑘 (𝜕G; 𝑦) 𝑑𝜔(𝑦) = 𝛿 𝑛𝑙 𝛿 𝑗 𝑘 . 𝜕G

Consequently, 𝑈 is representable in the form 𝑈 (𝑥) =

∞ 2𝑛+1 ∑︁ ∑︁ ∫ 𝑛=0 𝑗=1

 𝐹 (𝑦)𝐻 −𝑛−1, 𝑗 (𝜕G; 𝑦) 𝑑𝜔(𝑦) 𝐻 −𝑛−1, 𝑗 (𝜕G; 𝑥)

𝜕G

for all points 𝑥 ∈ K ⋐ G 𝑐 . Moreover, for each 𝑈 (𝑚) given by 𝑈

(𝑚)

(𝑥) =

𝑚 2𝑛+1 ∑︁ ∑︁ ∫ 𝑛=0 𝑗=1

 𝐹 (𝑦)𝐻 −𝑛−1, 𝑗 (𝜕G; 𝑦) 𝑑𝜔(𝑦) 𝐻 −𝑛−1, 𝑗 (𝜕G; 𝑥)

𝜕G

we have the estimate     sup ∇ (𝑘 ) 𝑈 (𝑥) − ∇ (𝑘 ) 𝑈 (𝑚) (𝑥) 𝑥∈K

1

2 2 ∫ 𝑚 2𝑛+1 ∑︁ ∑︁ ∫ ª © ® . |𝐹 (𝑦)| 2 𝑑𝜔(𝑦) − ≤𝐶­ 𝐹 (𝑦)𝐻 (𝜕G; 𝑦) 𝑑𝜔(𝑦) −𝑛−1, 𝑗 𝜕G 𝜕G 𝑛=0 𝑗=1 « ¬ + (ENP): For given 𝐹 ∈ C (0) (𝜕G), let 𝑈 be the solution of 𝑈 ∈ Pot (1) (G 𝑐 ), 𝜕𝑈 𝜕𝜈 𝜕G = 𝑅 𝐹. Corresponding to the countably infinite sequence {𝐻 −𝑛−1, 𝑗 } 𝑛∈N0 , 𝑗=1,...,2𝑛+1 there

exists {𝐻 −𝑛−1, 𝑗 (𝜕G; ·)} 𝑛∈N0 , 𝑗=1,...,2𝑛+1 ⊂ Pot (0) (R3 \Ωint 𝑅 ) such that the normal 𝜕 derivative { 𝜕𝜈 𝐻 −𝑛−1, 𝑗 (𝜕G; ·)} 𝑛∈N0 , 𝑗=1,...,2𝑛+1 is orthonormal in the sense that ∫ 𝜕 𝜕 𝐻 −𝑛−1, 𝑗 (𝜕G; 𝑦) 𝐻 −𝑙−1,𝑘 (𝜕G; 𝑦) 𝑑𝜔(𝑦) = 𝛿 𝑛𝑙 𝛿 𝑗 𝑘 . 𝜕𝜈 𝜕𝜈 𝜕G

Consequently, 𝑈 is representable in the form

606

12 Potentials in Euclidean Space

𝑈 (𝑥) =

∞ 2𝑛+1 ∑︁ ∑︁ ∫ 𝑛=0 𝑗=1

 𝜕 𝐹 (𝑦) 𝐻 −𝑛−1, 𝑗 (𝜕G, 𝑦) 𝑑𝜔(𝑦) 𝐻 −𝑛−1, 𝑗 (𝜕G; 𝑥) 𝜕𝜈 𝜕G

for all points 𝑥 ∈ K ⋐ G 𝑐 . Moreover, for each 𝑈 (𝑚) given by 𝑈 (𝑚) (𝑥) =

𝑚 2𝑛+1 ∑︁ ∑︁ ∫ 𝑛=0 𝑗=1

𝐹 (𝑦)

𝜕G

 𝜕 𝐻 −𝑛−1, 𝑗 (𝜕G; 𝑦) 𝑑𝜔(𝑦) 𝐻 −𝑛−1, 𝑗 (𝜕G; 𝑥) 𝜕𝜈

we have the estimate     sup ∇ (𝑘 ) 𝑈 (𝑥) − ∇ (𝑘 ) 𝑈 (𝑚) (𝑥) 𝑥∈K

1

∫ © ≤𝐶­ «

𝜕G

2 2 ª 𝜕𝐻 −𝑛−1, 𝑗 (𝜕G; 𝑦) 𝐹 (𝑦) 𝑑𝜔(𝑦) ® . 𝜕𝜈(𝑦) 𝜕G ¬

𝑚 2𝑛+1 ∑︁ ∑︁ ∫ |𝐹 (𝑦)| 2 𝑑𝜔(𝑦) − 𝑛=0 𝑗=1

Note that the orthonormalization procedure can be performed (e.g., by the wellknown Gram-Schmidt orthonormalization process) once and for all when the boundary surface 𝜕G of a regular region G is specified. In the same way, the inner boundaryvalue problems can be formulated by generalized Fourier expansions (orthogonal expansions) in terms of inner harmonics. Furthermore, locally uniform approximation by “generalized Fourier expansions” can be obtained not only for (the multipole system of inner/outer) harmonics, but also for mass point and related kernel representations. The details are omitted. Finally, we rewrite our generalized Fourier approach in a more abstract form. For that purpose we introduce the concept of Dirichlet and Neumann bases. Definition 12.47 (Dirichlet/Neumann Runge Basis) Let A, G ⊂ R3 be regular regions such that A ⋐ G holds true (cf. Fig. 12.7). A linearly independent system {𝐷 𝑛 } 𝑛=0,1,... , 𝐷 𝑛 ∈ Pot(A 𝑐 ) is called an (Pot(A 𝑐 )generated) L2 (𝜕G)-Dirichlet Runge basis , if span{𝐷 𝑛 |𝜕G}

∥ · ∥ L2 (𝜕G)

= L2 (𝜕G).

(12.240)

𝑛∈N0

A linearly independent system {𝑁 𝑛 } 𝑛=0,1,... , 𝑁 𝑛 ∈ Pot(A 𝑐 ) is called an (Pot(A 𝑐 )generated) L2 (𝜕G)-Neumann Runge basis, if  span 𝑛∈N0

 ∥ · ∥ L2 (𝜕G) 𝜕𝑁 𝑛 = L2 (𝜕G). 𝜕G 𝜕𝜈

(12.241)

12.9 Locally and Globally Uniform Approximation

607

G

G

A

A

Fig. 12.7: The geometric situation of an L2 (𝜕G)-Dirichlet/Neumann Runge basis (with A an arbitrary regular region such that A ⋐ G (left) and A an inner Runge ball (right)).

Corollary 12.48 Let A, G ⊂ R3 be regular regions such that A ⋐ G holds true. (EDP): Let {𝐷 ∗𝑛 } 𝑛=0,1,... , 𝐷 ∗𝑛 ∈ Pot(A 𝑐 ), be a system generated by (GramSchmidt) orthonormalization of an L2 (𝜕G)-Dirichlet Runge basis {𝐷 𝑛 } 𝑛=0,1,... , 𝐷 𝑛 ∈ Pot(A 𝑐 ), such that ∫ ∗ ∗ (𝐷 𝑛 , 𝐷 𝑚 )L2 (𝜕G) = 𝐷 ∗𝑛 (𝑥)𝐷 ∗𝑚 (𝑥) 𝑑𝜔(𝑥) = 𝛿 𝑛𝑚 . (12.242) 𝜕G

If 𝐹 ∈ C (0) (𝜕G), then ∫ lim 𝑚→∞

𝜕G

 12 2 (𝑚) 𝐹 (𝑥) − 𝐹 (𝑥) 𝑑𝜔(𝑥) = 0,

(12.243)

where 𝐹 (𝑚) =

𝑚 ∑︁

𝐹, 𝐷 ∗𝑛




L2 (𝜕G)

𝐷 ∗𝑛 |𝜕G.

(12.244)

𝑛=0

The potential 𝑈 ∈ Pot (0) (G 𝑐 ), 𝑈 + |𝜕G = 𝐹, can be represented in the form lim sup 𝑈 (𝑥) − 𝑈 (𝑚) (𝑥) = 0 (12.245) 𝑚→∞

𝑥∈K

with 𝑈 (𝑚) =

𝑚 ∑︁ 𝑛=0

for every K ⋐ G 𝑐 .

(𝐹, 𝐷 ∗𝑛 )L2 (𝜕G) 𝐷 ∗𝑛

(12.246)

608

12 Potentials in Euclidean Space

(ENP): Let {𝑁 𝑛∗ } 𝑛=0,1,... , 𝑁 𝑛∗ ∈ Pot(A), be a system generated by (Gram-Schmidt) orthonormalization of an L2 (𝜕G)-Neumann Runge basis {𝑁 𝑛 } 𝑛=0,1,... , 𝑁 𝑛 ∈ Pot(A 𝑐 ) such that  ∫ ∗  𝜕𝑁 𝑛∗ 𝜕𝑁 𝑚 𝜕𝑁 𝑛∗ 𝜕𝑁 ∗ , = (𝑥) 𝑚 (𝑥) 𝑑𝜔(𝑥) = 𝛿 𝑛𝑚 . (12.247) 𝜕𝜈 𝜕𝜈 L2 (𝜕G) 𝜕𝜈 𝜕G 𝜕𝜈 If 𝐹 ∈ C (0) (𝜕G), then ∫ lim 𝑚→∞

𝜕G

 12 2 (𝑚) 𝐹 (𝑥) − 𝐹 (𝑥) 𝑑𝜔(𝑥) = 0,

where 𝐹 (𝑚) =

𝑚  ∑︁

𝐹,

𝜕𝑁 𝑛∗ 𝜕𝜈



𝑛=0

The potential 𝑈 ∈ Pot (1) (G 𝑐 ),



𝜕𝑈 + 𝜕𝜈 𝜕G

L2 (𝜕G)

𝜕𝑁 𝑛∗ . 𝜕𝜈

(12.248)

(12.249)

= 𝐹, can be represented in the form

lim sup 𝑈 (𝑥) − 𝑈 (𝑚) (𝑥) = 0

(12.250)

𝜕𝑁 𝑛∗ 𝐹, 𝜕𝜈

(12.251)

𝑚→∞

𝑥∈K

with 𝑈

(𝑚)

=

𝑚  ∑︁ 𝑛=0



𝑁 𝑛∗ L2 (𝜕G)

for every K ⋐ G 𝑐 . The (generalized) Fourier expansions (12.244) and (12.249), indeed, are constructed so as to have the permanence property: The transition from 𝐹 (𝑚) to 𝐹 (𝑚+1) and, therefore, from 𝑈 (𝑚) to 𝑈 (𝑚+1) necessitates merely the addition of one more term, all the other terms obtained formerly remaining unchanged. This is characteristic for orthogonal expansions (generalized Fourier series). In connection with the L2 (𝜕G)-regularity theorems we find the following estimates: (EDP): For given 𝐹 ∈ C (0) (𝜕G), let 𝑈 satisfy 𝑈 ∈ Pot (0) (G 𝑐 ), 𝑈 + |𝜕G = 𝐹. Then 𝑚     ∑︁ (𝑘 ) sup ∇ 𝑈 (𝑥) − (𝐹, 𝐷 ∗𝑛 )L2 (𝜕G) ∇ (𝑘 ) 𝐷 ∗𝑛 (𝑥) 𝑥∈K 𝑛=0 ! 12 𝑚 ∑︁ 2 ∗ 2 ≤ 𝐶 ∥𝐹 ∥ L2 (𝜕G) − (𝐹, 𝐷 𝑛 )L2 (𝜕G) (12.252) 𝑛=0

holds for all 𝑘 ∈ N0 and all subsets K ⋐ G 𝑐 .

12.9 Locally and Globally Uniform Approximation

609

+ (ENP): For given 𝐹 ∈ C (0) (𝜕G), let 𝑈 satisfy 𝑈 ∈ Pot (1) (G 𝑐 ), 𝜕𝑈 𝜕𝜈 𝜕G = 𝐹. Then  𝑚      ∑︁ 𝜕𝑁 𝑛∗ (𝑘 ) sup ∇ 𝑈 (𝑥) − 𝐹, ∇ (𝑘 ) 𝑁 𝑛∗ (𝑥) 𝜕𝜈 L2 (𝜕G) 𝑥∈K 𝑛=0 ! 12  𝑚  ∑︁ 𝜕𝑁 𝑛∗ 2 2 ≤ 𝐶 ∥𝐹 ∥ L2 (𝜕G) − 𝐹, (12.253) 𝜕𝜈 L2 (𝜕G) 𝑛=0

holds for all 𝑘 ∈ N0 and all subsets K ⋐ G 𝑐 . In addition, Corollary 12.48 indicates that 𝐹 − 𝐹 (𝑚) is L2 (𝜕G)-orthogonal to all members of the L2 (𝜕G)-orthonormal Runge basis up to the index 𝑚. This observation is valid for the Dirichlet as well as the Neumann case. It leads us to the following result. Corollary 12.49 Let A, G ⊂ R3 be regular regions such that A ⋐ G holds true. (EDP): Let {𝐷 ∗𝑛 } 𝑛=0,1,... , 𝐷 ∗𝑛 ∈ Pot(A 𝑐 ), be an L2 (𝜕G)-Dirichlet Runge basis satisfying (12.242). If 𝐹 ∈ C (0) (𝜕G), then ∫ lim 𝑚→∞

𝜕G

 12 2 𝐹 (𝑥) − 𝐹 (𝑚) (𝑥) 𝑑𝜔(𝑥) = 0,

(12.254)

where the coefficients 𝑎 0𝑚 , . . . , 𝑎 𝑚 𝑚 of the function 𝐹 (𝑚) =

𝑚 ∑︁

𝑎𝑚 𝑛 𝐷𝑛

(12.255)

𝑛=0

satisfy the “normal equations” 𝑚 ∑︁

𝑎𝑚 𝑛 (𝐷 𝑘 , 𝐷 𝑛 )L2 (𝜕G) = (𝐷 𝑘 , 𝐹)L2 (𝜕G) ,

𝑘 = 0, . . . , 𝑚.

(12.256)

𝑛=0

Then, the potential 𝑈 ∈ Pot (0) (G 𝑐 ), 𝑈 + 𝜕G = 𝐹, can be represented in the form lim sup 𝑈 (𝑥) − 𝑈 (𝑚) (𝑥) = 0

𝑚→∞

(12.257)

𝑥∈K

with 𝑈 (𝑚) =

𝑚 ∑︁ 𝑛=0

for every K ⋐ G 𝑐 .

𝑎𝑚 𝑛 𝐷𝑛

(12.258)

610

12 Potentials in Euclidean Space

(ENP): Let {𝑁 𝑛∗ } 𝑛=0,1,... , 𝑁 𝑛∗ ∈ Pot(A), 𝑛 = 0, 1, . . . be an L2 (𝜕G)-Neumann Runge basis satisfying (12.247). If 𝐹 ∈ C (0) (𝜕G), then ∫ lim 𝑚→∞

𝜕G

 12 2 (𝑚) 𝐹 (𝑥) − 𝐹 (𝑥) 𝑑𝜔(𝑥) = 0,

(12.259)

where the coefficients 𝑎 0𝑚 , . . . , 𝑎 𝑚 𝑚 of the function 𝐹 (𝑚) =

𝑚 ∑︁

𝜕𝑁 𝑛 𝜕G 𝜕𝜈

𝑎𝑚 𝑛

(12.260)

𝑛=0

satisfy the “normal equations” 𝑚 ∑︁

 𝑎𝑚 𝑛

𝜕𝑁 𝑘 𝜕𝑁 𝑛 , 𝜕𝜈 𝜕𝜈

𝑛=0



 = L2 (𝜕G)

Then, the potential 𝑈 ∈ Pot (1) (G 𝑐 ),

𝜕𝑁 𝑘 ,𝐹 𝜕𝜈



𝜕𝑈 + 𝜕𝜈 𝜕G

,

𝑘 = 0, . . . , 𝑚.

= 𝐹, can be represented in the form

lim sup 𝑈 (𝑥) − 𝑈 (𝑚) (𝑥) = 0

𝑚→∞

(12.261)

L2 (𝜕G)

(12.262)

𝑥∈K

with 𝑈 (𝑚) =

𝑚 ∑︁

𝑎𝑚 𝑛 𝑁𝑛

(12.263)

𝑛=0

for every K ⋐ G 𝑐 . The approximation of boundary values and potential by the method of generalized Fourier expansion in terms of outer harmonics is achieved by superposition of functions with “oscillating character”. The oscillations grow in number, but they decrease in size with increasing truncation order. The oscillating character of the generalized Fourier expansions remains true (cf. W. Freeden (1983) for graphical illustrations) if other trial bases are used (for example, mass (mono)poles, certain kernel function representations such as Abel-Poisson and singularity kernels). Thus, generalized Fourier expansions provide least squares approximation by successive oscillations, which become larger and larger in number, but smaller and smaller in amplitude. It is therefore (as already A. Sommerfeld (1978) had pointed out) not a technique of “osculating character” (as, for example, interpolation or smoothing in reproducing Hilbert spaces by harmonic splines as, e.g., proposed by W. Freeden (1981c, 1987a), L. Shure et al. (1982)). As we know, a suitable superposition of outer harmonics leads to so-called kernel functions (such as the Abel-Poisson kernel and the singularity kernel) showing a reduced frequency but increased space localization on the reference sphere Ω𝑅 . The series conglomerates of outer harmonics, i.e., the kernel functions, are constructed

12.9 Locally and Globally Uniform Approximation

611

as to cover various spectral bands. Hence, they show certain intermediate stages of frequency- and space localization. A particular kernel in potential theory is the mass (single-)point kernel, i.e., the fundamental solution to the Laplacian (being the Kelvin transformed counterpart of the singularity kernel). This kernel interrelates the length of its spectral bands to the distance of the mass point from the reference sphere: the closer the point to the sphere, the more space localized and simultaneously the less frequency localized is the fundamental solution (Newton kernel). Closure in C (0) -Topology. From our considerations leading to locally uniform approximation we know for a regular region G with 𝑅 < inf 𝑥 ∈𝜕G |𝑥| that 𝐷+

∥ · ∥ L2 (𝜕G)

𝑅

span {𝐻 −𝑛−1, 𝑗 |𝜕G}

=

∥ · ∥ L2 (𝜕G)

= L2 (𝜕G),

(12.264)

𝑛=0,1,... 𝑗=1,...,2𝑛+1

∥·∥ 2 𝑁 + L (𝜕G)

 =

span 𝑛=0,1,... 𝑗=1,...,2𝑛+1

 ∥ · ∥ L2 (𝜕G) 𝜕 𝑅 𝐻 𝜕G = L2 (𝜕G). 𝜕𝜈 −𝑛−1, 𝑗

(12.265)

The same results remain valid when the regular surface 𝜕G is replaced by any inner parallel surface 𝜕G(−𝜏) of distance |𝜏| to 𝜕G (where |𝜏| is chosen sufficiently small. This fact will be exploited to verify the following closure properties (see W. Freeden (1980a) for more details, in particular the proofs). Theorem 12.50 Let 𝜕G be a regular region with 𝑅 < inf 𝑥 ∈𝜕G |𝑥|. Then the following statements are true: (i) {𝐻 −𝑛−1, 𝑗 |𝜕G} 𝑛∈N0 , 𝑗=1,...,2𝑛+1 is closed in 𝐷 + = C (0) (𝜕G): 𝑅

𝐷+ =

𝑅

span {𝐻 −𝑛−1, 𝑗 |𝜕G}

∥ · ∥ C (0) (𝜕G)

= 𝐷 + = C (0) (𝜕G),

(12.266)

𝑛=0,1,... 𝑗=1,...,2𝑛+1

(ii)



𝑅 𝜕 𝜕𝜈 𝐻 −𝑛−1, 𝑗 𝜕G 𝑛∈N0 , 𝑗=1,...,2𝑛+1  span 𝑛=0,1,... 𝑗=1,...,2𝑛+1

𝜕 𝑅 𝐻 −𝑛−1, 𝑗 𝜕G 𝜕𝜈

is closed in 𝑁 + = C (0) (𝜕G):

 ∥ · ∥ C (0) (𝜕G)

= 𝑁 + = C (0) (𝜕G).

(12.267)

Remark 12.51 The same arguments leading to the C (0) (𝜕G)-closure of outer harmonics on 𝜕G apply to all other systems for which L2 (𝜕G)-closure is known. Combining our results obtained by Theorem 12.50, we easily arrive at the following statement.

612

12 Potentials in Euclidean Space

Theorem 12.52 Let G ⊂ R3 be a regular region with 𝑅 < inf 𝑥 ∈𝜕G |𝑥|. Then the following statements are valid: (EDP): For a given function 𝐹 ∈ 𝐷 + = C (0) (𝜕G), let 𝑈 ∈ Pot (0) (G 𝑐 ) satisfy 𝑈 + |𝜕G = 𝐹. Then, to every 𝜀 > 0, there exist an integer 𝑚(= 𝑚(𝜀)) and a finite set of a real numbers 𝑎 𝑛, 𝑗 such that 𝑚 2𝑛+1 ∑︁ ∑︁ 𝑅 sup 𝑈 (𝑥) − 𝑎 𝑛, 𝑗 𝐻 −𝑛−1, 𝑗 (𝑥) 𝑥 ∈ G𝑐 𝑛=0 𝑗=1 𝑚 2𝑛+1 ∑︁ ∑︁ 𝑅 ≤ sup 𝐹 (𝑥) − 𝑎 𝑛, 𝑗 𝐻 −𝑛−1, 𝑗 (𝑥) 𝑥 ∈𝜕G 𝑛=0 𝑗=1 ≤ 𝜀.

(1) + (0) 𝑐 (ENP): For a given function 𝐹 ∈ 𝑁 = C (𝜕G), let 𝑈 ∈ Pot (G ) satisfy 𝜕𝑈 + 𝜕G = 𝐹. Then, to every 𝜀 > 0, there exist an integer 𝑚(= 𝑚(𝜀)) and a finite set 𝜕𝜈 of real numbers 𝑎 𝑛, 𝑗 such that 𝑚 2𝑛+1 ∑︁ ∑︁ 𝑅 sup 𝑈 (𝑥) − 𝑎 𝑛, 𝑗 𝐻 −𝑛−1, 𝑗 (𝑥) 𝑥 ∈ G𝑐 𝑛=0 𝑗=1 𝑚 2𝑛+1 ∑︁ ∑︁ 𝜕 𝑅 ≤ 𝐶 sup 𝐹 (𝑥) − 𝑎 𝑛, 𝑗 𝐻 −𝑛−1, 𝑗 (𝑥) 𝜕𝜈 𝑥 ∈𝜕G 𝑛=0 𝑗=1

≤ 𝐶𝜀.

12.10 Vector Outer Harmonics and the Gravitational Gradient

In what follows, we extend the results obtained for scalar outer harmonics to the vectorial case (cf. H. Nutz (2002)). It should be noted that the system of vector (𝑖) (𝑖) spherical harmonics { 𝑦˜ 𝑛,𝑚 } and not the system {𝑦 𝑛,𝑚 } is used to generate the set of (𝑖);𝑅 vector outer harmonics {ℎ −𝑛−1,𝑚 }. To be more concrete, the vectorial outer (solid spherical) harmonics (briefly called (𝑖);𝑅 vector outer harmonics) ℎ 𝑛,𝑚 of degree 𝑛, order 𝑚, and kind 𝑖 are given by (1);𝑅 ℎ −𝑛−1,𝑚 (𝑥) =

  𝑛+2   1 𝑅 𝑥 (1) 𝑦˜ 𝑛,𝑚 , 𝑅 |𝑥| |𝑥|

𝑛 = 0, 1, . . . , 𝑚 = 1, . . . 2𝑛 + 1,

12.10 Vector Outer Harmonics and the Gravitational Gradient

613

(12.268) (2);𝑅 ℎ −𝑛−1,𝑚 (𝑥) =



1 𝑅 𝑅 |𝑥|

𝑛

(2) 𝑦˜ 𝑛,𝑚





𝑥 , |𝑥|

𝑛 = 1, 2, . . . , 𝑚 = 1, . . . 2𝑛 + 1, (12.269)

(3);𝑅 ℎ −𝑛−1,𝑚 (𝑥) =



1 𝑅 𝑅 |𝑥|

 𝑛+1

(3) 𝑦˜ 𝑛,𝑚





𝑥 , |𝑥|

𝑛 = 1, 2, . . . , 𝑚 = 1, . . . 2𝑛 + 1, (12.270)

𝑥 ∈ R3 \{0}. (𝑖) From our results about the systems { 𝑦˜ 𝑛,𝑚 }, we are immediately able to deduce the following properties:

(𝑖);𝑅 • ℎ −𝑛−1,𝑚 is of class c (∞) (R3 \{0}), 𝑖 ∈ {1, 2, 3}, (𝑖);𝑅 • Δℎ −𝑛−1,𝑚 = 0 for all 𝑥 ∈ R3 \{0} and 𝑖 ∈ {1, 2, 3}, i.e., every component (𝑖) function ℎ −𝑛−1,𝑚 · 𝜀 𝑘 satisfies the Laplace equation, (𝑖);𝑅 • ℎ −𝑛−1,𝑚 |Ω𝑅 =

• •

1 (𝑖) 𝑅 𝑦˜ 𝑛,𝑚 , 𝑖 ∈ = 𝑂 (|𝑥| −1 ),

{1, 2, 3},

(𝑖);𝑅 |ℎ −𝑛−1,𝑚 (𝑥)| ( 𝑗 );𝑅 (𝑖);𝑅 (ℎ −𝑛−1,𝑚 (𝑥), ℎ −𝑙−1,𝑠 )l2 (Ω𝑅 )

|𝑥| → ∞, 𝑖 ∈ {1, 2, 3}, ∫ ( 𝑗 );𝑅 (𝑖);𝑅 = ℎ −𝑛−1,𝑚 (𝑥) · ℎ −𝑙−1,𝑠 (𝑥) 𝑑𝜔(𝑥) Ω𝑅

𝑖, 𝑗 ∈ {1, 2, 3}.

= 𝛿𝑖 𝑗 𝛿 𝑛𝑙 𝛿 𝑚𝑠 , (𝑖)

] 𝑛 -spaces to be As in the scalar case, we introduce the harm (𝑖)

] 𝑛 (Ωext ) = harm 𝑅

n

span

o (𝑖);𝑅 ext ℎ −𝑛−1,𝑚 Ω𝑅 .

(12.271)

𝑚=1,...,2𝑛+1

Furthermore, (1)

] 0 (Ωext ) = harm ] 0 (Ωext ), harm 𝑅 𝑅 ] 𝑛 (Ωext ) = harm 𝑅

3 Ê

(𝑖)

] 𝑛 (Ωext ). harm 𝑅

(12.272) (12.273)

𝑖=1 (𝑖)

] 𝑝,...,𝑞 (Ωext ), 0𝑖 ≤ 𝑝 ≤ 𝑞, denotes the space As usual, harm 𝑅 (𝑖)

] 𝑝,...,𝑞 (Ωext ) = harm 𝑅

𝑞 Ê 𝑛= 𝑝

(𝑖)

] 𝑛 (Ωext ). harm 𝑅

(12.274)

614

12 Potentials in Euclidean Space

We are now able to formulate the addition theorems for the vector outer harmonics. Again, we have two choices involving Legendre tensors or Legendre vectors, respectively. (𝑖);𝑅 Theorem 12.53 Let {ℎ −𝑛−1,𝑚 } 𝑚=1,...,2𝑛+1 be a system of vector outer harmonics of ext degree 𝑛, order 𝑚, and kind 𝑖. Then, for (𝑥, 𝑦) ∈ Ωext 𝑅 × Ω 𝑅 , the addition theorem for vector outer harmonics reads as follows: 2𝑛+1 ∑︁

𝑅2 |𝑥||𝑦|

 𝑛+2

𝑅2 |𝑥||𝑦|

𝑛 

𝑅2 |𝑥||𝑦|

 𝑛+1 

1 𝑅2



𝑅2 |𝑥||𝑦|

𝑛 

1 𝑅2



𝑅2

(2) ;𝑅 (3) ;𝑅 ℎ−𝑛−1,𝑚 ( 𝑥 ) ⊗ ℎ−𝑛−1,𝑚 ( 𝑦) =

1 𝑅2



(3) ;𝑅 (1);𝑅 ℎ−𝑛−1,𝑚 ( 𝑥 ) ⊗ ℎ−𝑛−1,𝑚 ( 𝑦) =

1 𝑅2

(1) ;𝑅 (1);𝑅 ℎ−𝑛−1,𝑚 ( 𝑥 ) ⊗ ℎ−𝑛−1,𝑚 ( 𝑦) =

1 𝑅2



1 𝑅2



1 𝑅2



𝑚=1

2𝑛+1 ∑︁

(1) ;𝑅 (2);𝑅 ℎ−𝑛−1,𝑚 ( 𝑥 ) ⊗ ℎ−𝑛−1,𝑚 ( 𝑦) =

  2𝑛 + 1 (1,1) 𝑥 𝑦 p˜ 𝑛 , , 4𝜋 |𝑥| |𝑦|

𝑅 |𝑥|

2

𝑚=1

2𝑛+1 ∑︁

(1) ;𝑅 (3) ;𝑅 ℎ−𝑛−1,𝑚 ( 𝑥 ) ⊗ ℎ−𝑛−1,𝑚 ( 𝑦) =

𝑚=1

2𝑛+1 ∑︁

(2) ;𝑅 (1);𝑅 ℎ−𝑛−1,𝑚 ( 𝑥 ) ⊗ ℎ−𝑛−1,𝑚 ( 𝑦) =

2𝑛 + 1 (1,2) p˜ 4𝜋 𝑛



𝑅 2𝑛 + 1 (1,3) p˜ |𝑥| 4𝜋 𝑛

𝑅 |𝑦|

2

𝑚=1

2𝑛 + 1 (2,1) p˜ 4𝜋 𝑛



(12.275)  𝑥 𝑦 , , |𝑥| |𝑦|





(12.276)  𝑥 𝑦 , , |𝑥| |𝑦|

(12.277)  𝑥 𝑦 , , |𝑥| |𝑦| (12.278)

2𝑛+1 ∑︁

(2) ;𝑅 (2);𝑅 ℎ−𝑛−1,𝑚 ( 𝑥 ) ⊗ ℎ−𝑛−1,𝑚 ( 𝑦) =

|𝑥||𝑦|

𝑚=1

2𝑛+1 ∑︁

(3) ;𝑅 (2) ;𝑅 ℎ−𝑛−1,𝑚 ( 𝑥 ) ⊗ ℎ−𝑛−1,𝑚 ( 𝑦) =



2𝑛 + 1 (2,2) 𝑥 𝑦 p˜ , , 4𝜋 𝑛 |𝑥| |𝑦|

(12.279)  𝑅 2𝑛 + 1 (2,3) 𝑥 𝑦 p˜ , , |𝑥||𝑦| |𝑦| 4𝜋 𝑛 |𝑥| |𝑦|   𝑛+1     𝑅2 𝑅 2𝑛 + 1 (3,1) 𝑥 𝑦 p˜ 𝑛 , , |𝑥||𝑦| |𝑦| 4𝜋 |𝑥| |𝑦| 𝑛 





1 𝑅2



𝑅2 |𝑥||𝑦|

𝑛 





1 𝑅2



𝑅2

 𝑛+1

𝑚=1

2𝑛+1 ∑︁



𝑅2

𝑚=1 2𝑛+1 ∑︁

𝑛

𝑚=1

𝑅 2𝑛 + 1 (3,2) p˜ |𝑥| 4𝜋 𝑛

(12.280)  𝑥 𝑦 , , |𝑥| |𝑦| (12.281)

2𝑛+1 ∑︁ 𝑚=1

(3) ;𝑅 (3) ;𝑅 ℎ−𝑛−1,𝑚 ( 𝑥 ) ⊗ ℎ−𝑛−1,𝑚 ( 𝑦) =

|𝑥||𝑦|





2𝑛 + 1 (3,3) 𝑥 𝑦 p˜ , . 4𝜋 𝑛 |𝑥| |𝑦| (12.282)

Combining scalar and vector outer harmonics and observing the concept of Legendre vectors, we are led to the following addition theorem:

12.10 Vector Outer Harmonics and the Gravitational Gradient

615

𝑅 Theorem 12.54 Let {𝐻 −𝑛−1,𝑚 } 𝑚=1,...,2𝑛+1 be a system of scalar outer harmonics (𝑖);𝑅 of degree 𝑛 and order 𝑚. Suppose that {ℎ −𝑛−1,𝑚 } 𝑚=1,...,2𝑛+1 forms the associated system of vector outer harmonics of degree 𝑛, order 𝑚, and kind 𝑖. Then, for ext (𝑥, 𝑦) ∈ Ωext 𝑅 × Ω 𝑅 , the addition theorem for scalar and vector outer harmonics reads as follows 2𝑛+1 ∑︁

(1) ;𝑅 𝑅 ℎ−𝑛−1,𝑚 ( 𝑥 ) 𝐻−𝑛−1,𝑚 ( 𝑦) =

𝑅2 |𝑥||𝑦|

 𝑛+1 

1 𝑅2



𝑅2 |𝑥||𝑦|

𝑛 

1 𝑅2



𝑅2

 𝑛+1

1 𝑅2



𝑚=1

   𝑅 2𝑛 + 1 (1) 𝑥 𝑦 𝑝˜ 𝑛 , , |𝑦| 4𝜋 |𝑥| |𝑦| (12.283)

2𝑛+1 ∑︁

(2) ;𝑅 𝑅 ℎ−𝑛−1,𝑚 ( 𝑥 ) 𝐻−𝑛−1,𝑚 ( 𝑦) =

𝑚=1







𝑅 2𝑛 + 1 (2) 𝑥 𝑦 𝑝˜ 𝑛 , , |𝑥| 4𝜋 |𝑥| |𝑦| (12.284)

2𝑛+1 ∑︁

(3) ;𝑅 𝑅 ℎ−𝑛−1,𝑚 ( 𝑥 ) 𝐻−𝑛−1,𝑚 ( 𝑦) =

|𝑥||𝑦|

𝑚=1





2𝑛 + 1 (3) 𝑥 𝑦 𝑝˜ 𝑛 , . 4𝜋 |𝑥| |𝑦| (12.285)

Our purpose now is to mention some important properties involving vector outer harmonics. Lemma 12.55 (Linear Independence of Vector Outer Harmonics) Assume that (𝑖);𝑅 {ℎ −𝑛−1,𝑚 }𝑖=1,2,3,𝑛=0𝑖 ,..., is a system of vector outer harmonics as defined by (12.268), 𝑚=1,...,2𝑛+1

(12.269) and (12.270). Then, for all 𝑟 > 0, the system (𝑖);𝑅 {ℎ −𝑛−1,𝑚 |Ω𝑟 }𝑖=1,2,3;𝑛=0𝑖 ,..., 𝑚=1,...,2𝑛+1

is linearly independent. Next we are interested in the completeness for vector outer harmonics on Ω𝑅 . Our results can be based on the corresponding theorems of the scalar theory. 𝑅 Lemma 12.56 Let {𝐻 −𝑛−1,𝑚 } 𝑛=0,1,...,𝑚=1,...,2𝑛+1 be a system of scalar outer harmonics. Then ∥ · ∥ l2 (Ω ) 𝑅 𝑅 span{𝐻 −𝑛−1,𝑚 𝜀 𝑖 |Ω𝑅 } = l2 (Ω𝑅 ),

and 𝑅 span{𝐻 −𝑛−1,𝑚 )𝜀 𝑖 |Ω𝑅 }

∥ · ∥ c(Ω 𝑅 )

= c(Ω𝑅 ).

Lemma 12.56 enables us to formulate the following theorem.

616

12 Potentials in Euclidean Space

(𝑖);𝑅 Theorem 12.57 Let {ℎ −𝑛−1,𝑚 }𝑖=1,2,3;𝑛=0𝑖 ,..., be a system of vector outer harmonics 𝑚=1,...,2𝑛+1

as defined by (12.268), (12.269) and (12.270). Then, for all 𝑟 > 0 the following statements hold true: l2 (Ω𝑟 ) = span 𝑖=1,2,3;𝑛=0𝑖 ,...,

(𝑖);𝑅 {ℎ −𝑛−1,𝑚 |Ω𝑟 }

∥ · ∥ l2 (Ω𝑟 )

,

𝑚=1,...,2𝑛+1

and c(Ω𝑟 ) = span 𝑖=1,2,3;𝑛=0𝑖 ,...,

(𝑖);𝑅 {ℎ −𝑛−1,𝑚 |Ω𝑅 }

∥ · ∥ c(Ω𝑟 )

.

𝑚=1,...,2𝑛+1

12.11 Satellite-to-Satellite Tracking

The hi-low SST problem can be described as follows (see W. Freeden (1999), W. Freeden et al. (1999), W. Freeden et al. (2002), and, for alternative approaches, ESA (1996), ESA (1998), ESA (1999) and the references therein): Let Ω𝑅 be the Earth’s surface, and let Ω𝑆 denote the orbital surface of the LEO under consideration. Let there be known the gradient vectors 𝑣(𝑥) = (∇𝑉)(𝑥),

𝑥 ∈ Ω𝑆 ,

(12.286)

at the flight positions of the LEO. Find the geopotential 𝑉 on Ωext 𝑅 , i.e., on and outside the Earth’s surface Ω𝑅 . The mathematical scenario of the lo-lo SST problem is characterized as follows (see the explanations in ESA (1996, 1998, 1999): Let there be known the vectors 𝑣(𝑥) = (∇𝑉)(𝑥) and 𝑣(𝑥 + ℎ(𝑥)) = (∇𝑉)(𝑥 + ℎ(𝑥)), 𝑥 ∈ Ω𝑆 . Find 𝑉 on Ωext 𝑅 from the values 𝑣(𝑥) − 𝑣(𝑥 + ℎ(𝑥)). We begin our considerations with the uniqueness of the SST problem from given vector values (in spherical geometry). Theorem 12.58 Suppose that X ⊂ Ω𝑆 (i.e., the subset of observational points on the satellite orbit Ω𝑆 , 𝑆 > 𝑅) is a dense system on Ω𝑆 . If 𝑣 satisfies ∇ · 𝑣 = 0, 𝐿 · 𝑣 = 0 in Ωext 𝑅 such that 𝑣(𝑥) = 0, 𝑥 ∈ X, then 𝑣 = 0 in Ωext 𝑅 . Proof Any field 𝑣 satisfying ∇ · 𝑣 = 0, 𝐿 · 𝑣 = 0 in Ωext 𝑅 can be expressed in the form ∇𝑉, hence, the coordinate functions 𝑣 · 𝜀 𝑖 , 𝑖 = 1, 2, 3, satisfy

12.11 Satellite-to-Satellite Tracking

617





Δ 𝑣 · 𝜀 𝑖 = Δ 𝜀 𝑖 · ∇ 𝑉 = 𝜀 𝑖 · ∇ Δ𝑉 = 0

(12.287)

ext in Ωext 𝑅 . Note that the harmonic function 𝑉 is arbitrarily often differentiable in Ω 𝑅 . 𝑖 Moreover, according to our assumption, (𝜀 · ∇)𝑉 (𝑥) = 0 for all points 𝑥 of the dense 𝑖 system X on Ω𝑆 , hence, X is a fundamental system in Ωext 𝑅 . This implies 𝑣 · 𝜀 = 0 ext in Ω𝑅 , 𝑖 = 1, 2, 3, as required. □

Furthermore, we are able to verify the following result. Theorem 12.59 Suppose that X ⊂ Ω𝑆 , 𝑆 > 𝑅, is a dense system of points on the satellite orbit Ω𝑆 , 𝑆 > 𝑅. If 𝑣 satisfies ∇ · 𝑣 = 0, 𝐿 · 𝑣 = 0 in Ωext 𝑅 with (−𝑥) · 𝑣(𝑥) = 0,

𝑥 ∈ X,

(12.288)

then 𝑣 = 0 in Ωext 𝑅 . Proof We again base our arguments on the identity 𝑣 = ∇𝑉. The potential 𝑉 |Ωext 𝑆 , 𝑆 > 𝑅, may be expanded in terms of outer harmonics 𝑉 (𝑥) =

∞ 2𝑛+1 ∑︁ ∑︁

𝑉

∧L2 (Ω

𝑆)

𝑆 (𝑛, 𝑘)𝐻 −𝑛−1,𝑘 (𝑥),

𝑥 ∈ Ωext 𝑆 ,

(12.289)

𝑛=0 𝑘=1

where 𝑉

∧L2 (Ω

𝑆)

(𝑛, 𝑘), 𝑛 = 0, 1, . . ., 𝑘 = 1, . . . , 2𝑛 + 1, are the expansion coefficients ∫ ∧ 𝑆 𝑉 L2 (Ω𝑆 ) (𝑛, 𝑘) = 𝑉 (𝑥)𝐻 −𝑛−1,𝑘 (𝑥) 𝑑𝜔(𝑥), (12.290) Ω𝑆

and the series expansion in (12.289) is absolutely and uniformly convergent in Ωext 𝑆 . It is not hard to see that ∞ 2𝑛+1

−

∑︁ ∑︁ 𝑛 + 1 ∧ 𝑥 𝑆 · (∇𝑉)(𝑥) = 𝑉 L2 (Ω𝑆 ) (𝑛, 𝑘)𝐻 −𝑛−1,𝑘 (𝑥), |𝑥| |𝑥| 𝑛=0 𝑘=1

𝑥 ∈ Ωext 𝑆 . (12.291)

Hence, 𝑥 ↦→ (−𝑥) · (∇𝑉)(𝑥),

𝑥 ∈ Ωext 𝑆 ,

(12.292)

ext ext is continuous in Ωext 𝑆 , twice continuously differentiable in Ω𝑆 , harmonic in Ω𝑆 , and regular at infinity. By virtue of (−𝑥) · (∇𝑉)(𝑥) = 0 for all 𝑥 ∈ X, we therefore obtain ∞ 2𝑛+1 ∑︁ ∑︁

𝑉

∧L2 (Ω

𝑆)

𝑆 (𝑛, 𝑘)(𝑛 + 1)𝐻 −𝑛−1,𝑘 (𝑥) = 0,

𝑥 ∈ X.

(12.293)

𝑛=0 𝑘=1

Since X is assumed to be a dense system on Ω𝑆 , 𝑆 > 𝑅, the identity (12.293) holds true for all 𝑥 ∈ Ωext 𝑆 . The completeness property of the theory of spherical harmonics

618

12 Potentials in Euclidean Space

then tells us that 𝑉

∧L2 (Ω

hence, 𝑉

𝑆)

(𝑛, 𝑘)(𝑛 + 1) = 0,

∧L2 (Ω

𝑆)

(𝑛, 𝑘) = 0

(12.294) (12.295)

for all 𝑛 = 0, 1, . . ., 𝑘 = 1, . . . , 2𝑛 + 1. This yields 𝑉 = 0 in Ωext 𝑆 . By analytical ext . This is the desired continuation, we obtain 𝑉 = 0 in Ωext , hence, 𝑣 = 0 in Ω 𝑅 𝑅 result. □ Theorem 12.59 mathematically means that the external gravitational field is uniquely recoverable from (negative) “radial derivatives” corresponding to a fundamental system X on the satellite orbit. In other words, the external gravitational field is uniquely detectable on and outside Ω𝑅 from SST data corresponding gradient vectors given on a dense system X on the satellite orbit Ω𝑆 . In conclusion, on the one side, the results concerning satellite-to-satellite tracking (SST) show that, for the problem of developing the gravitational potential in Ωext 𝑅 from given gradients in point systems on spherical orbits, it suffices to prescribe, e.g., the normal (i.e., radial) component (cf. Theorem 12.59). On the other side, the gravitational potential can be detected alternatively from a dense system of surface gradients on the satellite orbit. Both aspects of gravitational field determination from SST-data will be described finally in more detail: (2) (Ωext ) satisfying Δ𝑉 = 0 in Let the gravitational potential 𝑉 ∈ C(Ωext 𝑅 𝑅 ) ∩ C 1 1 Ωext 𝑅 , |𝑉 (𝑥)| = 𝑂 ( | 𝑥 | ), |∇𝑉 (𝑥)| = 𝑂 ( | 𝑥 | 2 ), |𝑥| → ∞, be given in the form

𝑉=

∞ 2𝑛+1 ∑︁ ∑︁

𝑉

∧L2 (Ω

𝑅)

𝑅 (𝑛, 𝑘)𝐻 −𝑛−1,𝑘 .

(12.296)

𝑛=0 𝑘=1

We have to derive 𝑉 from its gravitational gradient ∇𝑉 on the external sphere Ω𝑆 . For that purpose, we remember the decomposition of ∇𝑉 into its horizontal and tangential parts: 𝜕𝑉 (𝑟𝜉) 1 ∗ (∇𝑉)(𝑆𝜉) = 𝜉 + ∇ 𝜉 𝑉 (𝑟𝜉) , 𝑆 > 𝑅. (12.297) 𝜕𝑟 𝑟 𝑟=𝑆 𝑟=𝑆 First, we show that 𝑉 (as defined by (12.296)) is uniquely determined by its normal derivative on Ω𝑆 . Observing the identity 𝜕𝑉 (𝑟𝜉) 𝜕𝑟 𝑉 (𝑟𝜉) | 𝑟=𝑆 = (12.298) 𝜕𝑟 𝑟=𝑆 ∞ 2𝑛+1 ∑︁ ∑︁ ∧ 𝑆 = 𝑉 L2 (Ω𝑅 ) (𝑛, 𝑘)Λ𝜕 𝑅 (𝑛)𝐻 −𝑛−1,𝑘 (𝑆𝜉) 𝑆

𝑛=0 𝑘=1

12.11 Satellite-to-Satellite Tracking

619

with Λ𝜕 𝑅 (𝑛) = − 𝑆

 𝑛   𝑛+1 𝑛+1 𝑅 𝑛+1 𝑅 =− 𝑆 𝑆 𝑅 𝑆

(12.299)

we find 𝜕𝑟 𝑉 | 𝑟=𝑆 = 0 if and only if 𝑉 = 0. This means that 𝑉 is uniquely determined by its radial derivative 𝜕𝑟 𝑉 on Ω𝑆 . Second, it is not difficult to verify that 1 ∗ ∇ 𝜉 𝑉 (𝑟𝜉) 𝑟 𝑟=𝑆 2 ∑︁ ∞ 2𝑛+1 ∑︁ ∑︁ ∧ (1) = 𝑉 L2 (Ω𝑅 ) (𝑛, 𝑘)𝜆 ∇(𝑖)∗;𝑆 (𝑛)ℎ −𝑛−1,𝑘 (𝑆𝜉),

(12.300)

𝑖=1 𝑘=0𝑖 𝑘=1

where

( 𝜆 ∇(𝑖)∗;𝑆 (𝑛)

=


𝑅 𝑛 𝑆
𝑛+1 𝑅 𝑛 𝑆 𝑆

− 𝑆𝑛

√︃

𝑛+1

, 𝑖 = 1, 2𝑛+1 , 𝑖 = 2.

√︁ 2𝑛+1 𝑛

(12.301)

This leads us to the conclusion that 1 ∗ ∇ 𝑉 (𝑟𝜉)| 𝑟=𝑆 = 0 𝑟 𝜉

(12.302)

if and only if 𝑉 (𝑥) =

𝐶0,1 , |𝑥|

𝐶0,1 ∈ R.

(12.303)

Remark 12.60 Turning over from the gravitational potential 𝑉 to a disturbing potential 𝑇 we obtain the following results: 𝜕𝑟 𝑇 | 𝑟=𝑆 = 0 if and only if 𝑇 = 0, ∇∗;𝑆 𝑇 = 0 if and only if 𝑇 = 0. In other words, the disturbing potential is uniquely determined by its radial derivative or its surface gradient on the (orbital) surface Ω𝑆 , respectively. Finally, we remember √︃ √︁ (1);𝑅 (1);𝑅 𝑅 ∇𝐻 −𝑛−1,𝑘 = − 𝜇˜ 𝑛(1) ℎ −𝑛−1,𝑘 = − (𝑛 + 1)(2𝑛 + 1)ℎ −𝑛−1,𝑘 .

(12.304)

This enables us to write the gravitational field ∇𝑉 as follows: ∇𝑉 = −

∞ 2𝑛+1 ∑︁ ∑︁ 𝑛=0 𝑘=1

𝑉

∧L2 (Ω

𝑅)

√︃ (1);𝑅 (𝑛, 𝑘) 𝜇˜ 𝑛(1) ℎ −𝑛−1,𝑘 .

(12.305)

620

12 Potentials in Euclidean Space

Moreover, it is easily seen that ∫ Ω𝑆

∇𝑉 (𝑦) · ℎ −(1);𝑆 𝑝−1,𝑞 (𝑦) 𝑑𝜔(𝑦) = −

  𝑝+1 √︃ 𝑅 ∧ 𝑉 L2 (Ω𝑅 ) ( 𝑝, 𝑞) 𝜇˜ 𝑛(1) . 𝑆

(12.306)

Therefore, we finally obtain the following reformulation of 𝑉 from the series representation (12.296) 𝑉=

∞ 2𝑛+1 ∑︁ ∑︁ 



(1);𝑆 ∇𝑉, ℎ −𝑛−1,𝑘 l2 (Ω𝑆 )

( 𝜇˜ 𝑛(1) ) −1/2

  𝑛+1 𝑆 𝑅 𝐻 −𝑛−1,𝑘 . 𝑅

(12.307)

𝑛=0 𝑘=1

The last formula expresses the gravitational potential on Ωext 𝑅 in terms of the gravitational gradient on the satellite orbit Ω𝑆 . Essential tools are the vector outer harmonics. The same result holds true for the anomalous potential. Clearly, in (12.307), the equality on Ω𝑅 is understood in L2 (Ω𝑅 )-sense, while the convergence on each Ω𝑟 , 𝑟 > 𝑅 is understood in C(Ω𝑟 )-sense.

12.12 Tensor Outer Harmonics and the Gravitational Tensor

The extension of the vectorial theory of outer harmonics to the tensorial case is (𝑖,𝑘 );𝑅 straightforward. By use of the systems {y˜ 𝑛,𝑚 } of tensor spherical harmonics, (𝑖,𝑘 );𝑅 we are able to write down a set of tensor outer harmonics {h−𝑛−1,𝑚 }, where the arguments are quite similar to the vectorial case. The system of tensor outer harmonics of degree n, order m, and kind (i,k) (𝑖,𝑘 );𝑅 {h−𝑛−1,𝑚 (·)}𝑖,𝑘 ∈ {1,2,3} , 𝑛 = 0˜ 𝑖𝑘 , 0˜ 𝑖,𝑘 + 1, . . . , 𝑚 = 1, . . . , 2𝑛 + 1

(12.308)

is given by (1,1);𝑅 h−𝑛−1,𝑚 (𝑥) (1,2);𝑅 h−𝑛−1,𝑚 (𝑥) (2,1);𝑅 h−𝑛−1,𝑚 (𝑥) (2,2);𝑅 h−𝑛−1,𝑚 (𝑥) (3,3);𝑅 h−𝑛−1,𝑚 (𝑥)

  𝑛+3   1 𝑅 𝑥 (1,1) = y˜ 𝑛,𝑚 , 𝑅 |𝑥| |𝑥|   𝑛+1   1 𝑅 𝑥 (1,2) = y˜ 𝑛,𝑚 , 𝑅 |𝑥| |𝑥|   𝑛+1   1 𝑅 𝑥 (2,1) = y˜ 𝑛,𝑚 , 𝑅 |𝑥| |𝑥|   𝑛−1   1 𝑅 𝑥 (2,2) ˜ = y𝑛,𝑚 , 𝑅 |𝑥| |𝑥|   𝑛+1   1 𝑅 𝑥 (3,3) = y˜ 𝑛,𝑚 , 𝑅 |𝑥| |𝑥|

(12.309) (12.310) (12.311) (12.312) (12.313)

12.12 Tensor Outer Harmonics and the Gravitational Tensor (1,3);𝑅 h−𝑛−1,𝑚 (𝑥) =

621

  𝑛+2   1 𝑅 𝑥 (1,3) y˜ 𝑛,𝑚 , 𝑅 |𝑥| |𝑥|

 𝑛   1 𝑅 𝑥 (2,3) = y˜ 𝑢𝑛,𝑚 , 𝑅 |𝑥| |𝑥|   𝑛+2   1 𝑅 𝑥 (3,1);𝑅 (3,1) h−𝑛−1,𝑚 (𝑥) = y˜ 𝑛,𝑚 , 𝑅 |𝑥| |𝑥|  𝑛   1 𝑅 𝑥 (3,2);𝑅 (3,2) h−𝑛−1,𝑚 (𝑥) = y˜ 𝑛,𝑚 , 𝑅 |𝑥| |𝑥| (2,3);𝑅 h−𝑛−1,𝑚 (𝑥)

(12.314)

(12.315) (12.316) (12.317) (𝑖, 𝑗 )

𝑥 ∈ R3 \{0}, 𝑛 = 0˜ 𝑖𝑘 , 0˜ 𝑖𝑘 + 1, . . . , 𝑚 = 1, . . . , 2𝑛 + 1 (for the definition of the y˜ 𝑛,𝑚 , see Chapter 7). The following properties are valid: (𝑖,𝑘 );𝑅 • h−𝑛−1,𝑚 is of class c (∞) (R3 \{0}), (𝑖,𝑘 );𝑅 (𝑖,𝑘 );𝑅 • Δh−𝑛−1,𝑚 (𝑥) = 0 for 𝑥 ∈ R3 \{0}, i.e., the component functions of h𝑛,𝑚 fulfill the Laplace equation, (𝑖,𝑘 );𝑅 (𝑖,𝑘 ) • h−𝑛−1,𝑚 |Ω𝑅 = 𝑅1 y˜ 𝑛,𝑚 , (𝑖,𝑘 );𝑅 • |h−𝑛−1,𝑚 (𝑥)| = 𝑂 (|𝑥| −1 ),

•

( 𝑝,𝑞);𝑅 (𝑖,𝑘 );𝑅 (h−𝑛−1,𝑚 , h−𝑙−1,𝑠 )l2 (Ω𝑅 )

|𝑥| → ∞, = 𝛿𝑖 𝑝 𝛿 𝑘𝑞 𝛿 𝑛𝑙 𝛿 𝑚𝑠 .

In analogy to the vector case, we set n o (𝑖,𝑘 ) ] 𝑛 (Ωext ) = span h (𝑖,𝑘 );𝑅 (·)|Ωext 𝑚 = 1, . . . , 2𝑛 + 1 , harm 𝑅 𝑅 −𝑛−1,𝑚

(12.318)

and (𝑖,𝑘 )

] 𝑝,...,𝑞 (Ωext ) = harm 𝑅

𝑞 Ê

(𝑖,𝑘 )

]𝑛 harm

(Ωext 𝑅 ),

0˜ 𝑖𝑘 ≤ 𝑝 ≤ 𝑞.

(12.319)

𝑛= 𝑝

As in the case of spherical harmonics, we define ] 0 (Ωext ) = harm 𝑅

3 Ê

(𝑖,1)

(Ωext 𝑅 )

]0 harm

(12.320)

𝑖=1 3 Ê

] 1 (Ωext ) = harm 𝑅

(𝑖,𝑘 )

]1 harm

(Ωext 𝑅 ),

(12.321)

𝑖,𝑘=1 (𝑖,𝑘 )≠(2,2), (3,2)

] 𝑛 (Ωext ) = harm 𝑅

3 Ê 𝑖,𝑘=1

(𝑖,𝑘 )

]𝑛 harm

(Ωext 𝑅 ),

𝑛 ≥ 2.

(12.322)

622

12 Potentials in Euclidean Space

The addition theorems can be formulated in analogy to the vectorial case both for the tensor product of two tensor outer harmonics and for the product of a scalar and a tensor outer harmonic. They are very easy to derive, but lengthy to write down; so, we will omit them. In parallel to the vectorial case, we find the following lemma. Lemma 12.61 (Linear Independence of Tensor Outer Harmonics) Let (𝑖,𝑘 ); {h−𝑛−1,𝑚 }𝑖,𝑘=1,2,3;𝑛=0˜ 𝑖𝑘 ,..., be a system of tensor outer harmonics as defined in 𝑚=1,...,2𝑛+1

(12.309)–(12.317). Then, for all 𝑟 > 0 the system n o (𝑖,𝑘 ) h𝑛,𝑚 |Ω𝑟 𝑖,𝑘=1,2,3;𝑛=0˜ 𝑖𝑘 ,...,

(12.323)

𝑚=1,...,2𝑛+1

is linearly independent for all Ω𝑟 . It is obvious from the corresponding results of scalar outer harmonics that the following lemma holds true. 𝑅 Lemma 12.62 Let {𝐻 −𝑛−1,𝑚 } 𝑛=0,1,...,𝑚=1,...,2𝑛+1 be a system of scalar outer harmonics. Then 𝑅 span{𝐻 −𝑛−1,𝑚 𝜀 𝑖 ⊗ 𝜀 𝑘 |Ω𝑟 }

∥ · ∥ l2 (Ω𝑟 )

𝑅 span{𝐻 −𝑛−1,𝑚 𝜀 𝑖 ⊗ 𝜀 𝑘 |Ω𝑟 )}

∥ · ∥ c(Ω𝑟 )

= l2 (Ω𝑟 ),

(12.324)

= c(Ω𝑟 ).

(12.325)

Finally, we mention the following theorem. (𝑖,𝑘 );𝑅 Theorem 12.63 Let {h−𝑛−1,𝑚 }𝑖,𝑘=1,2,3,𝑛=0˜ 𝑖𝑘 ,..., be a system of tensor outer har𝑚=1,...,2𝑛+1

monics. Then, for all 𝑟 > 0, the following statements hold true: l2 (Ω𝑟 ) = span

𝑖,𝑘=1,2,3;𝑛=0˜ 𝑖𝑘 ,...,

(𝑖,𝑘 );𝑅 (h−𝑛−1,𝑚 )|Ω𝑟

∥ · ∥ l2 (Ω𝑟 )

,

(12.326)

.

(12.327)

𝑚=1,...,2𝑛+1

c(Ω𝑟 ) = span

𝑖,𝑘=1,2,3;𝑛=0˜ 𝑖𝑘 ,..., 𝑚=1,...,2𝑛+1

(𝑖,𝑘 );𝑅 (h−𝑛−1,𝑚 )|Ω𝑟

∥ · ∥ c(Ω𝑟 )

12.13 Satellite Gravitation Gradiometry

623

12.13 Satellite Gravitation Gradiometry

Our intent now is to provide an overview of satellite gravitational gradiometry (SGG), i.e., the measurement of the relative acceleration of test masses at different locations inside one satellite. In an idealized situation, free of non-gravitational influences, the acceleration vector of a proof mass in free fall at the center 𝑥 of mass of a space vehicle is, according to Newton’s law, equal to the gradient of the gravitational potential: 𝑣 = ∇𝑉. Considering now the motion of a second proof mass at 𝑦 close to 𝑥 relative to the first one, its acceleration is in the linearized sense 𝑣(𝑦) ≈ 𝑣(𝑥) + v(𝑥)(𝑦 − 𝑥). The matrix v(𝑥) = (∇𝑣)(𝑥) is the Hesse matrix v(𝑥) = ∇ ⊗ ∇𝑉 (𝑥) consisting of all second order derivatives of the gravitational potential 𝑉. Because of its tensor properties, v is called the gravitational tensor. In other words, measurements of the relative accelerations between two test masses provide information about the second order partial derivatives of the gravitational potential 𝑉. In an ideal observational situation, the full Hesse matrix is available by an array of test masses. An illustrative view on satellite gradiometry based on Newton’s theory of gravitation is as follows: Newton, when working on his law of gravitation, is said to have been inspired by a falling apple. Referring to the theory of gravitation as the tale of the falling apple, it would be appropriate to view gradiometry as the story of two falling apples. In C.W. Misner et al. (1973) this point is made clear. In one of their examples, it is shown that by measuring the relative distance between the shortest paths taken by two ants walking on the skin of an apple, from two adjacent beginning to two adjacent end points, the geometry of its curved surface can be derived. Translated to our case, shortest path means geodesic or free fall of two test particles (apples), from the relative motion of which the geometry of the curved space can be inferred, curved by the gravitational field of the Earth. Interpreting gravitation in terms of geometry in the sense of Einstein, when all nine observable gradient components are measured at a point, gradiometry shows the complete local geometry of the relative motion of adjacent proof masses in free fall. However, it is more practical to constrain their relative motion by highly sensitive springs and measure instead the tension and compression of the springs. This is equivalent to saying that a gradiometer is realized by a coupled system of highly sensitive micro accelerometers. The gravitational force acting on the test masses results in an elongation of the springs (see, e.g., R. Rummel (2015)), where (assuming linear stiffness) the elongation is proportional to the forces. Measuring the differences of the two elongations gives information on the differences of the forces, which is an approximation of the space derivative of the force field, and thus an approximation of the order derivative of the potential. In conclusion, the mathematical formulation of the SGG problem (after separating all non-gravitational influences) is as follows: Let there be known from the gravitational field 𝑣 on Ωext 𝑅 the gradients

624

12 Potentials in Euclidean Space

v = (∇𝑣) = ∇ ⊗ ∇𝑉,

(12.328)

on the orbital surface Ω𝑆 of the LEO satellite. Find the geopotential 𝑉 from the knowledge of the gravitational tensor ∇ ⊗ ∇𝑉 on Ω𝑆 . We first deal with the problem of uniqueness corresponding to the model situation of a system X ⊂ Ω𝑆 of known SGG-data. Theorem 12.64 Suppose that X ⊂ Ω𝑆 (i.e., the subset of observational points on the satellite orbit Ω𝑆 , 𝑆 > 𝑅) is a dense system on Ω𝑆 . If v satisfies ∇ · v = 0, L · v = 0 in Ωext 𝑅 such that v(𝑥) = 0, 𝑥 ∈ X, (2) then the associated field 𝑣 satisfying ∇ · 𝑣 = 0, 𝐿 · 𝑣 = 0 in Ωext 𝑅 with v = ∇𝑣 = ∇ 𝑉

satisfies 𝑣 = 0 in Ωext 𝑅 . Proof Any field v with ∇ · 𝑣 = 0, 𝐿 · 𝑣 = 0, in Ωext 𝑅 can be expressed in the (2) form ∇ 𝑉 = (∇ ⊗ ∇)𝑉. Furthermore, the coordinate functions v𝑖 𝑗 = 𝜀 𝑖 · v𝜀 𝑗 , ext 𝑖, 𝑗 ∈ {1, 2, 3}, satisfy Δv𝑖 𝑗 = 0 in Ωext 𝑅 . This implies v𝑖 𝑗 = 0 in Ω 𝑅 , 𝑖, 𝑗 ∈ {1, 2, 3}, ext (2) since X is a fundamental system in Ω𝑅 . From v = ∇ 𝑉 = (∇ ⊗ ∇)𝑉 = 0, we finally get 𝑉 = 0 in Ωext □ 𝑅 and, thus, 𝑣 = ∇𝑉 = 0, as required. In other words, the Earth’s external gravitational field 𝑣 is uniquely detectable on and outside the Earth’s surface Ω𝑅 if SGG data (i.e., second order derivatives of the Earth’s gravitational potential 𝑉) are given on a dense system X (on the satellite orbit Ω𝑆 ). Furthermore, we are able to verify the following result. Theorem 12.65 Suppose that X is a dense system of points on Ω𝑆 . If v satisfies ∇ · v = 0, 𝐿 · v = 0 in Ωext 𝑅 with 𝑥 · (v(𝑥)𝑥) = 0,

𝑥 ∈ X,

(2) then 𝑣 = 0 in Ωext 𝑅 (with v = ∇𝑣 = ∇ 𝑉).

Proof We base our arguments on the identity v(𝑥) = ∇ (2) 𝑉 (𝑥) = (∇ ⊗ ∇)𝑉 (𝑥),

𝑥 ∈ Ωext 𝑅 .

(12.329)

ext From our assumptions, it is clear that Ωext 𝑆 , 𝑆 > 𝑅, is a strict subset of Ω 𝑅 . Clearly, the potential 𝑉 |Ωext may be expanded in terms of outer harmonics 𝑆

𝑉 (𝑥) =

∞ 2𝑛+1 ∑︁ ∑︁ 𝑛=0 𝑘=1

𝑉

∧L2 (Ω

𝑆)

𝑆 (𝑛, 𝑘)𝐻 −𝑛−1,𝑘 (𝑥),

𝑥 ∈ Ωext 𝑆 ,

(12.330)

12.13 Satellite Gravitation Gradiometry

625

where 𝑉 ∧L2 (Ω𝑠 ) (𝑛, 𝑘) are the orthogonal coefficients (12.290). By elementary calculations, we obtain       𝑥 𝑥 𝑥 𝑥 · ∇ 𝑥(2) 𝑉 (𝑥) = · ∇𝑥 · ∇ 𝑥 𝑉 (𝑥) (12.331) |𝑥| |𝑥| |𝑥| |𝑥| ∞ 2𝑛+1 ∑︁ ∑︁ (𝑛 + 1)(𝑛 + 2) ∧ 𝑆 𝑉 L2 (Ω𝑆 ) (𝑛, 𝑘)𝐻 −𝑛−1,𝑘 = (𝑥), 2 |𝑥| 𝑛=0 𝑘=1 𝑥 ∈ Ωext 𝑆 . Hence,   𝑥 ↦→ 𝑥 · ∇ (2) 𝑉 (𝑥)𝑥,

𝑥 ∈ Ωext 𝑆 ,


(2) is a harmonic function in Ωext 𝑆 . In accordance with 𝑥 · (∇ 𝑉)(𝑥)𝑥 = 0, 𝑥 ∈ X, we thus obtain ∞ 2𝑛+1 ∑︁ ∑︁

𝑉

∧L2 (Ω

𝑆)

𝑆 (𝑛, 𝑘)(𝑛 + 1)(𝑛 + 2)𝐻 −𝑛−1,𝑘 (𝑥) = 0,

𝑥 ∈ X.

(12.332)

𝑛=0 𝑘=1 ext Since X is a fundamental system in Ωext 𝑅 , the identity (12.332) holds true in Ω 𝑅 . The theory of spherical harmonics then tells us that

𝑉

∧L2 (Ω

𝑅)

(𝑛, 𝑘)(𝑛 + 1)(𝑛 + 2) = 0,

(12.333)

∧

hence, 𝑉 L2 (Ω𝑆 ) (𝑛, 𝑘) = 0 for 𝑛 = 0, 1, . . ., 𝑘 = 1, . . . , 2𝑛 + 1. This yields 𝑉 = 0 in ext Ωext 𝑆 . By analytical continuation, we have 𝑉 = 0 in Ω𝑆 , and hence 𝑣 = ∇𝑉 = 0 in Ωext 𝑅 .

□

Theorem 12.65 means that the Earth’s external gravitational field is uniquely recoverable from “second radial derivatives” corresponding to a fundamental system X ⊂ Ω𝑆 . Next, we remember the decomposition of ∇ ⊗ ∇ into its well-known radial and tangential parts:     𝜕 1 𝜕 1 ∇⊗∇= 𝜉 + ∇∗ ⊗ 𝜉 + ∇∗ (12.334) 𝜕𝑟 𝑟 𝜕𝑟 𝑟 𝜕 𝜕 𝜕 1 = 𝜉⊗ 𝜉+ 𝜉 ⊗ ∇∗ 𝜉 𝜕𝑟 𝜕𝑟 𝜕𝑟 𝑟 1 ∗ 𝜕 1 ∗ + ∇ ⊗ 𝜉 + 2 ∇ ⊗ ∇∗ . 𝑟 𝜕𝑟 𝑟 It will be seen that the operators 𝜕/𝜕𝑟, (𝜕/𝜕𝑟) 2 , ∇∗ , 𝜕/𝜕𝑟 ∇∗ , and ∇∗ ⊗ ∇∗ form the constituting ingredients of the gravitational tensor ∇ ⊗ ∇𝑉. It should be noted that operators 𝜕/𝜕𝑟 and ∇∗ have already been described within the SST-context for

626

12 Potentials in Euclidean Space

specifying the gravitational gradient. Therefore, it remains to discuss the derivatives (𝜕/𝜕𝑟) 2 , 𝜕/𝜕𝑟 ∇∗ , and ∇∗ ⊗ ∇∗ . First, an easy calculation gives 2

(𝜕𝑟 ) 𝑉 (𝑟𝜉)| 𝑟=𝑆

𝜕 2𝑉 (𝑟𝜉) = 𝜕𝑟 2 𝑟=𝑆 ∞ 2𝑛+1 ∑︁ ∑︁ ∧ 𝑆 = 𝑉 L2 (Ω𝑅 ) (𝑛, 𝑘)Λ(𝑛)𝐻 −𝑛−1,𝑘 ,

(12.335)

𝑛=0 𝑘=1

where     𝑛   𝑛+2 𝑛+1 𝑛+2 𝑅 (𝑛 + 1)(𝑛 + 2) 𝑅 Λ(𝑛) = − − = . 𝑆 𝑆 𝑆 𝑆 𝑅2

(12.336)

Thus it follows that (𝜕 𝑅 ) 2𝑉 = 0 if and only if 𝑉 = 0. 𝑆

Second, we are able to write 1 1 𝜕 ∗ ∗ 𝜕 𝑅 ∇ 𝑉 (𝑟, 𝜉) = ∇ 𝑉 (𝑟𝜉) (12.337) 𝑟 𝑟 𝑟 𝜕𝑟 𝑟=𝑆 𝑟=𝑆 2 ∑︁ ∞ 2𝑛+1 ∑︁ ∑︁ ∧ (𝑖);𝑆 = 𝑉 L2 (Ω𝑅 ) (𝑛, 𝑘)𝜆 𝜕(𝑖)𝑟 ∇∗;𝑆 (𝑛)ℎ −𝑛−1,𝑘 (𝑆𝜉), 𝑖=1 𝑛=0𝑖 𝑘=1

where 𝜆 𝜕(𝑖)∇∗;𝑆 (𝑛) = 𝑟

This shows us that



   

√︃

𝑛(𝑛+1) 𝑛+2 𝑅 𝑛 2𝑛+1 ( 𝑆 ) , 𝑆2 2 √︁ 𝑛 𝑅 𝑛   − (𝑛+1) 2𝑛+1 ( 𝑆 ) , 𝑆2 

 1 𝜕 𝑅 ∇∗𝜉 𝑉 (𝑟𝜉) | 𝑟=𝑆 = 0 𝑟 𝑟

𝑖 = 1,

(12.338)

𝑖 = 2.

(12.339)

if and only if 𝑉 (𝑥) = Third, it can be shown that

𝐶 , |𝑥|

𝐶 ∈ R.

(12.340)

12.13 Satellite Gravitation Gradiometry

627

1 ∗ ∇ ⊗ ∇∗𝑉 (𝑟𝜉)| 𝑟=𝑆 (12.341) 𝑟2 2 ∞ 2𝑛+1 ∑︁ ∑︁ ∑︁ ∧ ) (𝑖,𝑘 );𝑆 = 𝑉 L2 (Ω𝑅 ) (𝑛, 𝑚)𝜆 ∇(𝑖,𝑘 ∗;𝑆 ∇ ∗;𝑆 (𝑛)h −𝑛−1,𝑚 (𝑆𝜉), 𝑖,𝑘=1 𝑛=0˜ 𝑖,𝑘 𝑚=1

where √︃   𝑛(𝑛+1)  𝜇 𝑛(1,1) 𝑆 2 (2𝑛+1) ( 𝑅 )𝑛,   (2𝑛+3) 𝑆 √︃    (𝑛−1)   − 𝜇 𝑛(1,2) 𝑆 2 (𝑛+1)  ( 𝑅 )𝑛, (𝑖,𝑘 ) (2𝑛−1) (2𝑛+1) 𝑆 √︃ 𝜆 ∇∗;𝑆 ⊗∇∗;𝑆 (𝑛) =  (2,1) 𝑛(𝑛+2)   − 𝜇 𝑛 𝑆 2 (2𝑛+3) (2𝑛+1) ( 𝑅𝑆 ) 𝑛 ,   √︃     𝜇 𝑛(2,2) 2𝑛(𝑛+1) (𝑛+2) ( 𝑅𝑆 ) 𝑛 , 𝑆 (2𝑛−1) (2𝑛+1) 

(𝑖, 𝑘) = (1, 1), (𝑖, 𝑘) = (1, 2),

(12.342)

(𝑖, 𝑘) = (2, 1), (𝑖, 𝑘) = (2, 2).

This implies 1 ∗ ∗ ∇ ⊗ ∇ 𝑉 (𝑟𝜉) =0 2 𝑟 𝑟=𝑆

(12.343)

if and only if 𝑉=

1 2𝑛+1 ∑︁ ∑︁

𝑅 𝐶𝑛,𝑚 𝐻 −𝑛−1,𝑚 ,

𝐶𝑛,𝑚 ∈ R.

(12.344)

𝑛=0 𝑚=1

Turning over from the gravitational potential 𝑉 to the disturbing potential 𝑇 (where the Fourier coefficients of order 0 and 1 are zero), we obtain the following result: Corollary 12.66 For the disturbing potential 𝑇, each of the following statements is equivalent to 𝑇 = 0: (i) 𝜕𝑟 𝑇 = 0 on a sphere at satellite’s height, (ii) (𝜕𝑟 ) 2𝑇 = 0 on a sphere at satellite’s height, (iii) ∇∗;𝑆 𝑇 = 0 on a sphere at satellite’s height, (iv) 𝜕𝑟 ∇∗;𝑆 𝑇 = 0 on a sphere at satellite’s height, (v) ∇∗;𝑆 ⊗ ∇∗;𝑆 𝑇 = 0 on a sphere at satellite’s height. Finally, we mention that √︃ 𝑅 ∇ ⊗ ∇𝐻 −𝑛−1,𝑘 =

=

√︁

(1,1);𝑅 𝜇˜ 𝑛(1,1) h−𝑛−1,𝑘

(1,1);𝑅 (𝑛 + 2)(𝑛 + 2)(2𝑛 − 3)(2𝑛 − 1)h−𝑛−1,𝑘 .

(12.345)

628

12 Potentials in Euclidean Space

Consequently, we find ∇ ⊗ ∇𝑉 =

∞ 2𝑛+1 ∑︁ ∑︁

𝑉

∧L2 (Ω

√︃

𝑅)

(1,1);𝑅 (𝑛, 𝑘) 𝜇˜ 𝑛(1,1) h−𝑛−1,𝑚 .

(12.346)

𝑘=0 𝑘=1

This enables us to verify that ∫ ∇ ⊗ ∇𝑉 (𝑦) · Ω𝑆

h−(1,1);𝑆 𝑝−1,𝑞 (𝑦)

  𝑝+2 √︃ 𝑅 ∧L2 (Ω ) 𝑅 𝑑𝜔(𝑦) = 𝑉 ( 𝑝, 𝑞) 𝜇˜ 𝑛(1,1) . (12.347) 𝑆

Therefore, from (12.296), we finally obtain the following reformulation of 𝑉, 𝑉=

∞ 2𝑛+1 ∑︁ ∑︁ 𝑛=0

𝑆 (1,1);𝑆 𝑅 (∇ ⊗ ∇𝑉; h−𝑛−1,𝑘 )l2 (Ω𝑆 ) ( 𝜇˜ 𝑛(1,1) ) −1/2 ( ) 𝑛+2 𝐻 −𝑛−1,𝑘 . 𝑅 𝑘=1

(12.348)

This formula expresses the gravitational potential 𝑉 in Ωext 𝑅 in terms of the gravitational tensor ∇ ⊗ ∇𝑉 on the satellite orbit Ω𝑆 . Essential tools are the tensor outer harmonics. The equality in (12.348) is understood in the standard sense.

12.14 Function Systems for Internal Elastic Fields

In the case of motion of material in the Earth’s interior, we are concerned with solid material in the interior of the Earth such that the assumption of a perfect fluid is not valid anymore. 3 For small displacements 𝑑 : Ωint 𝑅 × R → R , Euler’s equation of motion can be linearized in the following way:

𝜌(𝑥, 𝑡)

𝜕2 𝑑 (𝑥, 𝑡) = 𝑓 (𝑥, 𝑡) + ∇ 𝑥 · f (𝑥, 𝑡), 𝜕𝑡 2

(12.349)

where f (𝑥, 𝑡) = ( 𝑓𝑖, 𝑗 (𝑥, 𝑡))𝑖, 𝑗=1,2,3 with 𝑓𝑖, 𝑗 (𝑥, 𝑡) =

3 ∑︁ 3 ∑︁ 𝑘=1 𝑙=1

Ξ𝑖, 𝑗,𝑘,𝑙 (𝑥, 𝑡)

  1 𝜕𝑑 𝑘 (𝑥, 𝑡) 𝜕𝑑𝑙 (𝑥, 𝑡) + , 2 𝜕𝑥𝑙 𝜕𝑥 𝑘

(12.350)

𝑖, 𝑗 ∈ {1, 2, 3}. The occuring tensor of rank 4, 𝚵, is called elasticity tensor with the following symmetries Ξ𝑖, 𝑗,𝑘,𝑙 = Ξ 𝑘,𝑙,𝑖, 𝑗 = Ξ𝑖, 𝑗,𝑙,𝑘 . (12.351) An idealized case is an isotropic medium, where we have

12.14 Function Systems for Internal Elastic Fields

˜ 𝑡)𝛿𝑖 𝑗 𝛿 𝑘𝑙 + 𝜇(𝑥, Ξ𝑖, 𝑗,𝑘,𝑙 (𝑥, 𝑡) = 𝜆(𝑥, ˜ 𝑡)(𝛿𝑖𝑘 𝛿 𝑗𝑙 + 𝛿𝑖𝑙 𝛿 𝑗 𝑘 ),

629

(12.352)

where 𝜆˜ and 𝜇˜ are the so-called Lam`e parameters. Lemma 12.67 Under the assumption of an isotropic medium, we have for the Cauchy stress tensor   ˜ 𝑡)(∇ · 𝑑 (𝑥, 𝑡))i + 𝜇(𝑥, f (𝑥, 𝑡) = 𝜆(𝑥, ˜ 𝑡) ∇ ⊗ 𝑑 (𝑥, 𝑡) + (∇ ⊗ 𝑑 (𝑥, 𝑡)) 𝑇 . (12.353) Proof For the components of the Cauchy stress tensor, we have 𝑓𝑖, 𝑗 (𝑥, 𝑡)   3
𝜕𝑑 𝑘 (𝑥, 𝑡) 𝜕𝑑𝑙 (𝑥, 𝑡) 1 ∑︁ ˜ = 𝜆(𝑥, 𝑡)𝛿𝑖 𝑗 𝛿 𝑘𝑙 + 𝜇(𝑥, ˜ 𝑡)(𝛿𝑖𝑘 𝛿 𝑗𝑙 + 𝛿𝑖𝑙 𝛿 𝑗 𝑘 ) + 2 𝑘,𝑙=1 𝜕𝑥𝑙 𝜕𝑥 𝑘  3  ∑︁ 1 ˜ 𝜕𝑑 𝑘 (𝑥, 𝑡) 𝜕𝑑 𝑘 (𝑥, 𝑡) = 𝜆(𝑥, 𝑡)𝛿𝑖 𝑗 + 2 𝜕𝑥 𝑘 𝜕𝑥 𝑘 𝑘=1   𝜕𝑑𝑖 (𝑥, 𝑡) 𝜕𝑑 𝑗 (𝑥, 𝑡) 𝜕𝑑 𝑗 (𝑥, 𝑡) 𝜕𝑑𝑖 (𝑥, 𝑡) + 𝜇(𝑥, ˜ 𝑡) + + + 𝜕𝑥 𝑗 𝜕𝑥𝑖 𝜕𝑥𝑖 𝜕𝑥 𝑗   𝜕𝑑𝑖 (𝑥, 𝑡) 𝜕𝑑 𝑗 (𝑥, 𝑡) ˜ = 𝜆(𝑥, 𝑡)𝛿𝑖 𝑗 (∇ · 𝑑 (𝑥, 𝑡)) + 𝜇(𝑥, ˜ 𝑡) + . 𝜕𝑥 𝑗 𝜕𝑥𝑖

Lemma 12.68 Under the assumption of an isotropic medium, Euler’s equation of motion becomes 𝜌(𝑥)

𝜕 2 𝑑 (𝑥, 𝑡) = 𝑓 (𝑥, 𝑡) + ( 𝜇(𝑥, ˜ 𝑡) + 𝜇(𝑥, ˜ 𝑡))∇(∇ · 𝑑 (𝑥, 𝑡)) 𝜕𝑡 2 ˜ 𝑡) + 𝜇(𝑥, + (∇ · 𝑑 (𝑥, 𝑡))∇𝜆(𝑥, ˜ 𝑡)Δ𝑑 (𝑥, 𝑡) + (∇ ⊗ 𝑑 (𝑥, 𝑡) + (∇ ⊗ 𝑑 (𝑥, 𝑡)) 𝑇 )∇𝜇(𝑥, 𝑡).

Proof For the proof of this assertion, we have to calculate the divergence of the Cauchy stress tensor. We split this calculation into two parts and for the first part we obtain ˜ 𝑡)(∇ · 𝑑 (𝑥, 𝑡))i) ∇ · (𝜆(𝑥, ˜ 𝑡)(∇ · 𝑑 (𝑥, 𝑡)) 0 𝜆(𝑥, 0 © ª ˜ 𝑡)(∇ · 𝑑 (𝑥, 𝑡)) 0 𝜆(𝑥, = ∇ · ­0 ® ˜ 0 0 𝜆(𝑥, 𝑡)(∇ · 𝑑 (𝑥, 𝑡)) « ¬ ˜ 𝑡)∇ · 𝑑 (𝑥, 𝑡)) = ∇(𝜆(𝑥, ˜ 𝑡) + 𝜆(𝑥, ˜ 𝑡)∇(∇ · 𝑑 (𝑥, 𝑡)). = (∇𝑑 (𝑥, 𝑡))∇𝜆(𝑥, For the second part, we first get

630

12 Potentials in Euclidean Space



∇ · ∇ ⊗ 𝑑 (𝑥, 𝑡) + (∇ ⊗ 𝑑 (𝑥, 𝑡)) 𝑇



  3 ©∑︁ 𝜕 𝜕𝑑 𝑗 (𝑥, 𝑡) 𝜕𝑑𝑖 (𝑥, 𝑡) ª =­ + ® 𝜕𝑥 𝑗 𝜕𝑥𝑖 𝜕𝑥 𝑗 « 𝑗=1 ¬𝑖=1,2,3 3 3 ©∑︁ 𝜕 𝜕𝑑 𝑗 (𝑥, 𝑡) ª ©∑︁ 𝜕 2 𝑑𝑖 (𝑥, 𝑡) ª =­ +­ ® ® 𝜕𝑥 𝑗 𝜕𝑥𝑖 𝜕𝑥 2𝑗 « 𝑗=1 ¬𝑖=1,2,3 « 𝑗=1 ¬𝑖=1,2,3 = ∇(∇ · 𝑑 (𝑥, 𝑡)) + Δ𝑑 (𝑥, 𝑡).

Thus, we obtain for the second part of ∇ · f, ∇ · ( 𝜇(𝑥, ˜ 𝑡)(∇ ⊗ 𝑑 (𝑥, 𝑡) + (∇ ⊗ 𝑑 (𝑥, 𝑡)) 𝑇 )) = 𝜇(𝑥, ˜ 𝑡) (∇(∇ · 𝑑 (𝑥, 𝑡)) + Δ𝑑 (𝑥, 𝑡))   + ∇ ⊗ 𝑑 (𝑥, 𝑡) + (∇ ⊗ 𝑑 (𝑥, 𝑡)) 𝑇 ∇ 𝜇(𝑥, ˜ 𝑡), □

which finishes the proof.

If the Lam𝑒´ constants 𝜆˜ and 𝜇˜ are real constants that are not dependent on the spatial variable 𝑥, then we are confronted with a homogeneous medium. If we, furthermore, neglect the body forces 𝑓 in the Cauchy-Navier equation, we obtain a simplified version of this equation given by 𝜌(𝑥)

𝜕2 𝑑 (𝑥, 𝑡) = (𝜆˜ + 𝜇)∇(∇ ˜ · 𝑑 (𝑥, 𝑡)) + 𝜇Δ𝑑 ˜ (𝑥, 𝑡). 𝜕𝑡 2

(12.354)

Moreover, when we treat equilibrium problems of an isotropic homogeneous elastic body, the field equations reduce to the Navier equation (also called the CauchyNavier equation)
𝜇Δ𝑑 ˜ (𝑥) + 𝜆˜ + 𝜇˜ ∇(∇ · 𝑑 (𝑥)) = 0, 𝑥 ∈ Ωint (12.355) 𝑅 . This equation plays in the theory of elasticity the same part as the Laplace equation in the theory of harmonic functions, and it formally reduces to it for 𝜇˜ = 1, 𝜆˜ = −1. The Cauchy-Navier equation admits the equivalent formulation ♦𝑑 (𝑥) = Δ𝑑 (𝑥) + 𝜏∇(∇ ˜ · 𝑑 (𝑥)) = 0, 𝑥 ∈ Ωint 𝑅, where 𝜏˜ =

1 𝜆˜ , 𝛿˜ = ˜ 1 − 2𝛿 2(𝜆 + 𝜇) ˜

(𝛿˜ is called Poisson’s ratio). Since

(12.356)

(12.357)

12.14 Function Systems for Internal Elastic Fields

631

Δ𝑑 (𝑥) = ∇(∇ · 𝑑 (𝑥)) − ∇ ∧ (∇ ∧ 𝑑 (𝑥)), 𝑥 ∈ Ωint 𝑅,

(12.358)

we equivalently have
♦𝑑 (𝑥) = 𝜆˜ + 2 𝜇˜ ∇(∇ · 𝑑 (𝑥)) − 𝜇∇ ˜ ∧ (∇ ∧ 𝑑 (𝑥)), 𝑥 ∈ Ωint 𝑅 .

(12.359)

Suppose now that 𝑑 is a (sufficiently often differentiable) vector field satisfying the Navier equation. Then it follows that

0 = 𝜇∇ ˜ · (♦𝑑 (𝑥)) = ∇ · 𝜇Δ𝑑 ˜ (𝑥) + 𝜆˜ + 𝜇˜ ∇ · (∇(∇ · 𝑑 (𝑥)))
= 𝜇Δ(∇ ˜ · 𝑑 (𝑥)) + 𝜆˜ + 𝜇˜ Δ(∇ · 𝑑 (𝑥))
= 𝜆˜ + 2 𝜇˜ Δ(∇ · 𝑑 (𝑥)), (12.360)
0 = 𝜇∇ ˜ ∧ (♦𝑑 (𝑥)) = 𝜇Δ(∇ ˜ ∧ 𝑑 (𝑥)) + 𝜆˜ + 𝜇˜ ∇ ∧ (∇(∇ · 𝑑 (𝑥))) = 𝜇Δ(∇ ˜ ∧ 𝑑 (𝑥)),

(12.361)


0 = 𝜇Δ(♦𝑑) ˜ = 𝜇ΔΔ𝑑 ˜ (𝑥) + 𝜆˜ + 𝜇˜ ∇(Δ(∇ · 𝑑 (𝑥))) = 𝜇ΔΔ𝑑 ˜ (𝑥).

(12.362)

Summarizing our results, we therefore obtain for a sufficiently often differentiable 3 int field 𝑑 : Ωint 𝑅 → R satisfying ♦𝑑 (𝑥) = 0 , 𝑥 ∈ Ω 𝑅 : Δ(∇ · 𝑑 (𝑥)) = 0, 𝑥 ∈ Ωint 𝑅,

(12.363)

Ωint 𝑅, int Ω𝑅 .

(12.364)

Δ(∇ ∧ 𝑑 (𝑥)) = 0, 𝑥 ∈ Δ(Δ𝑑 (𝑥)) = 0, 𝑥 ∈

(12.365)

In other words, our considerations have led to the conclusions that the displacement field 𝑑 is biharmonic, and its divergence and curl are harmonic. This shows a deep relation between linear elasticity and potential theory. Moreover it should be noted that, according to the invariance of the differential operators ∇, Δ with respect to orthogonal transformations, we are able to derive that ♦𝑑 = 0 is
equivalent to ♦ t𝑇 𝑑 (t·) = 0 for all orthogonal transformations t (for 𝑑 ∈ c (2) (Ωint 𝑅 )). Let nav𝑛 (more explicitly: nav𝑛 (R3 )) be the class of homogeneous vector polynomials of degree 𝑛 satisfying Navier’s equations in R3 :   𝜆˜ + 𝜇˜ nav𝑛 = 𝑢 ∈ hom𝑛 ♦𝑢 = Δ𝑢 + 𝜏∇(∇ · 𝑢) = 0, 𝜏 = . (12.366) 𝜇˜

632

12 Potentials in Euclidean Space

Remark 12.69 If 𝜏 = 0, (12.366) leads back to the space of harm𝑛 (R3 ) of vectorial harmonic polynomials (well known from W. Freeden et al. (1994)). Every vector field 𝑢 ∈ nav𝑛 can be written in the form 𝑢(𝑥) =

𝑛 ∑︁

𝑐 𝑛− 𝑗 (𝑥1 , 𝑥2 ) 𝑥 3 , 𝑥 ∈ R3 , 𝑗

𝑥 = (𝑥1 , 𝑥2 , 𝑥3 ) 𝑇 ,

(12.367)

𝑗=0

where 𝑐 𝑛− 𝑗 : R2 → R3 denote homogeneous vector polynomials of degree 𝑛 − 𝑗. It readily can be seen that ♦𝑢 allows the following representation: ♦𝑢(𝑥) = Δ𝑢(𝑥) + 𝜏∇(∇ · 𝑢(𝑥)) =a

𝜕2 𝜕𝑥 32

𝑢(𝑥) + b 𝑥

(12.368)

𝜕 𝑢(𝑥) + c 𝑥 , 𝑥 ∈ R3 , 𝜕𝑥3

where we have used the matrix operators a, b 𝑥 , c 𝑥 given by 10 0 © ª a = ­0 1 0 ®, «0 0 1 + 𝜏¬ 0 0 𝜏 𝜕𝑥𝜕 1 © 0 𝜏 𝜕𝑥𝜕 2 b 𝑥 = ­­ 0 𝜕 𝜕 « 𝜏 𝜕𝑥1 𝜏 𝜕𝑥2 0 2

𝜕 © (1 + 𝜏) 𝜕𝑥12 + ­ 𝜏 𝜕𝑥𝜕 2 𝜕𝑥𝜕 1 c 𝑥 = ­­ ­ 0 «

𝜕2 𝜕𝑥22

ª ®, ® ¬ 𝜏 𝜕𝑥𝜕 1 𝜕2 𝜕𝑥12

𝜕 𝜕𝑥2

+ (1 + 0

𝜕2 𝜏) 𝜕𝑥 2 2

0 ª ® ®. ® ® 2 𝜕 + 𝜕𝑥 2 . 2 ¬ 0

𝜕2 𝜕𝑥12

Observing the fact that 𝑛−1

∑︁ 𝜕𝑢 𝑗 (𝑥1 , 𝑥2 , 𝑥3 ) = ( 𝑗 + 1)𝑐 𝑛− 𝑗 −1 (𝑥1 , 𝑥2 )𝑥3 , 𝜕𝑥3 𝑗=0

(12.369)

𝑛−2

∑︁ 𝜕2𝑢 𝑗 (𝑥 , 𝑥 , 𝑥 ) = ( 𝑗 + 2)( 𝑗 + 1)𝑐 𝑛− 𝑗 −2 (𝑥1 , 𝑥2 )𝑥3 1 2 3 2 𝜕𝑥3 𝑗=0

(12.370)

we obtain from (12.367) the recursion relation ( 𝑗 + 2)( 𝑗 + 1)a𝑐 𝑛− 𝑗 −2 ( 𝑥) ˜ + ( 𝑗 + 1)b 𝑥 𝑐 𝑛− 𝑗 −1 ( 𝑥) ˜ + c 𝑥 𝑐 𝑛− 𝑗 ( 𝑥) ˜ = 0,

(12.371)

𝑥˜ = (𝑥1 , 𝑥2 ) 𝑇 , 𝑗 = 0, ..., 𝑛 − 2. Since the matrix a is regular (notice that 𝜏 ≠ −1), all polynomials 𝑐 𝑗 are determined provided that 𝑐 𝑛 and 𝑐 𝑛−1 are known.

12.14 Function Systems for Internal Elastic Fields

633

By summarizing our results, we obtain the following theorem. Theorem 12.70 Let 𝑐 𝑛 , 𝑐 𝑛−1 : R2 → R3 be homogeneous polynomials of degree 𝑛, 𝑛 − 1, respectively. For 𝑗 = 0, ..., 𝑛 − 2 we define recursively a𝑐 𝑛− 𝑗 −2 (𝑥1 , 𝑥2 ) = −


1 ( 𝑗 + 1)b 𝑥 𝑐 𝑛− 𝑗 −1 (𝑥1 , 𝑥2 ) + c 𝑥 𝑐 𝑛− 𝑗 (𝑥 1 , 𝑥2 ) . ( 𝑗 + 2)( 𝑗 + 1)

Then 𝑢 𝑛 : R3 → R3 given by 𝑢 𝑛 (𝑥1 , 𝑥2 , 𝑥3 ) =

𝑛 ∑︁

𝑗

𝑐 𝑛− 𝑗 (𝑥1 , 𝑥2 )𝑥3

𝑗=0

is a homogeneous polynomial of degree 𝑛 in R3 satisfying the Navier equation ♦ 𝑥 𝑢 𝑛 (𝑥) = 0, 𝑥 ∈ R3 . Moreover, the number of linearly independent homogeneous polynomials is equal to the total number of coefficients of 𝑐 𝑛 and 𝑐 𝑛−1 , that is d(nav𝑛 ) = 3(2𝑛 + 1).

(12.372)

Remark 12.71 We know (see, e.g., W. Freeden et al. (1994))) that homogeneous harmonic polynomials of different degree are orthogonal (in the l2 -sense). This fact, however, is not true for the spaces nav𝑛 (𝜏 ≠ 0), as the following example shows. The vector fields 𝑥1 𝑥2 0 © ª © 𝜏 (𝑥 12 + 𝑥22 + 𝑥32 ) ª® 𝑢 0 (𝑥) = ­ 1 ® , 𝑢 2 (𝑥) = ­ − 2( 𝜏+3) 0 «0¬ « ¬ are elements of nav0 and nav2 , respectively. However, it follows by an easy calculation that ∫ 2𝜋𝜏 𝑢 0 (𝜉) · 𝑢 2 (𝜉) 𝑑𝜔(𝜉) = − ≠ 0. 𝜏+3 Ω

Nevertheless, we are able to prove the following result. Theorem 12.72 Let 𝑢 𝑛 ∈ nav𝑛 , 𝑢 𝑚 ∈ nav𝑚 . Then ∫ (𝑢 𝑛 , 𝑢 𝑚 )l2 (Ω) = 𝑢 𝑛 (𝜉) · 𝑢 𝑚 (𝜉) 𝑑𝜔(𝜉) = 0 Ω

if |𝑛 − 𝑚| ≠ 2 and 𝑛 ≠ 𝑚. Proof Applying Green’s formula and the Gauss theorem, we see that ∫ 0= (𝑢 𝑛 (𝑥) · ♦𝑢 𝑚 (𝑥) − 𝑢 𝑚 (𝑥) · ♦𝑢 𝑛 (𝑥)) 𝑑𝑥 | 𝑥 | ≤1

634

12 Potentials in Euclidean Space

∫ (𝑢 𝑛 (𝑥) · Δ𝑢 𝑚 (𝑥) − 𝑢 𝑚 (𝑥) · Δ𝑢 𝑛 (𝑥)) 𝑑𝑥 (12.373) ∫ +𝜏 (𝑢 𝑛 (𝑥) · ∇(∇ · 𝑢 𝑚 (𝑥)) − 𝑢 𝑚 (𝑥) · ∇(∇ · 𝑢 𝑛 (𝑥))) 𝑑𝑥 | 𝑥 | ≤1 ∫ = (𝑚 − 𝑛) 𝑢 𝑛 (𝑥) · 𝑢 𝑚 (𝑥) 𝑑𝜔(𝑥) | 𝑥 |=1 ∫ +𝜏 (∇ · (𝑢 𝑛 (𝑥)∇ · 𝑢 𝑚 (𝑥)) − ∇ · (𝑢 𝑚 (𝑥)∇ · 𝑢 𝑛 (𝑥))) 𝑑𝑥 | 𝑥 | ≤1 ∫ = (𝑚 − 𝑛) 𝑢 𝑛 (𝑥) · 𝑢 𝑚 (𝑥) 𝑑𝜔(𝑥) | 𝑥 |=1 ∫ +𝜏 ((𝑥 · 𝑢 𝑛 (𝑥))∇ · 𝑢 𝑚 (𝑥) − (𝑥 · 𝑢 𝑚 (𝑥))∇ · 𝑢 𝑛 (𝑥)) 𝑑𝜔(𝑥). =

| 𝑥 | ≤1

| 𝑥 |=1

The functions 𝑥 ↦→ (∇ · 𝑢 𝑚 )(𝑥) and 𝑥 ↦→ (∇ · 𝑢 𝑛 )(𝑥), 𝑥 ∈ R3 , are harmonic and the functions 𝑥 ↦→ 𝑥 · 𝑢 𝑛 (𝑥) and 𝑥 ↦→ 𝑥 · 𝑢 𝑚 (𝑥), 𝑥 ∈ R3 , are biharmonic. For example, Δ(𝑥 · 𝑢 𝑛 (𝑥)) = 𝑥 · Δ𝑢 𝑛 (𝑥) + 2∇ · 𝑢 𝑛 (𝑥)

(12.374)

= 2∇ · 𝑢 𝑛 (𝑥) − 𝜏(𝑥 · ∇)∇ · 𝑢 𝑛 (𝑥) = (2 − 𝜏(𝑛 − 1))∇ · 𝑢 𝑛 (𝑥) so that we have ΔΔ(𝑥 · 𝑢 𝑛 (𝑥)) = 0, 𝑥 ∈ R3 . In an analogous way, it follows that ΔΔ(𝑥 · 𝑢 𝑚 (𝑥)) = 0, 𝑥 ∈ R3 . Therefore (see W. Freeden et al. (1994)), there exist scalar homogeneous harmonic polynomials 𝐻𝑛−1 , 𝐻𝑛+1 , 𝐻𝑚−1 , and 𝐻𝑚+1 of degree 𝑛 − 1, 𝑛 + 1, 𝑚 − 1, and 𝑚 + 1, respectively, with 𝑥 · 𝑢 𝑛 (𝑥) = 𝐻𝑛+1 (𝑥) + |𝑥| 2 𝐻𝑛−1 (𝑥)

(12.375)

𝑥 · 𝑢 𝑚 (𝑥) = 𝐻𝑚+1 (𝑥) + |𝑥| 2 𝐻𝑚−1 (𝑥).

(12.376)

and According to our assumptions, we have 𝑚 − 1 ≠ 𝑛 + 1 and 𝑚 + 1 ≠ 𝑛 − 1. Thus we find ∫ (𝑥 · 𝑢 𝑛 (𝑥))∇ · 𝑢 𝑚 (𝑥) 𝑑𝜔(𝑥) = 0, (12.377) | 𝑥 |=1 ∫ (𝑥 · 𝑢 𝑚 (𝑥))∇ · 𝑢 𝑛 (𝑥) 𝑑𝜔(𝑥) = 0. (12.378) | 𝑥 |=1

Hence, Equation (12.373) reduces to ∫ 0= 𝑢 𝑛 (𝜉) · 𝑢 𝑚 (𝜉) 𝑑𝜔(𝜉) Ω

(12.379)

12.14 Function Systems for Internal Elastic Fields

635

□

if 𝑛 ≠ 𝑚. This is the required result.

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 example, by using the recursion formula (12.371). However, we are also able to use known information about scalar homogeneous harmonic polynomials. We start with a preparatory lemma. Lemma 12.73 Let 𝐻𝑛 : R3 → R, 𝑛 ≥ 0, be a scalar homogeneous harmonic polynomial of degree 𝑛. Then (𝑖) Δ(𝐻𝑛 (𝑥)𝑥) = 2∇𝐻𝑛 (𝑥), (𝑖𝑖) Δ(|𝑥| 𝑚 𝐻𝑛 (𝑥)) = 𝑚(𝑚 + 2𝑛 + 1)|𝑥| 𝑚−2 𝐻𝑛 (𝑥), 𝑚 ≥ 2,
(𝑖𝑖𝑖) Δ 𝑥 2 ∇𝐻𝑛 (𝑥) = 2(2𝑛 + 1)∇𝐻𝑛 (𝑥). Proof The formulas (i), (ii), and (iii) can be obtained by straightforward calculations. □ We are now interested in the following lemma. Lemma 12.74 Let 𝐻𝑛 : R3 → R be a homogeneous harmonic polynomial of degree 𝑛. Then the following identities are valid: (i) For all 𝑥 ∈ R3 , ♦(∇𝐻𝑛 (𝑥)) = 0. (ii) For all 𝑥 ∈ R3 , ♦(𝑥 ∧ ∇𝐻𝑛 (𝑥)) = 0. (iii) For all 𝑥 ∈ R3 , ♦(𝑥𝐻𝑛 (𝑥) + 𝛼𝑛 |𝑥| 2 ∇𝐻𝑛 (𝑥)) = 0, where 𝛼𝑛 = −

˜ + 𝑛) + 𝜇(5 𝜆(3 ˜ + 𝑛)
. ˜ 2 𝑛𝜆 + 𝜇(3𝑛 ˜ + 1)

(12.380)

(iv) For all 𝑥 ∈ R3 , ♦(𝐻𝑛 (𝑥)𝜀 𝑘 + 𝛽𝑛 |𝑥| 2 ∇∇ · (𝐻𝑛 (𝑥)𝜀 𝑘 )) = 0 , where 𝛽𝑛 = −

𝜆˜ + 𝜇˜ .
2𝜆˜ + 6 𝜇˜ 𝑛 − 2𝜆˜ − 4 𝜇˜

(12.381)


(v) For all 𝑥 ∈ R3 , ♦ 𝐻𝑛 (𝑥)𝜀 𝑘 + 𝛾𝑛 (𝜀 𝑘 · ∇𝐻𝑛 (𝑥))𝑥 = 0, where 𝛾𝑛 = −

𝜆˜ + 𝜇˜ . (𝑛 + 2) 𝜆˜ + (𝑛 + 4) 𝜇˜

(12.382)

636

12 Potentials in Euclidean Space

Proof The formulas can be obtained by elementary calculations.

□

Fig. 12.8: Reference (left) and deformed (right) configurations of Ω associated to the displacement function 𝑥 ↦→ 𝛼0 𝑥𝐻1 (𝑥) + 𝛼1 |𝑥| 2 ∇𝐻1 (𝑥), 𝐻1 (𝑥) = 𝑥 · 𝜀 2 , 𝑥 ∈ R3 . Lemma 12.74 enables us to develop three important systems of polynomial solutions of the Navier equation. Lemma 12.75 Let {𝐻𝑛, 𝑗 } 𝑗=1,...,2𝑛+1 be a linearly independent system of scalar homogeneous harmonic polynomials of degree n. Then the functions 𝑤𝑛, 𝑗,𝑘 : R3 → R3 , 𝑘 = 1, 2, 3, defined by    𝑤𝑛, 𝑗,𝑘 (𝑥) = 𝐻𝑛, 𝑗 (𝑥)𝜀 𝑘 + 𝛽𝑛 |𝑥| 2 ∇ ∇ · 𝐻𝑛, 𝑗 (𝑥)𝜀 𝑘 , 𝑥 ∈ R3 , (12.383) form a set of 3(2𝑛 + 1) linearly independent elements of nav𝑛 (R3 ), where 𝛽𝑛 is given by (12.381). Lemma 12.76 Let {𝐻𝑛, 𝑗 } 𝑗=1,...,2𝑛+1 be a linearly independent system of scalar homogeneous harmonic polynomials of degree 𝑛. Then the functions 𝑣𝑛, 𝑗,𝑘 : R3 → R3 , 𝑘 = 1, 2, 3, defined by   𝑣𝑛, 𝑗,𝑘 (𝑥) = 𝐻𝑛, 𝑗 (𝑥)𝜀 𝑘 + 𝛾𝑛 𝜀 𝑘 · ∇𝐻𝑛, 𝑗 (𝑥) 𝑥, 𝑥 ∈ R3 , (12.384) form a set of 3(2𝑛 + 1) linearly independent elements of nav𝑛 (R3 ), where 𝛾𝑛 is given by (12.382). Remark 12.77 The system (12.383) can be found in A. Lurje (1963), while the system (12.384) has been discussed in H. Bauch (1981), W. Freeden, R. Reuter

12.14 Function Systems for Internal Elastic Fields

637

(1990b). Unfortunately, both systems are not orthogonal invariant, that is, t𝑇 𝑣𝑛, 𝑗,𝑘 (t·) (resp. t𝑇 𝑤𝑛, 𝑗,𝑘 (t·)) generally is not a member of the span of the system {𝑣𝑛, 𝑗,𝑘 } (resp. {𝑤𝑛, 𝑗,𝑘 }). A polynomial system showing this property will be listed now (see Figs. 12.8, and 12.9). Lemma 12.78 Let {𝐻 𝑘, 𝑗 } 𝑘=𝑛−1,𝑛,𝑛+1 be a linearly independent system of scalar ho𝑗=1,...,2𝑛+1

(𝑖) 3 3 mogeneous harmonic polynomials. Then, the functions 𝑢 𝑛, 𝑗 : R → R , 𝑖 = 1, 2, 3, defined by (1) 2 𝑢 𝑛, 𝑗 (𝑥) = 𝐻 𝑛−1, 𝑗 (𝑥)𝑥 + 𝛼𝑛−1 |𝑥| ∇𝐻 𝑛−1, 𝑗 (𝑥),

(12.385)

𝑛 = 1, 2, ..., 𝑗 = 1, ..., 2𝑛 − 1, (2) 𝑢 𝑛, 𝑗 (𝑥) (3) 𝑢 𝑛, 𝑗 (𝑥)

= ∇𝐻𝑛+1, 𝑗 (𝑥), 𝑛 = 0, 1, ..., 𝑗 = 1, ..., 2𝑛 + 3,

(12.386)

= 𝑥 ∧ ∇𝐻𝑛, 𝑗 (𝑥), 𝑛 = 1, 2, ..., 𝑗 = 1, ..., 2𝑛 + 1,

(12.387)

form a set of 3(2𝑛 + 1) linearly independent elements of nav𝑛 (R3 ), where 𝛼𝑛 is given by (12.380). (2) (3) The functions 𝑢 𝑛, 𝑗 , 𝑢 𝑛, 𝑗 are characterized by the properties: (2) (2) ∇ · 𝑢 𝑛, 𝑗 (𝑥) = 0, ∇ ∧ 𝑢 𝑛, 𝑗 (𝑥) = 0,

𝑥·

(3) 𝑢 𝑛, 𝑗 (𝑥)

= 0, ∇ ·

(3) 𝑢 𝑛, 𝑗 (𝑥)

(12.388)

= 0.

(12.389)

(2) From a physical point of view, this means that 𝑢 𝑛, 𝑗 is a poloidal field (i.e., a vector (3) field free of dilatation and torsion), while 𝑢 𝑛, 𝑗 is a toroidal field. Only the functions (1) 𝑢 𝑛, 𝑗 are responsible for volume change.

Remark 12.79 There is a very interesting relation between the systems {𝑤𝑛, 𝑗,𝑘 }, (𝑖) {𝑣𝑛, 𝑗,𝑘 }, 𝑘 = 1, 2, 3, introduced above and the system {𝑢 𝑛+1, 𝑗 }, 𝑖 = 1, 2, 3. Replacing Í3 𝑘 𝑘 𝐻𝑛−1, 𝑗 by 𝑘=1 𝜀 · ∇𝐻𝑛, 𝑗 (note that 𝜀 · ∇𝐻𝑛, 𝑗 is a homogeneous harmonic polynomial of degree 𝑛 − 1 due to a result in W. Freeden et al. (1994)) in the representation (1) of 𝑢 𝑛, 𝑗 , we obtain a field 𝑧 𝑛, 𝑗 defined as follows: 𝑧 𝑛, 𝑗 (𝑥) =

3  ∑︁

𝜀 𝑘 · ∇𝐻𝑛, 𝑗 (𝑥)

𝑘=1



! 𝑥 + 𝛼𝑛−1 𝑥 2 ∇

3  ∑︁

𝜀 𝑘 · ∇𝐻𝑛, 𝑗 (𝑥)



! .

(12.390)

𝑘=1

It is clear that 𝑧 𝑛, 𝑗 satisfies the Navier equation. Moreover, it is easy to see that 𝛾𝑛 = (−𝛽𝑛 )/𝛼𝑛−1 . However, this shows that 𝑣𝑛, 𝑗 = 𝑤𝑛, 𝑗 −

𝛽𝑛 𝑧 𝑛, 𝑗 , 𝑛 = 0, 1, . . . , 𝑗 = 1, . . . , 2𝑛 + 1, 𝛼𝑛−1

(12.391)

638

12 Potentials in Euclidean Space

Fig. 12.9: Radial (color) and tangential (arrows) displacements of Ω associated to the displacement function 𝑥 ↦→ 𝛼0 𝑥𝐻1 (𝑥) + 𝛼1 |𝑥| 2 ∇𝐻1 (𝑥), 𝐻1 (𝑥) = 𝑥 · 𝜀 2 , 𝑥 ∈ R3 . where we have used the abbreviations 𝑣𝑛, 𝑗 =

3 ∑︁

𝑣𝑛, 𝑗,𝑘 ,

𝑤𝑛, 𝑗 =

𝑘=1

3 ∑︁

𝑤𝑛, 𝑗,𝑘 .

(12.392)

𝑘=1

Assuming that the scalar system {𝐻𝑛, 𝑗 } 𝑛=0,1,..., 𝑗=1,...,2𝑛+1 forms an orthonormal system of homogeneous harmonic polynomials with respect to L2 (Ω), the following orthogonal relations can be guaranteed: ∫ (𝑖) (𝑘 ) 𝑢 𝑛, (12.393) 𝑗 (𝜉) · 𝑢 𝑛,𝑙 (𝜉) 𝑑𝜔(𝜉) = 0 if 𝑖 ≠ 𝑘 or 𝑗 ≠ 𝑙, Ω ∫ (𝑖) (𝑖) 𝑢 𝑛, (12.394) 𝑗 (𝜉) · 𝑢 𝑚,𝑘 (𝜉) 𝑑𝜔(𝜉) = 0 if 𝑛 ≠ 𝑚 or 𝑗 ≠ 𝑘, 𝑖 = 1, 2, 3, Ω ∫ (3) (𝑖) 𝑢 𝑛, . (12.395) 𝑗 (𝜉) · 𝑢 𝑚,𝑘 (𝜉) 𝑑𝜔(𝜉) = 0 if 𝑖 = 1, 2 Ω

This shows us the following lemma. Lemma 12.80 The space nav𝑛 , 𝑛 > 0, defined by (12.366) can be decomposed into three subspaces nav𝑛(𝑖) , 𝑖 = 1, 2, 3, given by n o (𝑖) nav𝑛(𝑖) = span 𝑢 𝑛, (12.396) 𝑗 𝑗=1,...,2𝑛+1

such that

12.14 Function Systems for Internal Elastic Fields

639

nav𝑛 = nav𝑛(1) ⊕ nav𝑛(2) ⊕ nav𝑛(3) . Moreover, we have the following dimensions:       𝑑 nav𝑛(1) = 2𝑛 − 1, 𝑑 nav𝑛(2) = 2𝑛 + 3, 𝑑 nav𝑛(3) = 2𝑛 + 1. For 𝑛 = 0,

(2) (nav0 ) = 3. nav0 = nav0(2) = span 𝑢 0, 𝑗, 𝑑

(12.397)

(12.398)

(12.399)

𝑗=1,2,3

As mentioned above, the spaces nav𝑛(𝑖) , 𝑖 = 1, 2, 3, are orthogonal invariant in the sense that 𝑢 ∈ nav𝑛(𝑖) is equivalent to t𝑇 𝑢(t·) ∈ nav𝑛(𝑖) , 𝑖 = 1, 2, 3, for every orthogonal transformation t. Thus, we have found a decomposition of nav𝑛 into three invariant subspaces. Next, assume that 𝑤𝑛(𝑖) is a member of nav𝑛(𝑖) . Consider the space ℎ 𝑛(𝑖) of all linear combinations of functions 𝑤𝑛(𝑖) (t·), where t is an orthogonal transformation: n o ℎ 𝑛(𝑖) = span 𝑤𝑛(𝑖) (t·) t ∈ 𝑂 (3) . (12.400) Then it is clear that 0 < 𝑑 (ℎ 𝑛(𝑖) ) ≤ 𝑑 (nav𝑛(𝑖) ). Moreover, it can be shown that there exists no orthogonal invariant subspace in nav𝑛(𝑖) . Thus, it follows immediately that nav𝑛(𝑖) = ℎ 𝑛(𝑖) . This leads us to the following lemma. Lemma 12.81 Let 𝑤𝑛(𝑖) be of class nav𝑛(𝑖) . Then, there exist 𝑑 (nav𝑛(𝑖) ) orthogonal transformations t 𝑗 , 𝑗 = 1, ..., 𝑑 (nav𝑛(𝑖) ), such that any element 𝑢 (𝑖) ∈ nav𝑛(𝑖) can be written in the form   (𝑖)

𝑑 nav𝑛

𝑢 (𝑖) =

∑︁

𝑇 (𝑖) 𝑐 (𝑖) 𝑗 t 𝑗 𝑤𝑛 (t 𝑗 ·),

(12.401)

𝑗=1

where 𝑐 (𝑖) 𝑗 are real numbers. (𝑘 ) Finally, we formulate the addition theorem for the system {𝑢 𝑛, 𝑗 } developed in Lemma 12.78. By separation of radial and angular tangential components, we first obtain after simple calculations (1) (1) (1) (1) (2) 𝑢 𝑛, 𝑗 (𝑥) = 𝛾 𝑛 (|𝑥|)𝑦 𝑛−1, 𝑗 (𝜉) + 𝛿 𝑛 (|𝑥|)𝑦 𝑛−1, 𝑗 (𝜉),

(12.402)

(2) 𝑢 𝑛, 𝑗 (𝑥)

(12.403)

(3) 𝑢 𝑛, 𝑗 (𝑥)

=

(1) 𝛾𝑛(2) (|𝑥|)𝑦 𝑛+1, 𝑗 (𝜉)

=

(3) 𝛾𝑛(3) (|𝑥|)𝑦 𝑛, 𝑗,

+

(2) 𝛿 𝑛(2) (|𝑥|)𝑦 𝑛+1, 𝑗 (𝜉),

(12.404)

640

12 Potentials in Euclidean Space

where we have used the abbreviations 𝛾𝑛(1) (|𝑥|) = |𝑥| 𝑛 (1 + (𝑛 − 1)𝛼𝑛−1 ), √︁ 𝛿 𝑛(1) (|𝑥|) = |𝑥| 𝑛 𝛼𝑛−1 (𝑛 − 1)𝑛,

(12.405)

𝛾𝑛(2) (|𝑥|)

(12.407)

𝑛

= |𝑥| (1 + 𝑛), √︁ = |𝑥| 𝑛 (𝑛 + 1)(𝑛 + 2), √︁ 𝛾𝑛(3) (|𝑥|) = |𝑥| 𝑛 𝑛(𝑛 + 1). 𝛿 𝑛(2) (|𝑥|)

(12.406)

(12.408) (12.409)

Remembering the addition theorem for vector spherical harmonics (see Theorem 6.31), we obtain the following theorem. Theorem 12.82 For 𝑥, 𝑦 ∈ R3 , 𝑥 = 𝑟𝜉, 𝑦 = 𝜌𝜂, 𝑟 = |𝑥|, 𝜌 = |𝑦|, 2𝑛−1 ∑︁

(1) (1) 𝑢 𝑛, 𝑗 (𝑥) ⊗ 𝑢 𝑛, 𝑗 (𝑦)

𝑗=1 (1,1) (1,2) = 𝛾𝑛(1) (𝑟)𝛾𝑛(1) (𝜌)p𝑛−1 (𝜉, 𝜂) + 𝛾𝑛(1) (𝑟)𝛿 𝑛(1) (𝜌)p𝑛−1 (𝜉, 𝜂) (2,1) (2,2) + 𝛿 𝑛(1) (𝑟)𝛾𝑛(1) (𝜌)p𝑛−1 (𝜉, 𝜂) + 𝛿 𝑛(1) (𝑟)𝛿 𝑛(1) (𝜌)p𝑛−1 (𝜉, 𝜂), 2𝑛+3 ∑︁

(2) (2) 𝑢 𝑛, 𝑗 (𝑥) ⊗ 𝑢 𝑛, 𝑗 (𝑦)

𝑗=1 (1,1) (1,2) = 𝛾𝑛(2) (𝑟)𝛾𝑛(2) (𝜌)p𝑛+1 (𝜉, 𝜂), +𝛾𝑛(2) (𝑟)𝛿 𝑛(2) (𝜌)p𝑛+1 (𝜉, 𝜂) (2,1) (2,2) + 𝛿 𝑛(2) (𝑟)𝛾𝑛(2) (𝜌)p𝑛+1 (𝜉, 𝜂) + 𝛿 𝑛(2) (𝑟)𝛿 𝑛(2) (𝜌)p𝑛+1 (𝜉, 𝜂), 2𝑛+1 ∑︁

(3) (3) (3) (3) (3,3) 𝑢 𝑛, (𝜉, 𝜂). 𝑗 (𝑥) ⊗ 𝑢 𝑛, 𝑗 (𝑦) = 𝛾 𝑛 (𝑟)𝛾 𝑛 (𝜌)p𝑛

𝑗=1

In particular, we find the following result. Lemma 12.83 If 𝑥 ∈ R3 , 𝑟 = |𝑥|, 𝑥 = 𝑟𝜉, then 2𝑛−1 ∑︁ 𝑗=1

  2𝑛 − 1 (1) 2 2 𝑛(𝑛 − 1) , 𝑢 𝑛, 𝑗 (𝑥) = 𝑟 2𝑛 (1 + (𝑛 − 1)𝛼𝑛−1 ) 2 + 𝛼𝑛−1 4𝜋

2𝑛+3 ∑︁ 𝑗=1

(𝑛 + 1)(2𝑛 + 3) 2 (2) 2 , 𝑢 𝑛, 𝑗 (𝑥) = 𝑟 2𝑛 4𝜋

2𝑛+1 ∑︁ 𝑗=1

𝑛(𝑛 + 1)(2𝑛 + 1) (3) 2 . 𝑢 𝑛, 𝑗 (𝑥) = 𝑟 2𝑛 4𝜋

12.14 Function Systems for Internal Elastic Fields

641

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 12.70 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) 3 3 Lemma 12.84 Let 𝑣𝑛, 𝑗 , R → R , 𝑖 = 1, 2, 3, be defined by

  (1) 2 𝑣𝑛, (𝑥) = 𝐻 (𝑥)𝑥 + 𝛼 𝑥 − 1 ∇𝐻𝑛, 𝑗 (𝑥), 𝑛, 𝑗 𝑛 𝑗 (2) 𝑣𝑛, 𝑗 (𝑥)

𝑛 = 0, 1, ..., 𝑗 = 1, ..., 2𝑛 + 1,   1 (1) = (𝑛(𝑛 + 1)) − 2 ∇𝐻𝑛, 𝑗 (𝑥) − 𝑛𝑣𝑛, (𝑥) , 𝑗

(12.410)

(12.411)

𝑛 = 1, 2, ..., 𝑗 = 1, ..., 2𝑛 + 1, 1

(3) −2 𝑣𝑛, 𝑥 ∧ ∇𝐻𝑛, 𝑗 (𝑥), 𝑗 (𝑥) = (𝑛(𝑛 + 1))

(12.412)

𝑛 = 1, 2, ..., 𝑗 = 1, ..., 2𝑛 + 1, where 𝛼𝑛 = −

𝑛𝜏 + 2 + 3𝜏 , 2(𝑛(𝜏 + 2) + 1)

𝐻𝑛, 𝑗 (𝑥) = |𝑥| 𝑛𝑌𝑛, 𝑗 (𝜉),

𝑥 = |𝑥|𝜉, 𝜉 ∈ Ω.

(12.413) (12.414)

(𝑖) (𝑖) int with 𝑣 (𝑖) |Ω = Then 𝑣𝑛, 𝑗 satisfies the Cauchy-Navier equation ♦𝑣𝑛, 𝑗 (𝑥) = 0 in Ω 𝑛, 𝑗 (𝑖) 𝑦 𝑛, 𝑗 .

Proof It is not hard to see that (1) ♦𝑣𝑛, 𝑗 (𝑥) = 2∇𝐻 𝑛, 𝑗 (𝑥) + 𝜏(3 + 𝑛)∇𝐻 𝑛, 𝑗 (𝑥)

(12.415)

+ 𝛼𝑛 ((6 + 4(𝑛 − 1))∇𝐻𝑛, 𝑗 (𝑥) + 2𝑛𝜏∇𝐻𝑛, 𝑗 (𝑥)) = 0, (2) ♦𝑣𝑛, 𝑗 (𝑥)

1

= (𝑛(𝑛 + 1)) − 2 (♦∇𝐻𝑛, 𝑗 (𝑥))   1 (1) − 𝑛(𝑛(𝑛 + 1)) − 2 (♦)𝑣𝑛, 𝑗 (𝑥)

(12.416)

= 0, (3) ♦𝑣𝑛, 𝑗 (𝑥)

1

= (𝑛(𝑛 + 1)) − 2 ♦(𝑥 ∧ ∇𝐻𝑛, 𝑗 (𝑥)) = −2∇ ∧ ∇𝐻𝑛, 𝑗 (𝑥) = 0.

(12.417)

642

12 Potentials in Euclidean Space

Using the polar coordinates 𝑥 = 𝑟𝜉, 𝑟 = |𝑥|, 𝜉 ∈ Ω, we obtain after simple calculations (1) (1) (1) (1) (2) 𝑣𝑛, 𝑗 (𝑥) = 𝜎𝑛 (𝑟)𝑦 𝑛, 𝑗 (𝜉) + 𝜏𝑛 (𝑟)𝑦 𝑛, 𝑗 (𝜉),

(12.418)

(2) (2) (1) (2) (2) 𝑣𝑛, 𝑗 (𝑥) = 𝜎𝑛 (𝑟)𝑦 𝑛, 𝑗 (𝜉) + 𝜏𝑛 (𝑟)𝑦 𝑛, 𝑗 (𝜉),

(12.419)

(3) 𝑣𝑛, 𝑗 (𝑥)

(12.420)

=

(3) 𝜎𝑛(3) (𝑟)𝑦 𝑛, 𝑗 (𝜉),

where    𝜎𝑛(1) (𝑟) = 𝑟 𝑛−1 𝑟 2 + 𝑛𝛼𝑛 𝑟 2 − 1 , 𝜎𝑛(2) (𝑟) = (𝑛(𝑛 + 1))

− 12

(12.421) 



𝑛(1 + 𝑛𝛼𝑛 )𝑟 𝑛−1 1 − 𝑟 2 ,

𝜎𝑛(3) (𝑟) = 𝑟 𝑛 ,

(12.422) (12.423)

+ 12





𝜏𝑛(1) (𝑟) = 𝛼𝑛 (𝑛(𝑛 + 1)) 𝑟 𝑛−1 𝑟 2 − 1 ,    𝜏𝑛(2) (𝑟) = 𝑟 𝑛−1 1 − 𝑛𝛼𝑛 𝑟 2 − 1 .

(12.424)

(𝑖) − (𝑖) (𝑖) This shows us that 𝑣𝑛, 𝑗 = 𝑣𝑛, 𝑗 |Ω = 𝑦 𝑛, 𝑗 , as required.

□

(12.425)

It should be mentioned that (1) (1) 𝑣𝑛, 𝑗 = 𝑢 𝑛+1, 𝑗 − 𝛼𝑛 ∇𝐻 𝑛, 𝑗 , 𝑛 = 0, 1, ..., 𝑗 = 1, ..., 2𝑛 + 1,   (2) (2) (1) − 12 𝑣𝑛, = (𝑛(𝑛 + 1)) 𝑢 − 𝑛𝑣 𝑗 𝑛, 𝑗 , 𝑛 = 1, 2, ..., 𝑗 = 1, ..., 2𝑛 + 1. 𝑛−1, 𝑗 (𝑖) (𝑖) Thus the polynomial solution 𝑣𝑛, 𝑗 , 𝑖 = 1, 2 corresponding to 𝑦 𝑛, 𝑗 on Ω is not homogeneous.

Remark 12.85 Observe that, under the assumption 3𝜆˜ + 2 𝜇˜ > 0, 𝜇˜ > 0, it follows that 1 𝜆˜ + 𝜇˜ 1 3𝜆˜ + 2 𝜇˜ 𝜏= = + > . (12.426) 𝜇˜ 3 3 𝜇˜ 3 Therefore, it is not difficult to deduce that for all 𝑛 ≥ 3 1 𝑛𝜏 + 3𝜏 + 2 2 𝑛𝜏 + 2𝑛 + 1 3𝜏 2 1 1 + 𝑛𝜏 + 𝑛𝜏 12+ = ≤ 2𝑛 1 2 1 + 𝑛𝜏 21+ + 𝑛𝜏

|𝛼𝑛 | =

while for all 𝑛 ≥ 1

(12.427) 2 𝑛𝜏 1 𝑛𝜏

≤ 1,

(12.428)

3

|𝛼𝑛 | ≤

1 + 2𝑛 2 ≤ 2. 2 1 + 𝑛𝜏

(12.429)

12.14 Function Systems for Internal Elastic Fields

643

The sequence (𝛼𝑛 ) therefore is uniformly bounded with respect to 𝜏. (𝑖);𝑅 3 3 Remark 12.86 Let us denote by 𝑣𝑛, 𝑗 : R → R the vector fields (1);𝑅 𝑣𝑛, 𝑗 (𝑥)

=

𝜎𝑛(1)

(2);𝑅 (2) 𝑣𝑛, 𝑗 (𝑥) = 𝜎𝑛 (3);𝑅 (3) 𝑣𝑛, 𝑗 (𝑥) = 𝜎𝑛



 |𝑥| (1) 𝑦 𝑛, 𝑗 (𝜉) + 𝜏𝑛(1) 𝑅   |𝑥| (1) 𝑦 𝑛, 𝑗 (𝜉) + 𝜏𝑛(2) 𝑅   |𝑥| (3) 𝑦 𝑛, 𝑗 (𝜉), 𝑅



 |𝑥| (2) 𝑦 𝑛, 𝑗 (𝜉), 𝑅   |𝑥| (2) 𝑦 𝑛, 𝑗 (𝜉), 𝑅

(12.430) (12.431) (12.432)

where 𝑥 = |𝑥|𝜉, |𝑥| ≤ 𝑅 and 𝜎𝑛(1) , 𝜏𝑛(1) are given as follows 𝜎𝑛(1) 𝜏𝑛(1) 𝜎𝑛(2)



 |𝑥| = 𝑅   |𝑥| = 𝑅   |𝑥| = 𝑅











|𝑥| 𝑅

 𝑛−1 

|𝑥| 𝑅

 𝑛−1

|𝑥| 𝑅

 𝑛−1

|𝑥| 𝑅

2

 + 𝑛𝛼𝑛

𝛼𝑛 (𝑛(𝑛 + 1))

1/2

|𝑥| 𝑅 

!

2

−1 , |𝑥| 𝑅

𝑛(1 + 𝑛𝛼𝑛 )(𝑛(𝑛 + 1))

(12.433) !

2

−1 , 

|𝑥| 1− 𝑅

1/2

(12.434) 2! , (12.435)

𝜏𝑛(2) 𝜎𝑛(3)



 𝑛−1

|𝑥| |𝑥| = 𝑅 𝑅    𝑛 |𝑥| |𝑥| = 𝑅 𝑅

 1 − 𝑛𝛼𝑛

|𝑥| 𝑅

!!

2 −1

,

(12.436) (12.437)

with 𝜏=

𝜆˜ + 𝜇˜ . 𝜇˜

(12.438)

(𝑖);𝑅 Then 𝑣𝑛, 𝑗 is the unique solution of the first boundary-value problem

    (𝑖);𝑅 int ∩ c (2) Ωint , ♢𝑣 (𝑖);𝑅 = 0 in Ωint , 𝑣𝑛, ∈ c Ω 𝑗 𝑛, 𝑗 𝑅 𝑅 𝑅

(12.439)

corresponding the boundary values (𝑖);𝑅 (𝑖) 𝑣𝑛, 𝑗 |Ω 𝑅 = 𝑦 𝑛, 𝑗 .

We easily obtain the following theorem (see T. Gervens (1989)).

(12.440)

644

12 Potentials in Euclidean Space

Theorem 12.87 Suppose that 𝑓 is of class c(Ω). Then, the unique solution 𝑢 of the Dirichlet problem 𝑢 ∈ c (2) (Ωint ) ∩ c(Ωint ), ♦𝑢 = 0 in Ωint 𝑢|Ω = 𝑓 is representable in the form 3 ∑︁ ∞ 2𝑛+1 ∑︁ ∑︁ (𝑖) 𝑢(𝑥) = ( 𝑓 (𝑖) ) ∧ (𝑛, 𝑗) 𝑣𝑛, 𝑗 (𝑥) 𝑖=1 𝑛=0𝑖 𝑗=1

for all 𝑥 ∈ 𝐾 with 𝐾 ⊂ Ωint and dist(𝐾, Ω) > 0, where ( 𝑓 (𝑖) ) ∧ (𝑛, 𝑗) are the Fourier (𝑖) coefficients of 𝑓 with respect to the system {𝑦 𝑛, 𝑗} (𝑓

(𝑖) ∧

) (𝑛, 𝑗) =





(𝑖) 𝑓 , 𝑦 𝑛, 𝑗 l2 (Ω)

∫ =

(𝑖) 𝑓 (𝜂) · 𝑦 𝑛, 𝑗 (𝜂) 𝑑𝜔(𝜂).

Ω

(𝑖) From Lemma 12.84, it is not difficult to determine the stress vector field 𝑇𝜈 (𝑣𝑛, 𝑗 )(𝑥) for any point 𝑥 ∈ Ωint :  

(1) ˜ + 3) + 𝛼𝑛 𝜆˜ + 𝜇˜ 𝐻𝑛, 𝑗 (𝑥)𝑥 |𝑥|𝑇𝜈 𝑣𝑛, ˜ + 2) + 𝜆(𝑛 𝑗 (𝑥) = 𝜇(𝑛

+ ( 𝜇˜ + 2 𝜇𝑛𝛼 ˜ 𝑛 ) 𝑥 2 ∇𝐻𝑛, 𝑗 (𝑥) − 2𝛼𝑛 𝜇(𝑛 ˜ − 1)∇𝐻𝑛, 𝑗 (𝑥), 𝑛 = 0, 1, ..., 𝑗 = 1, ..., 2𝑛 + 1, 







(2) |𝑥|𝑇𝜈 𝑣𝑛, 𝑗

  1 (1) (𝑥) = (𝑛(𝑛 + 1)) − 2 (2 𝜇(𝑛 ˜ − 1))∇𝐻𝑛, 𝑗 (𝑥) − 𝑛𝑇𝜈 𝑣𝑛, 𝑗 (𝑥) , 𝑛 = 1, 2, ..., 𝑗 = 1, ..., 2𝑛 + 1,

(3) |𝑥|𝑇𝜈 𝑣𝑛, 𝑗

1

(𝑥) = (𝑛(𝑛 + 1)) − 2 𝜇(𝑛 ˜ − 1)𝑥 ∧ ∇𝐻𝑛, 𝑗 (𝑥), 𝑛 = 1, 2, ..., 𝑗 = 1, ..., 2𝑛 + 1.

This leads us to the following theorem. Theorem 12.88 Let 𝑓 be of class c(Ω). Suppose that 𝑢 is the solution of the inner Dirichlet problem 𝑢 of the Dirichlet problem 𝑢 ∈ c (2) (Ωint ) ∩ c(Ωint ), ♦𝑢 = 0 in Ωint , 𝑢|Ω = 𝑓 . Then |𝑥|𝑇𝜈 (𝑢)(𝑥) =

3 ∑︁ ∞ 2𝑛+1 ∑︁ ∑︁

  (𝑖) ( 𝑓 (𝑖) ) ∧ (𝑛, 𝑗)𝑇𝜈 𝑣𝑛, 𝑗 (𝑥)

𝑖=1 𝑛=0𝑖 𝑗=1

for each 𝑥 ∈ Ωint . (𝑖) Next, we note that the fields 𝑣𝑛, 𝑗 admit a decomposition into curl-free and divergencefree parts. For that purpose, we formulate the following lemma (see T. Gervens (1989)).

Lemma 12.89 Under the assumptions of Lemma 12.84

12.14 Function Systems for Internal Elastic Fields



645





  (1) 2 2 𝑣𝑛, (𝑥) = 𝛿 ∇ 𝑥 𝐻 (𝑥) + 𝜀 ∇ ∧ ∇ ∧ 𝑥 𝐻 (𝑥) 𝑥 , 𝑛 𝑛, 𝑗 𝑛 𝑛, 𝑗 𝑗   (2) − 12 𝑣𝑛, ∇ 𝐻𝑛, 𝑗 (𝑥) − 𝑛𝛿 𝑛 𝑥 2 𝐻𝑛, 𝑗 (𝑥) 𝑗 (𝑥) = (𝑛(𝑛 + 1))    1 −(𝑛(𝑛 + 1)) − 2 𝑛𝜀 𝑛 ∇ ∧ ∇ ∧ 𝑥 2 𝐻𝑛, 𝑗 (𝑥) 𝑥 , 1

(3) −2 𝑣𝑛, ∇ ∧ (𝐻𝑛, 𝑗 (𝑥)𝑥), 𝑗 (𝑥) = −(𝑛(𝑛 + 1))

where 𝛿𝑛 =

𝑛 + 3 + 2𝑛𝛼𝑛 2𝑛𝛼𝑛 − 1 , 𝜀𝑛 = . 2(2𝑛 + 3) 2(2𝑛 + 3)

(12.441)

Proof Elementary calculations show us that   ∇ 𝑥 2 𝐻𝑛, 𝑗 (𝑥) = 2𝐻𝑛, 𝑗 (𝑥)𝑥 + ∇𝐻𝑛, 𝑗 (𝑥),

(12.442)

and   𝑥 2 𝐻𝑛, 𝑗 (𝑥) 𝑥   = −∇ 𝑥 2 𝑥 ∧ ∇𝐻𝑛, 𝑗 (𝑥)

= −2𝑥 ∧ 𝑥 ∧ ∇𝐻𝑛, 𝑗 (𝑥) + 𝑥 2 ∇ ∧ ∇ ∧ 𝐻𝑛, 𝑗 (𝑥)𝑥

∇∧∇∧



= −2𝑛𝐻𝑛, 𝑗 (𝑥)𝑥 + 2𝑥 2 ∇𝐻𝑛, 𝑗 (𝑥) + 𝑥 2 ∇ ∇ · 𝐻𝑛, 𝑗 (𝑥)𝑥
− 𝑥 2 Δ 𝐻𝑛, 𝑗 (𝑥)𝑥

(12.443)




= −2𝑛𝐻𝑛, 𝑗 (𝑥)𝑥 + (𝑛 + 3)𝑥 2 ∇𝐻𝑛, 𝑗 (𝑥). This implies 𝐻𝑛, 𝑗 (𝑥)𝑥 = (2(2𝑛+3)) −1



(12.444)  (𝑛+3)∇ 𝑥 2 𝐻𝑛, 𝑗 (𝑥) −∇ ∧ ∇ ∧ 𝑥 2 𝐻𝑛, 𝑗 (𝑥)𝑥 , 





𝑥 2 ∇𝐻𝑛, 𝑗 (𝑥)      = (2𝑛+3) −1 𝑛∇ 𝑥 2 𝐻𝑛, 𝑗 (𝑥) +∇ ∧ ∇ ∧ 𝑥 2 𝐻𝑛, 𝑗 (𝑥)𝑥 . (𝑖) Therefore, the vector fields 𝑣𝑛, 𝑗 , 𝑖 = 1, 2, 3, can be written as indicated by Lemma 12.89. □

Lemma 12.89 leads us to the following result. Theorem 12.90 For given 𝑓 ∈ c(Ω), the uniquely determined solution 𝑢 of the Dirichlet problem 𝑢 of the Dirichlet problem 𝑢 ∈ c (2) (Ωint ) ∩ c(Ωint ), ♦𝑢 = 0 in Ωint , 𝑢|Ω = 𝑓 is given by

646

12 Potentials in Euclidean Space





𝑢(𝑥) = ∇𝑍1 (𝑥) + ∇ ∧ ∇ ∧ 𝑥 2 𝑍2 (𝑥)𝑥 + ∇ ∧ (𝑍3 (𝑥)𝑥) for all 𝑥 ∈ 𝐾 with 𝐾 ⊂ Ωint and dist(𝐾, Ω) > 0, where the functions 𝑍𝑖 , 𝑖 = 1, 2, 3, can be written as follows: 𝑍1 (𝑥) =

∞ 2𝑛+1 ∑︁ ∑︁

( 𝑓 (1) ) ∧ (𝑛, 𝑗)𝛿 𝑛 𝑥 2 𝐻𝑛, 𝑗 (𝑥)

𝑛=0 𝑗=1



( 𝑓 (2) ) ∧ (𝑛, 𝑗)𝜎𝑛 𝐻𝑛, 𝑗 (𝑥) − 𝑛𝛿 𝑛 𝑥 2 𝐻𝑛, 𝑗 (𝑥) , √︁ 𝑛(𝑛 + 1) ∞ 2𝑛+1 ∑︁ ∑︁
𝑍2 (𝑥) = ( 𝑓 (1) ) ∧ (𝑛, 𝑗) − 𝑛 (𝑛(𝑛 + 1)) −1/2 ( 𝑓 (2) ) ∧ (𝑛, 𝑗) +

𝑛=0 𝑗=1


𝜀 𝑛 𝐻𝑛, 𝑗 (𝑥) , ∞ 2𝑛+1 ∑︁ ∑︁ 𝑍3 (𝑥) = − (𝑛(𝑛 + 1)) −1/2 ( 𝑓 (3) ) ∧ (𝑛, 𝑗)𝐻𝑛, 𝑗 (𝑥), 𝑛=1 𝑗=1

where 𝜎0 = 0 and 𝜎𝑛 = 1 for 𝑛 > 0. Obviously, the vector fields 𝑢 𝑖 , 𝑖 = 1, 2, 3, given by 𝑢 1 (𝑥) = ∇𝑍1 (𝑥),

(12.445) 

2



𝑢 2 (𝑥) = ∇ ∧ ∇ ∧ 𝑥 𝑍2 (𝑥)𝑥 ,

(12.446)

𝑢 3 (𝑥) = ∇ ∧ (𝑍3 (𝑥)𝑥)

(12.447)

satisfy ∇ ∧ 𝑢 1 (𝑥) = 0,

(12.448)

∇ · 𝑢 2 (𝑥) = 0, 𝑥 · ∇ ∧ 𝑢 2 (𝑥) = 0,

(12.449)

∇ · 𝑢 3 (𝑥) = 0, 𝑥 · 𝑢 3 (𝑥) = 0,

(12.450)

for all 𝑥 ∈ 𝐾 ⊂ Ωint with dist(𝐾, Ω) > 0. The vector field 𝑢 2 is of poloidal type, while 𝑢 3 is of toroidal type. Finally, we discuss the Neumann problem of determining polynomial solutions from given surface tractions on the unit sphere (see T. Gervens (1989)). (𝑖) Lemma 12.91 The vector fields 𝑤𝑛, 𝑗 , 𝑖 = 1, 2, 3, defined by (1) 𝑤𝑛, 𝑗 (𝑥)

 = 𝜁𝑛

 1 + 2𝑛𝛼𝑛 𝐻𝑛, 𝑗 (𝑥)𝑥 + 𝛼𝑛 𝑥 ∇𝐻𝑛, 𝑗 (𝑥) − ∇𝐻𝑛, 𝑗 (𝑥) , 2(𝑛 − 1) 2

12.14 Function Systems for Internal Elastic Fields

(1) 𝑤1, 𝑗 (𝑥)

647

𝑛 = 0, 2, 3, ..., 𝑗 = 1, ..., 2𝑛 + 1,   = 3𝜁1 𝐻1, 𝑗 (𝑥)𝑥 + 𝛼1 𝑥 2 ∇𝐻1, 𝑗 (𝑥) , 𝑗 = 1, 2, 3, 1

1

(2) (1) −2 𝑤𝑛, (2 𝜇(𝑛 ˜ − 1)) −1/2 ∇𝐻𝑛, 𝑗 (𝑥) − (𝑛(𝑛 + 1)) − 2 𝑛𝑤𝑛, 𝑗 (𝑥) = (𝑛(𝑛 + 1)) 𝑗 (𝑥),

𝑛 = 2, 3, ..., 𝑗 = 1, ..., 2𝑛 + 1, (3) 𝑤𝑛, 𝑗 (𝑥)

1

= (𝑛(𝑛 + 1)) − 2 ( 𝜇(𝑛 ˜ − 1)) −1 𝑥 ∧ ∇𝐻𝑛, 𝑗 (𝑥), 𝑛 = 2, 3, ..., 𝑗 = 1, ..., 2𝑛 + 1,

where 𝛼𝑛 = −

𝑛𝜏 + 2 + 3𝜏 1 , 𝜁𝑛 = , 2(𝑛(𝜏 + 2) + 1) (𝜆˜ + 𝜇)(3 ˜ + 𝑛 + 2𝑛𝛼𝑛 ) − 𝜇˜ 𝐻𝑛, 𝑗 (𝑥) = |𝑥| 𝑛𝑌𝑛, 𝑗 (𝜉), 𝑥 = |𝑥|𝜉, 𝜉 ∈ Ω,

(𝑖) (2) (Ωint ) ∩ c(Ωint ), ♦𝑢 = 0 in Ωint , and satisfy 𝑤𝑛, 𝑗 ∈ c

 − (1) (1) 𝑇𝜈 𝑤𝑛, = 𝑦 𝑛, 𝑗 𝑗 , 𝑛 = 0, 2, 3, ..., 𝑗 = 1, ..., 2𝑛 + 1,  − √ (2) (1) (1) 𝑇𝜈 𝑤1, = 2𝑦 𝑛, 𝑗 − 2 𝑦 𝑛, 𝑗 , 𝑗 = 1, 2, 3, 𝑗  − (𝑖) (𝑖) 𝑇𝜈 𝑤𝑛, = 𝑦 𝑛, 𝑗 𝑗 , 𝑖 = 2, 3; 𝑛 = 2, 3, ..., 𝑗 = 1, ..., 2𝑛 + 1.

Note that 𝜁 𝑛 is well defined for all 𝑛 ≥ 1 provided that 3𝜆˜ + 2 𝜇˜ > 0, 𝜇˜ > 0. We conclude our considerations with the following theorem. Theorem 12.92 Suppose that 𝑓 is of class c (0,𝛾) (Ω), i.e., 𝛾-H¨older continuous on Ω satisfying the conditions ∫ ∫ 𝑓 (𝜉) 𝑑𝜔(𝜉) = 0, ( 𝑓 (𝜉) ∧ 𝜉) 𝑑𝜔(𝜉) = 0. (12.451) Ω

Ω

Then the series (1) 𝑢 = ( 𝑓 (1) ) ∧ (0, 1)𝑤0,1 +

3 ∑︁ 𝑗=1

(1) ( 𝑓 (1) ) ∧ (1, 𝑗)𝑤1, 𝑗 +

3 ∑︁ ∞ 2𝑛+1 ∑︁ ∑︁

(𝑖) ( 𝑓 (𝑖) ) ∧ (𝑛, 𝑗)𝑤𝑛, 𝑗

𝑖=1 𝑛=2 𝑗=1

solves Neumann’s problem 𝑢 ∈ c (2) (Ωint ) ∩ c (1,𝛾) (Ωint ), ♦𝑢 = 0 in Ωint , 𝑇𝜈 (𝑢) = 𝑓 on every K ⊂ Ωint with dist(K, Ω) > 0. The extension of our results to the sphere Ω𝑅 around the origin with radius 𝑅 is obvious (see W. Freeden et al. (1990)).

648

12 Potentials in Euclidean Space

12.15 Bibliographical Notes

In the theory of harmonic functions, a result first motivated by C. Runge (1885) in one-dimensional complex analysis and later generalized, e.g., by J.L. Walsh (1929), I.N. Vekua (1953), and L. H¨ormander (1975) to potential theory in three-dimensional Euclidean space R3 is of basic interest. For geodetically relevant obligations (see, e.g., T. Krarup (1969), H. Moritz (1977, 2015), F. Sans`o (1982), and the references therein) it may be formulated in accordance with H. Moritz (1980): Let G ⊂ R3 be a regular region. Any harmonic function in G 𝑐 = R3 \G that is regular at infinity can be approximated by a function that is harmonic outside an arbitrarily given Runge (i.e., in geodetic nomenclature sometimes called Bjerhammar) ball A ⋐ G, i.e., A ⊂ G with dist(A, 𝜕G) > 0 in the sense (cf. Fig. 12.7) that, for any given 𝜀 > 0, the absolute error between the two functions is smaller than 𝜀 for all points outside and on any closed surface completely surrounding 𝜕G in its outer space. The value 𝜀 may be arbitrarily small, and the surrounding surface may be arbitrarily close to the surface 𝜕G. Obviously, the Runge-Walsh theorem in the preceding geodetic formulation (with G, e.g., chosen as the interior of the actual Earth) represents a pure existence theorem. It guarantees only the existence of an approximating function and does not provide a constructive method to find it. Nothing is said about the approximation procedure and the computational structure and methodology of the special function systems involved in the approximation. The theorem merely describes the theoretical background for the approximation of a potential by another potential defined on a larger harmonicity domain, i.e., the Runge region outside the sphere 𝜕A. The situation, however, is completely different if spherical geometries are exclusively involved in the Runge concept. Assuming that both A, G are concentric balls around the origin with A ⋐ G, a constructive approximation of a potential in the outer space G 𝑐 is available, e.g., by outer harmonic (orthogonal) expansions (see, e.g., A.M. Legendre (1785), P.S. de Laplace (1785), C.F. Gauss (1838), A. Wangerin (1921), F. Neumann (1887), O.D. Kellogg (1929)). More concretely, within the classical context of a twofold spherical configuration, a constructive version of the Runge-Walsh theorem can be guaranteed by finite truncations of Fourier expansions in terms of outer harmonics, where the 𝐿 2 (𝜕G)-convergence of the Fourier series implies uniform convergence on any point set K ⋐ G 𝑐 . The Fourier coefficients are obtained by integration over the sphere 𝜕G. The gravitational potential is available (in spectral sense) by tables of the Fourier coefficients. Nowadays, in fact, outer harmonic expansions constitute the conventional geodetic tools in globally reflected approximation of the Earth’s gravitational potential and its observables (see, e.g., F.G. Lemoine et al. (1998), N.K. Pavlis et al. (2008)). From a superficial point of view, one could suggest that the standard approximation by truncated series expansions in terms of outer harmonics is closely related to spherical geometries 𝜕A, 𝜕G. Following the work W. Freeden (1980a), W. Freeden, V. Michel (2004), M.A. Augustin et al. (2018), however, we know that the essential steps to a constructive Fourier approach can be extended to any regular, i.e., not-necessarily spherical region G and to any

12.15 Bibliographical Notes

649

regular, i.e., not-necessarily spherical Runge region A ⋐ G (see Fig. 12.7, left illustration). As a matter of fact, the Runge-Walsh approach enables us to avoid any calamities with the convergence to the gravitational potential by the generalized Fourier series for arbitrary sets K ⋐ G 𝑐 . In analogy to the spherical case, however, it likewise does not help to specify convergence inside A 𝑐 \G 𝑐 , so that any attempts (see A. Bjerhammar (1962)) to reduce gravitational information via infinite Fourier series downward from 𝜕G to the surface 𝜕A are not justifiable by the Runge-Walsh framework. In summary, the Runge-Walsh concept as presented in this work reflects constructive approximation capabilities of the Earth’s gravitational (and not gravity) potential (cf. E.W. Grafarend (2015)) even if geoscientifically realistic (i.e., not necessarily spherical) geometries come into play. Mathematically, it should be pointed out that the main techniques for assuring the notnecessarily spherical results are the limit and jump relations and their formulations of potential theory in the Hilbert space nomenclature of (𝐿 2 (𝜕G), ∥ · ∥ 𝐿 2 (𝜕G) ) (cf. W. Freeden (1980a), W. Freeden, C. Gerhards (2013)) . The special function systems for use in constructive Runge-Walsh theorems are manifold, for example, outer harmonics, ellipsoidal harmonics, certain single- and multipole systems, certain kernel representations aso. as already described in W. Freeden, M. Schreiner (2009). More explicitly (cf. M.A. Augustin et al. (2018)), all harmonic functions systems that are regular at infinity can be taken into account, whose restrictions to the boundary 𝜕G of a regular region G form an 𝐿 2 (𝜕G)-complete system. Early numerical studies involving certain monopole and multipole configurations are provided by W. Freeden (1983, 1987a,c), R. Litzenberger (2001). In more detail, seen from methodological point of view, Runge (Fourier) approaches using a sequence of kernel functions corresponding to an inner fundamental system can be realized in equivalent manner to outer harmonics expansions in the L2 (𝜕G)-topology for recovering a gravitational potential. In fact, a sequence of kernel functions is conceptually easier to implement than outer harmonic expansions, as far as the kernels are available in closed form as elementary functions. Thus, kernel function approximations have a long history. Early attempts to make the “method of fundamental solutions” become reality date back to the first decades of the last century (cf. W. Ritz (1909), E. Trefftz (1926)). Further ideas are, e.g., due to I.N. Vekua (1953), V. Kupradze, M. Aleksidze (1964), V.D. Kupradze (1965), E. Kita, N. Kamiya (1995), M. Golberg (1995). The line to the Fourier approach as presented in this work is designed by J.L. Walsh (1929), I.N. Vekua (1953), W. Freeden (1981c, 1983, 1987a), W. Freeden, H. Kersten (1980, 1981, 1982), M. Golberg, C. Chen (1998), W. Freeden, C. Mayer (2006), W. Freeden, C. Gerhards (2013), M.A. Augustin et al. (2018). In the meantime generalized Fourier expansions are theoretically established and practically applied not only to the Laplace equation, but also to more general elliptic partial differential equations, e.g., the reduced (Helmholtz) wave equation, the Cauchy-Navier equation, (reduced) Maxwell equations, the (linear) Stokes equations, poroelastic equations, etc. (see, e.g., W. Freeden, F. Schneider (1998a, 1999), W. Freeden et al. (2003), W. Freeden, R. Reuter (1990b),

650

12 Potentials in Euclidean Space

M.K. Abeyratne et al. (2003), W. Freeden, C. Mayer (2003, 2007), W. Freeden, M. Schreiner (2009), M.A. Augustin (2015), W. Freeden, H. Nutz (2015) for a more detailed list of references). The drawback of the numerical realization is the adequate selection of a finite number of points out of the infinite inner fundamental system, which is both computationally efficient and physically relevant. An optimal strategy for positioning the finite system remains a great challenge for future work. A constructive procedure of determining best approximate coefficients in the C (0) (𝜕G)-topology seems to be unknown. Therefore, in W. Freeden, V. Michel (2004) based on W. Freeden (1981b,c, 1982c, 1987a), harmonic splines are introduced in reproducing Hilbert subspaces of Pot (0) (R3 \Ωint 𝑅 ) (characterized by variational principles), so that the spline method can be regarded as an immediate extension of the method of generalized Fourier expansions to reproducing kernel subspaces of Pot (0) (R3 \Ωint 𝑅 ), hence, providing coefficients optimal in a different (Sobolev like) norm.

Chapter 13

Potentials on the Sphere

The property Δ∗𝜂 𝐺 (Δ∗ ; 𝜉 · 𝜂) = −

1 , 4𝜋

𝜂 ∈ Ω \ {𝜉},

(13.1)

denotes the major difference between the fundamental solution 𝐺 (Δ; ·) for the Laplace operator and its spherical counterpart 𝐺 (Δ∗ ; ·). While 𝐺 (Δ; ·) generates a “true” Dirac distribution in the sense that Δ 𝑦 𝐺 (Δ; |𝑥 − 𝑦|) = 0, 𝑦 ∈ R3 \ {𝑥}, the fundamental solution 𝐺 (Δ∗ ; ·) given by 𝐺 (Δ∗ ; 𝑡) =

1 1 ln(1 − 𝑡) + (1 − ln(2)), 4𝜋 4𝜋

𝑡 ∈ [−1, 1),

(13.2)

only generates a Dirac distribution up to an additive constant (reflecting the nullspace of the Beltrami operator Δ∗ ). Thus, potential theory on the sphere Ω ⊂ R3 with respect to Δ∗ shows some aspects different from potential theory in Euclidean space R3 .

13.1 Green’s Formulas

Let Γ ⊂ Ω be a regular region with continuously differentiable boundary curve 𝜕Γ. The solid angle 𝛼 of the regular region is defined such that 𝛼(𝜉) = 2𝜋 for 𝜉 ∈ Γ, 𝛼(𝜉) = 𝜋 for 𝜉 ∈ 𝜕Γ, and 𝛼(𝜉) = 0 for 𝜉 ∈ Γ 𝑐 = Ω\Γ. Applying Green’s formulas the properties of 𝐺 (Δ∗ ; ·) lead to the following already known integral representations.

© The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2022 W. Freeden and M. Schreiner, Spherical Functions of Mathematical Geosciences, Geosystems Mathematics, https://doi.org/10.1007/978-3-662-65692-1_13

651

652

13 Potentials on the Sphere

(i) If 𝐹 is of class C (2) (Γ), then we have for 𝜉 ∈ Ω, ∫ 𝛼(𝜉) 1 𝐹 (𝜉) = 𝐹 (𝜂) 𝑑𝜔(𝜂) 2𝜋 4𝜋 Γ ∫ + 𝐺 (Δ∗ ; 𝜉 · 𝜂)Δ∗𝜂 𝐹 (𝜂) 𝑑𝜔(𝜂) ∫Γ 𝜕 + 𝐹 (𝜂) 𝐺 (Δ∗ ; 𝜉 · 𝜂) 𝑑𝜎(𝜂) 𝜕𝜈(𝜂) 𝜕Γ ∫ 𝜕 − 𝐺 (Δ∗ ; 𝜉 · 𝜂) 𝐹 (𝜂) 𝑑𝜎(𝜂). 𝜕𝜈(𝜂) 𝜕Γ

(13.3)

(ii) If 𝐹 is of class C (1) (Γ), then we have for 𝜉 ∈ Ω, ∫ ∫ 𝛼(𝜉) 1 𝐹 (𝜉) = 𝐹 (𝜂)𝑑𝜔(𝜂) − ∇∗𝜂 𝐺 (Δ∗ ; 𝜉 · 𝜂) · ∇∗𝜂 𝐹 (𝜂)𝑑𝜔(𝜂) 2𝜋 4𝜋 Γ Γ (13.4) ∫

𝜕 + 𝐹 (𝜂) 𝐺 (Δ∗ ; 𝜉 · 𝜂)𝑑𝜎(𝜂), 𝜕𝜈(𝜂) 𝜕Γ ∫ ∫ 1 = 𝐹 (𝜂)𝑑𝜔(𝜂) − 𝐿 ∗𝜂 𝐺 (Δ∗ ; 𝜉 · 𝜂) · 𝐿 ∗𝜂 𝐹 (𝜂)𝑑𝜔(𝜂) 4𝜋 Γ Γ ∫ 𝜕 + 𝐹 (𝜂) 𝐺 (Δ∗ ; 𝜉 · 𝜂)𝑑𝜎(𝜂). 𝜕𝜈(𝜂) 𝜕Γ Choosing 𝐹 = 1 in any of the formulas implies the relation ∫ 𝛼(𝜉) ∥Γ∥ 𝜕 = + 𝐺 (Δ∗ ; 𝜉 · 𝜂) 𝑑𝜎(𝜂), 2𝜋 4𝜋 𝜕𝜈(𝜂) 𝜕Γ i.e., ∫ 𝜕Γ

 ∥Γ∥   4 𝜋 , 𝜉 ∈ Γ,  1 − ∥Γ∥  𝜕 ∗ 𝐺 (Δ ; 𝜉 · 𝜂) 𝑑𝜎(𝜂) = 12 − 4 𝜋 , 𝜉 ∈ 𝜕Γ,  𝜕𝜈(𝜂)   − ∥Γ∥ , 𝜉 ∈ Γ 𝑐 ,  4𝜋

(13.5)

where ∥Γ∥ denotes the surface area of Γ. The behavior of (13.5) across the boundary 𝜕Γ states a first hint at the limit and jump relations of the layer potentials in Section 13.4. Apart from the additive constant ∥4Γ∥ 𝜋 , they are analogous to the Euclidean setting.

13.2 Surface Harmonicity

653

13.2 Surface Harmonicity

Now we turn towards functions that are harmonic (with respect to Beltrami operator) in Γ, i.e., functions 𝑈 of class C (2) (Γ) that satisfy Δ∗𝑈 = 0

in Γ.

(13.6)

If no confusion with the Euclidean case is likely to arise, we just say that 𝑈 is harmonic. Plugging such functions into (13.3), together with the choice of Γ being a spherical cap Γ𝜌 (𝜉) = {𝜂 ∈ Ω|1 − 𝜉 · 𝜂 < 𝜌} with center 𝜉 ∈ Ω and radius 𝜌 ∈ (0, 2), we end up with the mean value property for harmonic functions. Theorem 13.1 (Mean Value Property I) A function 𝑈 of class C (0) (Γ) is harmonic if and only if √︁ ∫ ∫ 2−𝜌 1 𝑈 (𝜉) = 𝑈 (𝜂)𝑑𝜔(𝜂) + 𝑈 (𝜂)𝑑𝜎(𝜂), √ 4𝜋 Γ𝜌 ( 𝜉 ) 4𝜋 𝜌 𝜕Γ𝜌 ( 𝜉 )

𝜉 ∈ Γ,

(13.7)

for any spherical cap Γ𝜌 (𝜉) ⊂ Γ. The mean value property above contains the typical additive term for spherical problems. However, we can get rid of this additive constant when using Green’s functions for spherical caps as described later. We are led to the following representation which resembles a Mean Value Property that is more closely related to the Euclidean case of functions that are harmonic with respect to the Laplace operator (cf. C. Gerhards (2018)). Theorem 13.2 (Mean Value Property II) A function 𝑈 of class C (0) (Γ) is harmonic if and only if ∫ 1 𝑈 (𝜉) = √︁ 𝑈 (𝜂)𝑑𝜎(𝜂), 2𝜋 𝜌(2 − 𝜌) 𝜕Γ𝜌 ( 𝜉 )

𝜉 ∈ Γ,

(13.8)

for any spherical cap Γ𝜌 (𝜉) ⊂ Γ. Once a Mean Value Property is established, it can be used to derive a maximum principle (for details, we refer to W. Freeden, C. Gerhards (2013)).

654

13 Potentials on the Sphere

Theorem 13.3 (Maximum Principle) If 𝑈 of class C (2) (Γ) ∩ C (0) (Γ) is harmonic, then sup |𝑈 (𝜉)| ≤ sup |𝑈 (𝜉)|. 𝜉 ∈Γ

(13.9)

𝜉 ∈𝜕Γ

13.3 Surface Potentials

Analogously to the Euclidean setting, we can define a Newton potential and layer potentials for the spherical setting which take over the corresponding roles. The obvious difference is that now the Newton potential is a surface potential and the layer potentials represent curve potentials. Throughout this section, we take a closer look at the surface potential ∫ 𝑈 (𝜉) = 𝐺 (Δ∗ ; 𝜉 · 𝜂)𝐻 (𝜂) 𝑑𝜔(𝜂), 𝜉 ∈ Ω,

(13.10)

Γ

From the properties of the fundamental solution for the Beltrami operator it becomes directly clear that 𝑈 is of class C (2) (Γ 𝑐 ) and that ∫ ∫ 1 ∗ ∗ Δ𝜉 𝐺 (Δ ; 𝜉 · 𝜂)𝐻 (𝜂) 𝑑𝜔(𝜂) = − 𝐻 (𝜂) 𝑑𝜔(𝜂), 𝜉 ∈ Γ 𝑐 . (13.11) 4𝜋 Γ Γ Yet, the interesting question is what happens if 𝜉 ∈ Γ, i.e., when the integration region contains the logarithmic singularity of 𝐺 (Δ∗ ; ·). Theorem 13.4 If 𝐻 is of class C (0) (Γ) and 𝑈 is given by (13.10), then 𝑈 is of class C (1) (Ω) and ∫ ∫ ∇∗𝜉 𝐺 (Δ∗ ; 𝜉 · 𝜂)𝐻 (𝜂) 𝑑𝜔(𝜂) = ∇∗𝜉 𝐺 (Δ∗ ; 𝜉 · 𝜂)𝐻 (𝜂) 𝑑𝜔(𝜂), 𝜉 ∈ Ω. Γ

Γ

(13.12) The proof of the theorem above can be based on a mollifier regularization of the fundamental solution 𝐺 (Δ∗ ; ·). This approach also works for the application of the Beltrami operator to 𝑈. However, we find that Δ∗𝑈 is not continuous across 𝜕Γ anymore (compare equation (13.11) and Theorem 13.5). For brevity, we do not supply the proofs at this point but refer the reader, e.g., to W. Freeden, C. Gerhards

13.3 Surface Potentials

655

(2013). A related mollifier regularized Green’s function plays an important role in the applications and is explained in more detail later on. Theorem 13.5 If 𝐻 is of class C (1) (Γ) and 𝑈 is given by (13.10), then 𝑈 is of class C (2) (Γ) and satisfies ∫ ∫ 1 ∗ ∗ Δ𝜉 𝐺 (Δ ; 𝜉 · 𝜂)𝐻 (𝜂) 𝑑𝜔(𝜂) = 𝐻 (𝜉) − 𝐻 (𝜂) 𝑑𝜔(𝜂), 𝜉 ∈ Γ. (13.13) 4𝜋 Γ Γ Poisson Problem for the Beltrami Operator. Let 𝐻 be of class C (1) (Γ), then we are looking for a function 𝑈 of class C (2) (Γ) such that Δ∗𝑈 (𝜉) = 𝐻 (𝜉), If we choose 𝐻¯ = 𝐻 − Theorem 13.5, that

1 ∥Γ∥

𝑈¯ (𝜉) =

∫ Γ

∫

𝜉 ∈ Γ.

𝐻 (𝜂) 𝑑𝜔(𝜂), we find that

𝐺 (Δ∗ ; 𝜉 · 𝜂) 𝐻¯ (𝜂) 𝑑𝜔(𝜂),

(13.14) ∫ Γ

𝐻¯ (𝜂) 𝑑𝜔(𝜂) = 0 and, by

𝜉 ∈ Γ,

(13.15)

Γ

satisfies Δ∗𝑈¯ (𝜉) = 𝐻¯ (𝜉), for 𝜉 ∈ Γ. Setting 𝑈 (𝜉) = 𝑈¯ (𝜉) −

1 ¯ ln(1 − 𝜉 · 𝜉) ∥Γ∥

∫ 𝐻 (𝜂) 𝑑𝜔(𝜂),

(13.16)

Γ

for some fixed 𝜉¯ ∈ Γ 𝑐 , we eventually obtain the desired solution satisfying ∫ 1 Δ∗𝑈 (𝜉) = 𝐻¯ (𝜉) + 𝐻 (𝜂) 𝑑𝜔(𝜂) = 𝐻 (𝜉), 𝜉 ∈ Γ. (13.17) ∥Γ∥ Γ The solution of (13.14), however, is not unique. Prescribing further boundary values on 𝑈, e.g., Dirichlet boundary values 𝑈 − (𝜉) = 𝐹 (𝜉), for 𝜉 ∈ 𝜕Γ, it is possible to obtain uniqueness. Letting 𝑈˜ denote the function 𝑈 from (13.17) that we constructed before, we can formulate the boundary-value problem of finding a function 𝑈˜ that solves Δ∗𝑈˜ (𝜉) = 0, 𝜉 ∈ Γ, 𝑈˜ − (𝜉) = 𝐹 (𝜉) − 𝑈˜ − (𝜉),

(13.18) 𝜉 ∈ 𝜕Γ.

(13.19)

The newly obtained function 𝑈 = 𝑈˜ + 𝑈˜ would then satisfy the desired differential equation (13.14) and the desired Dirichlet boundary values. Boundary-value problems such as (13.18), (13.19) are studied in more detail in the upcoming sections.

656

13 Potentials on the Sphere

13.4 Curve Potentials

While surface potentials are useful to deal with the Poisson problem (PP), curve potentials are particularly useful when dealing with functions that are harmonic (with respect to the Beltrami operator). More precisely, we take a closer look at the two layer potentials ∫ 𝑈1 (𝜉) = 𝑈1 [𝑄] (𝜉) = 𝐺 (Δ∗ ; 𝜉 · 𝜂)𝑄(𝜂) 𝑑𝜎(𝜂), 𝜉 ∈ Γ, (13.20) 𝜕Γ

and ∫ 𝑈2 (𝜉) = 𝑈2 [𝑄] (𝜉) = 𝜕Γ



 𝜕 𝐺 (Δ∗ ; 𝜉 · 𝜂) 𝑄(𝜂) 𝑑𝜎(𝜂), 𝜕𝜈(𝜂)

𝜉 ∈ Γ.

(13.21)

From the properties of the fundamental solution 𝐺 (Δ∗ ; ·) it can be seen that the socalled double-layer potential 𝑈2 [𝑄] is harmonic in Γ for any 𝑄 of class C (0) (𝜕Γ). ˜ is harmonic in Γ if 𝑄 is of class C (0) (𝜕Γ) and if The single-layer potential 𝑈1 [𝑄] ∫ the integral over 𝜕Γ vanishes, i.e., if 𝜕Γ 𝑄(𝜂)𝑑𝜎(𝜂) = 0 (we say that 𝑄 is of class C0(0) (𝜕Γ)). Therefore, these two potentials represent good candidates for solutions to the boundary-value problems (DP) and (NP). The aim of the present section is to investigate the behaviour of the single- and double-layer potentials 𝑈1 [𝑄] and ˜ respectively, when they approach the boundary 𝜕Γ. The essential behaviour 𝑈2 [𝑄], of the double-layer potential 𝑈2 [𝑄] is already reflected by the relation (13.5). Based on this relation and a set of several more technical estimates, one can prove the following set of limit and jump relations at the boundary 𝜕Γ.

13.5 Limit Formulas and Jump Relations Theorem 13.6 Let 𝑄 be of class C (0) (𝜕Γ) and 𝑈1 , 𝑈2 be given as in (13.20) and (13.21), respectively. Furthermore, let 𝜉 ∈ 𝜕Γ. (i) For the single-layer potential, we have the limit relations   𝜉 ± 𝜏𝜈(𝜉) lim 𝑈1 √ − 𝑈1 (𝜉) = 0, 𝜏→0+ 1 + 𝜏2      𝜕 𝜉 ± 𝜏𝜈(𝜉) 𝜕 1 lim 𝑈1 − 𝑈 √ 1 (𝜉) = ± 𝑄(𝜉). 2 𝜏→0+ 𝜕𝜈 𝜕𝜈 2 1+𝜏 For the double-layer potential, we have

(13.22) (13.23)

13.6 Boundary-Value Problems for the Beltrami Operator

 lim 𝑈2 𝜏→0+

657



𝜉 ± 𝜏𝜈(𝜉) 1 − 𝑈2 (𝜉) = ∓ 𝑄(𝜉). √ 2 1 + 𝜏2

(ii) For the single-layer potential, we have the jump relations      𝜉 + 𝜏𝜈(𝜉) 𝜉 − 𝜏𝜈(𝜉) lim 𝑈1 √ − 𝑈1 √ = 0, 𝜏→0+ 1 + 𝜏2 1 + 𝜏2       𝜕 𝜉 + 𝜏𝜈(𝜉) 𝜕 𝜉 − 𝜏𝜈(𝜉) lim 𝑈1 − 𝑈1 = 𝑄(𝜉). √ √ 2 𝜏→0+ 𝜕𝜈 𝜕𝜈 1+𝜏 1 + 𝜏2 For the double-layer potential, we have      𝜉 + 𝜏𝜈(𝜉) 𝜉 − 𝜏𝜈(𝜉) lim 𝑈2 √ − 𝑈2 √ = −𝑄(𝜉), 𝜏→0+ 1 + 𝜏2 1 + 𝜏2       𝜕 𝜉 + 𝜏𝜈(𝜉) 𝜕 𝜉 − 𝜏𝜈(𝜉) lim 𝑈2 − 𝑈2 = 0. √ √ 2 𝜏→0+ 𝜕𝜈 𝜕𝜈 1+𝜏 1 + 𝜏2

(13.24)

(13.25) (13.26)

(13.27) (13.28)

The jump relations from part (b) are a direct consequence of part (a) of Theorem 13.6. The second relation in (b) has no analogue in (a) for functions 𝑄 of class C (0) (𝜕Γ). However, if 𝑄 is assumed to have additional differentiability properties, a limit relation can be derived as well. Theorem 13.6 essentially tells us that the single-layer potential 𝑈1 and the normal 𝜕 derivative of the double-layer potential 𝜕𝜈 𝑈2 are continuous across the boundary 𝜕Γ, while the double-layer potential 𝑈2 and the normal derivative of the single𝜕 𝜕 layer potential 𝜕𝜈 𝑈1 are not. However, one has to be careful about 𝜕𝜈 𝑈2 : it is only well-defined on 𝜕Γ under higher smoothness assumptions on 𝑄 than just C (0) (𝜕Γ). Therefore, we only supplied the jump relation for this particular case but not the limit relation, which is sufficient for most theoretical considerations. Remark 13.7 The relations in Theorem 13.6 may be formulated with respect to the uniform topology for 𝑄 ∈ C (0) (Ω). However, they can also be formulated with respect to the L2 (Ω)-topology for 𝑄 ∈ L2 (Ω).

13.6 Boundary-Value Problems for the Beltrami Operator

Next we investigate boundary-value problems. Our goal is to obtain integral representations of their solutions.

658

13 Potentials on the Sphere

• Dirichlet Problem (DP): Let 𝐹 be of class C (0) (𝜕Γ). We are looking for a function 𝑈 of class C (2) (Γ) ∩ C (0) (Γ) such that Δ∗𝑈 (𝜉) = 0, −

𝑈 (𝜉) = 𝐹 (𝜉),

𝜉 ∈ Γ,

(13.29)

𝜉 ∈ 𝜕Γ.

(13.30)

• Neumann Problem (NP): Let 𝐹 be of class C (0) (𝜕Γ). We are looking for a function 𝜕 𝑈 of class C (2) (Γ) ∩ C (0) (Γ), with a well-defined normal derivative 𝜕𝜈 𝑈 − on 𝜕Γ, such that Δ∗𝑈 (𝜉) = 0, 𝜕 − 𝑈 (𝜉) = 𝐹 (𝜉), 𝜕𝜈

𝜉 ∈ Γ,

(13.31)

𝜉 ∈ 𝜕Γ.

(13.32)

The term 𝑈 − (𝜉) for 𝜉 ∈ 𝜕Γ is meant in the sense   𝜉 − 𝜏𝜈(𝜉) − 𝑈 (𝜉) = lim 𝑈 √ , 𝜏→0+ 1 + 𝜏2

(13.33)

i.e., we approach the boundary 𝜕Γ in normal direction from the inside of Γ. The term 𝑈 + (𝜉) is meant in the sense   𝜉 + 𝜏𝜈(𝜉) + 𝑈 (𝜉) = lim 𝑈 √ , (13.34) 𝜏→0+ 1 + 𝜏2 i.e., we approach the boundary 𝜕Γ in normal direction from the outside of Γ (or, 𝜕 ± in other words, from within Γ 𝑐 ). The expressions 𝜕𝜈 𝑈 (𝜉) are meant analogously. We can already see the connection to the limit and jump relations given in Theorem 13.6. More precisely, understanding 𝑈 as 𝑈2 for the Dirichlet problem (DP) and 𝑈 as 𝑈1 for the Neumann problem (NP), Theorem 13.6 leads to the following closely related problems: • Integral Dirichlet Problem (IDP): Let 𝐹 be of class C (0) (𝜕Γ). We are looking for some 𝑄 of class C (0) (𝜕Γ) that satisfies 1 𝐹 (𝜉) = 𝑈2 [𝑄] (𝜉) + 𝑄(𝜉), 2

𝜉 ∈ 𝜕Γ.

(13.35)

• Integral Neumann Problem (INP): Let 𝐹 be of class C (0) (𝜕Γ). We are looking for some 𝑄 of class C0(0) (𝜕Γ) that satisfies 1 𝐹 (𝜉) = 𝑈1 [𝑄] (𝜉) − 𝑄(𝜉), 2

𝜉 ∈ 𝜕Γ.

(13.36)

13.6 Boundary-Value Problems for the Beltrami Operator

659

In other words, the Dirichlet problem (DP) and the Neumann problem (NP) have been reduced to Fredholm integral equations (IDP) and (INP). These boundary integral formulations have been used, e.g., in S. Gemmrich et al. (2008), M.C.A. Kropinski, N. Nigam (2014) to solve the original boundary-value problems for the Beltrami operator numerically. In what follows, however, we are mainly interested in (IDP) and (INP) as tools to derive the existence of solutions to (DP) and (NP), respectively, via the Fredholm alternative (similarly to the Euclidean case). Uniqueness of the solutions can be obtained by virtue of the maximum principle from Theorem 13.3 and the Green’s formulas (cf. W. Freeden, C. Gerhards (2013)). Remark 13.8 There are two noteworthy differences in comparison to the Euclidean case: First, considerations on the sphere do not require a distinction between interior and exterior problems since the open complement Γ 𝑐 of a regular region Γ ⊂ Ω is again a regular region. Second, the single-layer potential 𝑈1 is only harmonic if 𝑄 ∈ C0(0) (𝜕Γ). A solution of (13.36) in C0(0) (𝜕Γ) exists if and only if 𝐹 is of class C0(0) (𝜕Γ), which suits the general necessary condition for the existence of a solution to (NP) that can be obtained from Green’s formulas. However, it should be mentioned that the integral equation (13.36) additionally has a unique solution 𝑄 ∈ C (0) (𝜕Γ), if 𝐹 is of class C (0) (𝜕Γ). This is not true for the Euclidean counterpart. Summarizing we obtain the following results (for more details, the reader is again referred to W. Freeden, C. Gerhards (2013)). Theorem 13.9 (Uniqueness)

(i) A solution of (DP) is uniquely determined. (ii) A solution of (NP) is uniquely determined up to an additive constant. Theorem 13.10 (Existence for Generalized (DP)) Let 𝐹 be of class C (0) (𝜕Γ) and 𝐻 of class C (1) (Γ). Then there exists a unique solution 𝑈 of class C (2) (Γ) ∩ C (0) (Γ) of the Dirichlet problem Δ∗𝑈 (𝜉) = 𝐻 (𝜉), −

𝑈 (𝜉) = 𝐹 (𝜉),

𝜉 ∈ Γ,

(13.37)

𝜉 ∈ 𝜕Γ.

(13.38)

660

13 Potentials on the Sphere

Theorem 13.11 (Existence for Generalized (NP)) Let 𝐹 be of class C (0) (𝜕Γ) and 𝐻 of class C (1) (Γ). Then there exists an up to an additive constant uniquely determined solution 𝑈 of class C (2) (Γ) ∩ C (0) (Γ), with 𝜕 a well-defined normal derivative 𝜕𝜈 𝑈 − on 𝜕Γ, to the Neumann problem Δ∗𝑈 (𝜉) = 𝐻 (𝜉), 𝜕 − 𝑈 (𝜉) = 𝐹 (𝜉), 𝜕𝜈

𝜉 ∈ Γ,

(13.39)

𝜉 ∈ 𝜕Γ,

(13.40)

𝐻 (𝜂) 𝑑𝜔(𝜂) = 0.

(13.41)

if and only if ∫

∫ 𝐹 (𝜂) 𝑑𝜎(𝜂) −

𝜕Γ

Γ

Proof The condition (13.41) is an immediate consequence from ∫ ∫ ∫ ∫ 𝜕 𝐻 (𝜂) 𝑑𝜔(𝜂) = Δ∗𝑈 (𝜂) 𝑑𝜔(𝜂) = 𝑈 (𝜂) 𝑑𝜎(𝜂) = 𝐹 (𝜂) 𝑑𝜎(𝜂), Γ Γ 𝜕Γ 𝜕𝜈 𝜕Γ (13.42) where Green’s formulas have been used for the second equation. The existence follows from the application of the Fredholm alternative to (INP). □ Dirichlet and Neumann Green’s Function. We are interested in the representation of a solution to (DP) and (NP). A solution strategy is indicated by Green’s formula (13.3). However, this representation requires the simultaneous knowledge of 𝑈 and 𝜕 𝜕𝜈 𝑈 on the boundary 𝜕Γ, which is not necessary and can be problematic since the two quantities are not independent from each other. As a remedy, Green’s functions for Dirichlet and Neumann boundary values can be used. In more detail, a function 𝐺 𝐷 (Δ∗ ; ·, ·) is called a Dirichlet Green’s function (with respect to the Beltrami operator), if it can be decomposed in the form 𝐺 𝐷 (Δ∗ ; 𝜉, 𝜂) = 𝐺 (Δ∗ ; 𝜉 · 𝜂) − Φ𝐷 (𝜉, 𝜂),

𝜂 ∈ Γ, 𝜉 ∈ Γ, 𝜉 ≠ 𝜂,

(13.43)

where Φ𝐷 (𝜉, ·) is of class C (2) (Γ) ∩ C (1) (Γ) satisfying 1 , 𝜂 ∈ Γ, 4𝜋 − Φ𝐷 (𝜉, 𝜂) = 𝐺 (Δ∗ ; 𝜉 · 𝜂), 𝜂 ∈ 𝜕Γ,

Δ∗𝜂 Φ𝐷 (𝜉, 𝜂) = −

(13.44) (13.45)

for every 𝜉 ∈ Γ. Analogously, a function 𝐺 𝑁 (Δ∗ ; ·, ·) is called a Neumann Green’s function (with respect to the Beltrami operator), if it can be decomposed in the form 𝐺 𝑁 (Δ∗ ; 𝜉, 𝜂) = 𝐺 (Δ∗ ; 𝜉 · 𝜂) − Φ 𝑁 (𝜉, 𝜂),

𝜂 ∈ Γ, 𝜉 ∈ Γ, 𝜉 ≠ 𝜂,

(13.46)

13.6 Boundary-Value Problems for the Beltrami Operator

661

where Φ 𝑁 (𝜉, ·) is of class C (2) (Γ) ∩ C (1) (Γ) and satisfies the conditions 1 1 − , 𝜂 ∈ Γ, ∥Γ∥ 4𝜋 𝜕 𝜕 Φ−𝑁 (𝜉, 𝜂) = 𝐺 (Δ∗ ; 𝜉 · 𝜂), 𝜂 ∈ 𝜕Γ, 𝜕𝜈(𝜂) 𝜕𝜈(𝜂) Δ∗𝜂 Φ 𝑁 (𝜉, 𝜂) =

(13.47) (13.48)

for every 𝜉 ∈ Γ. For Φ𝐷 and Φ 𝑁 , we eventually achieve the representations ∫ ∫ 𝜕 𝑈 (𝜉) = 𝐺 𝐷 (Δ∗ ; 𝜉, 𝜂)Δ∗𝜂 𝑈 (𝜂) 𝑑𝜔(𝜂) + 𝑈 (𝜂) 𝐺 𝐷 (Δ∗ ; 𝜉, 𝜂) 𝑑𝜎(𝜂) 𝜕𝜈(𝜂) Γ 𝜕Γ (13.49) and 𝑈 (𝜉) =

1 ∥Γ∥ ∫ −

∫

∫ 𝑈 (𝜂) 𝑑𝜔(𝜂) +

Γ

(13.50)

Γ

𝐺 𝑁 (Δ∗ ; 𝜉, 𝜂)

𝜕Γ

𝐺 𝑁 (Δ∗ ; 𝜉, 𝜂)Δ∗𝜂 𝑈 (𝜂) 𝑑𝜔(𝜂)

𝜕 𝑈 (𝜂) 𝑑𝜎(𝜂), 𝜕𝜈(𝜂)

which yield integral representations for solutions to (DP) and (NP), respectively, under the condition that 𝑈 is of class C (2) (Γ). It remains to construct the auxiliary functions Φ𝐷 and Φ 𝑁 . Some general construction principles on the sphere can be found, e.g., in E. Gutkin, K.P. Newton (2004), R. Kidambi, K.P. Newton (2000). In our approach we focus on spherical caps Γ𝜌 (𝜁). The procedure is similar to the construction of a Dirichlet Green’s function for a disc in R2 . For 𝜉 ∈ Γ𝜌 (𝜁), we need to find a reflection point 𝜉ˇ ∈ (Γ𝜌 (𝜁)) 𝑐 and a scaling factor 𝑟ˇ ∈ R such that
1 − 𝜉 · 𝜂 = 𝑟ˇ 1 − 𝜉ˇ · 𝜂 , 𝜂 ∈ 𝜕Γ𝜌 (𝜁), 𝜉 ∈ Γ𝜌 (𝜁). (13.51) Indeed, under this assumption, we see that Φ𝐷 (𝜉, 𝜂) =

1 1 ln(𝑟ˇ (1 − 𝜉ˇ · 𝜂)) + (1 − ln(2)) 4𝜋 4𝜋

(13.52)

satisfies the desired conditions (13.44) and (13.45). The reflection point 𝜉ˇ can be obtained by a stereographic projection of 𝜉 onto R2 , then applying a Kelvin transform to the projection point, and eventually projecting it back to the sphere (cf. Fig. 13.1 for an illustration). The point 𝜉ˇ represents the spherical Kelvin transformation of 𝜉. The scaling factor 𝑟ˇ is obtained by solving (13.51). Alternatively, the entire Dirichlet Green’s function 𝐺 𝐷 (Δ∗ ; ·, ·) can be obtained from a stereographic projection of the Dirichlet Green’s function for the Laplace operator on a disc in R2 . However, this route would not supply us with a spherical counterpart to the Kelvin transform.

662

13 Potentials on the Sphere

Summarizing our considerations we obtain the following theorem (see also T. Fehlinger (2009), C. Gerhards (2011), W. Freeden, C. Gerhards (2013), C. Gerhards (2018)). Kelvin transformation

𝜁 R2 Γ𝜌 ( 𝜁 )

𝑝stereo ( 𝜉 )

𝑝stereo ( 𝜉ˇ )

𝜉 𝜂

𝜉ˇ

−𝜁

ˇ Fig. 13.1: Schematic description of the construction of the reflection point 𝜉.

Theorem 13.12 Let Γ = Γ𝜌 (𝜁) be a spherical cap with center 𝜁 ∈ Ω and radius 𝜌 ∈ (0, 2). Furthermore, for 𝜉 ∈ Γ𝜌 (𝜁) we set 1 𝑟ˇ − 1 𝜉ˇ = 𝜉 − 𝜁, 𝑟ˇ 𝑟ˇ (𝜌 − 1) 1 + 2𝜉 · 𝜁 (𝜌 − 1) + (𝜌 − 1) 2 𝑟ˇ = − . 𝜌(𝜌 − 2)

(13.53) (13.54)

Then 𝐺 𝐷 (Δ∗ ; 𝜉, 𝜂) =

1 1 ln(1 − 𝜉 · 𝜂) − ln(𝑟ˇ (1 − 𝜉ˇ · 𝜂)), 4𝜋 4𝜋

(13.55)

and a solution 𝑈 ∈ C (2) (Γ) of the Dirichlet problem (DP) can be represented by ∫ 1 𝜉 ·𝜁 +𝜌−1 1 𝑈 (𝜉) = 𝐹 (𝜂) 𝑑𝜎(𝜂), 𝜉 ∈ Γ𝜌 (𝜁). (13.56) √︁ 2𝜋 𝜌(2 − 𝜌) 𝜕Γ𝜌 (𝜁 ) 1 − 𝜉 · 𝜂 Remark 13.13 Applying Theorem 13.12 for 𝜁 = 𝜉 leads to the Mean Value Property II from Theorem 13.2.

13.7 Complete Function Systems

663

A Neumann Green’s function for the Beltrami operator is not obtainable by a simple stereographic projection of the Neumann Green’s function for the Laplace operator on a disc in R2 . However, some computations based on the previously obtained auxiliary function Φ𝐷 lead to the following theorem (see C. Gerhards (2011), W. Freeden, C. Gerhards (2013), C. Gerhards (2018)). Theorem 13.14 Let Γ = Γ𝜌 (𝜁) be a spherical cap with center 𝜁 ∈ Ω and radius 𝜌 ∈ (0, 2). Furthermore, let 𝜉ˇ and 𝑟ˇ be given as in Theorem 13.12. Then, a Neumann Green’s function is given by 𝐺 𝑁 (Δ∗ ; 𝜉, 𝜂) =

1 1 1−𝜌 ln(1 − 𝜉 · 𝜂) + ln(𝑟ˇ (1 − 𝜉ˇ · 𝜂)) + ln(1 + 𝜁 · 𝜂). 4𝜋 4𝜋 2𝜋𝜌 (13.57)

A solution solution 𝑈 ∈ C (2) (Γ) of the Neumann problem (NP) can be represented by ∫ 1 𝑈 (𝜉) = 𝑈 (𝜂) 𝑑𝜔(𝜂) (13.58) 2𝜋𝜌 Γ𝜌 (𝜁 )   ∫ 1 1−𝜌 − ln(1 − 𝜉 · 𝜂) + ln(2 − 𝜌) 𝐹 (𝜂) 𝑑𝜎(𝜂), 𝜉 ∈ Γ𝜌 (𝜁). 2𝜋𝜌 𝜕Γ𝜌 (𝜁 ) 2𝜋

13.7 Complete Function Systems

Keeping the results of the previous sections in mind, we are now able to formulate analogous completeness results for function systems on regular curves 𝜕Γ ⊂ Ω. In fact, we obtain completeness for certain function systems in L2 (𝜕Γ), whose harmonic extensions into Γ ⊂ Ω are particularly well-suited for the approximation of functions that are harmonic with respect to the Beltrami operator. First, we need to explain the notion of a fundamental system: Suppose that {𝜉 𝑘 } 𝑘 ∈N ⊂ Γ is a set of points satisfying dist({𝜉 𝑘 } 𝑘 ∈N , 𝜕Γ) > 0.

(13.59)

If, for any harmonic function 𝐹 in Γ, the condition 𝐹 (𝜉 𝑘 ) = 0, 𝑘 ∈ N, implies that 𝐹 (𝜉) = 0 for all 𝜉 ∈ Γ, then we call {𝜉 𝑘 } 𝑘 ∈N a fundamental system (with respect to Γ).

664

13 Potentials on the Sphere

Assuming that Σ ⊂ Γ is a regular region with dist(Σ, 𝜕Γ) > 0, an example for such a fundamental system is given by a dense point set {𝜉 𝑘 } 𝑘 ∈N ⊂ 𝜕Σ. A particularly simple choice for Σ is a spherical cap within Γ (cf. Fig. 13.2).

· 𝜉𝑘 𝜕Γ

Fig. 13.2: Example for a fundamental system {𝜉 𝑘 } 𝑘 ∈N (with respect to Γ). We begin with the completeness of function systems based on the fundamental solution for the Beltrami operator. Theorem 13.15 Let {𝜉 𝑘 } 𝑘 ∈N be a fundamental system with respect to Γ. Then the following statements hold true: (i) The function system {𝐺 𝑘 } 𝑘 ∈N0 given by 𝐺 𝑘 (𝜉) =

1 ln(1 − 𝜉 𝑘 · 𝜉), 𝑘 ∈ N, 4𝜋

𝐺 0 (𝜉) =

1 , 4𝜋

is complete, and hence closed in L2 (𝜕Γ). (ii) The function system {𝐺˜ 𝑘 } 𝑘 ∈N0 , given by 1 𝜕 𝐺˜ 𝑘 (𝜉) = ln(1 − 𝜉 𝑘 · 𝜉), 𝑘 ∈ N, 4𝜋 𝜕𝜈(𝜉)

1 𝐺˜ 0 (𝜉) = , 4𝜋

is complete, and hence closed in L2 (𝜕Γ). Remark 13.16 Let us assume that {𝜉 𝑘 } 𝑘 ∈N is a fundamental system with respect to Γ 𝑐 . Then the functions 𝐺˜ 𝑘 from Theorem 13.15 are harmonic in Γ and, thus, particularly suitably for the approximation of harmonic functions in Γ. The functions 𝐺 𝑘 from Theorem 13.15 need to be modified since they only satisfy Δ∗ 𝐺 𝑘 (𝜉) = − 41𝜋 , for 𝜉 ∈ Γ and 𝑘 ∈ N. Any auxiliary function 𝐺 of class C (2) (Γ) that satisfies Δ∗ 𝐺 (𝜉) = 41𝜋 , 𝜉 ∈ Γ, can be added to 𝐺 𝑘 without changing the completeness property. In other words, e.g.,

13.7 Complete Function Systems

𝐺 𝑘(mod) (𝜉) = 𝐺 𝑘 (𝜉) −

1 ¯ 𝑘 ∈ N, ln(1 − 𝜉 · 𝜉), 4𝜋

665

𝐺 0(mod) (𝜉) = 𝐺 0 (𝜉),

with a fixed 𝜉¯ ∈ Γ 𝑐 , forms a complete function system in L2 (𝜕Γ) that additionally satisfies Δ∗ 𝐺 𝑘(mod) (𝜉) = 0, 𝜉 ∈ Γ. Next, we want to transfer the results from Theorem 13.15 to inner harmonics for spherical caps (see C. Gerhards (2018)). In order to achieve this, we first need to clarify what we mean by inner harmonics for spherical caps. The sine and cosine functions obviously take the role of spherical harmonics on a circle in R2 . Their harmonic continuations into the disc {𝑥 ∈ R2 | |𝑥| < 𝑅} with radius 𝑅 > 0 and into its exterior {𝑥 ∈ R2 | |𝑥| > 𝑅} (the so-called inner and outer harmonics, respectively) are given by 1  𝑟 𝑛 cos(𝑛𝜑), 𝑛 ∈ N0 , |𝑥| < 𝑅, √ 𝑅 𝜋 𝑅 1  𝑟 𝑛 (int) 𝐻𝑛,2 (𝑅; 𝑥) = √ sin(𝑛𝜑), 𝑛 ∈ N, |𝑥| < 𝑅, 𝑅 𝜋 𝑅  𝑛 1 𝑅 (ext) 𝐻𝑛,1 (𝑅; 𝑥) = √ cos(𝑛𝜑), 𝑛 ∈ N0 , |𝑥| > 𝑅, 𝑅 𝜋 𝑟  𝑛 1 𝑅 (ext) 𝐻𝑛,2 (𝑅; 𝑥) = √ sin(𝑛𝜑), 𝑛 ∈ N, |𝑥| > 𝑅, 𝑅 𝜋 𝑟 (int) 𝐻𝑛,1 (𝑅; 𝑥) =

(13.60) (13.61) (13.62) (13.63)

where 𝑥 = (𝑟 cos(𝜑), 𝑟 sin(𝜑)) 𝑇 , 𝑟 ≥ 0, 𝜑 ∈ [0, 2𝜋). Inner harmonics on a spherical cap Γ𝜌 (𝜁) with radius 𝜌 ∈ (0, 2) and center 𝜁 ∈ Ω can then be obtained by a simple stereographic projection. More precisely,  1  1 𝜌,𝜁 (int) 𝐻𝑛,𝑘 (𝜉) = 𝐻𝑛,𝑘 𝜌 4 (2 − 𝜌) 4 ; 𝑝 stereo (𝜁; 𝜉) , 𝜉 ∈ Γ𝜌 (𝜁), (13.64) denotes an inner harmonic (of degree 𝑛 and order 𝑘) on Γ𝜌 (𝜁). The applied stereographic projection 𝑝 stereo (𝜁; ·) : Ω \ {−𝜁 } → R2 is defined via   2𝜉 · (t𝜀 1 ) 2𝜉 · (t𝜀 2 ) 𝑝 stereo (𝜁; 𝜉) = , , (13.65) 1+𝜉 ·𝜁 1+𝜉 ·𝜁 where 𝜀 1 = (1, 0, 0) 𝑇 , 𝜀 2 = (0, 1, 0) 𝑇 , 𝜀 3 = (0, 0, 1) 𝑇 denotes the canonical basis in R3 and t ∈ R3×3 a rotation matrix with t𝜀 3 = 𝜁. From the harmonicity of 𝜌,𝜁 (int) 𝐻𝑛,𝑘 (𝑅; ·) in {𝑥 ∈ R2 | |𝑥| < 𝑅} it follows that 𝐻𝑛,𝑘 is harmonic in Γ𝜌 (𝜁). Note that, as always, harmonicity in the Euclidean space R2 is meant with respect to the Laplace operator while it is meant with respect to the Beltrami operator when we are intrinsic on the sphere Ω. Opposed to the Euclidean case, outer harmonics for spherical caps do not play a distinct role. Actually, for a spherical cap Γ𝜌 (𝜁), the corresponding
𝑐 outer harmonics coincide with the inner harmonics for the spherical cap Γ𝜌 (𝜁) = Γ2−𝜌 (−𝜁), which is why we do not consider them separately. The

666

13 Potentials on the Sphere

relation ln(1 − 𝜉 · 𝜂) = − ln(2) + ln(1 + 𝜉 · 𝜁) + ln(1 − 𝜂 · 𝜁) −

√︁

𝜌(2 − 𝜌)𝜋

∞ ∑︁ 2 ∑︁ 2

𝑛

𝜌,𝜁

2−𝜌,−𝜁

𝐻𝑛,𝑘 (𝜉)𝐻𝑛,𝑘

(13.66) (𝜂),

𝑛=1 𝑘=1

for 𝜉 ∈ Ω \ {−𝜁 }, 𝜂 ∈ Ω \ {𝜁 }, and | 𝑝 stereo (𝜁; 𝜉)| < | 𝑝 stereo (𝜁; 𝜂)|, eventually allows to transfer the completeness results from Theorem 13.15 to inner harmonics on spherical caps. Theorem 13.17 Let Γ𝜌 (𝜁) be a spherical cap with Γ ⊂ Γ𝜌 (𝜁). Then the following statements hold true:  𝜌,𝜁  𝜌,𝜁 (i) The inner harmonics 𝐻0,1 ∪ 𝐻𝑛,𝑘 𝑛∈N,𝑘=1,2 form a complete, and hence closed function system in L2 (𝜕Γ). (ii) The normal derivatives of the inner harmonics, i.e., 

𝜌,𝜁

𝐻0,1

∪

 𝜕 𝜌,𝜁 𝐻 𝜕𝜈 𝑛,𝑘 𝑛∈N,𝑘=1,2

(13.67)

form a complete and hence closed function system in L2 (𝜕Γ). We conclude this section by stating the use of the function systems from above for the approximation of solutions to the spherical boundary-value problems (DP) and (NP). Theorem 13.18 Let {Φ 𝑘 } 𝑘 ∈N0 denote one of the function systems introduced in Theorem 13.15(b), Remark 13.16, or Theorem 13.17, and 𝑈 ∈ C (2) (Γ) ∩ C (0) (Γ) be a solution of one of the boundary-value problems (DP) or (NP). Then, for every 𝜀 > 0, there exist 𝑀 ∈ N0 and coefficients 𝑎 𝑘 ∈ R, 𝑘 = 0, 1, . . . , 𝑀, such that

𝑀

∑︁

𝑎 𝑘 Φ𝑘 < 𝜀. (13.68)

𝑈 −

2 𝑘=0

L (Γ)

The choice of 𝑀 and the coefficients 𝑎 𝑘 , 𝑘 = 1, . . . , 𝑀, can be based solely on an 𝜕 approximation of 𝑈 or 𝜕𝜈 𝑈 on the boundary 𝜕Γ. Remark 13.19 All closure and approximation results that were obtained in this section in an L2 -context also hold true in a C (0) -context with respect to the uniform topology.

13.8 Bibliographical Notes

667

13.8 Bibliographical Notes

In the geomathematically motivated monograph on potential theory, W. Freeden, C. Gerhards (2013) are concerned with the development of the Runge-Walsh theory, known for regular regions in Euclidean space R3 , for regular regions on the sphere Ω. Instead of the Laplace equation the Beltrami equation is considered, the (Runge-)Bjerhammar sphere is replaced by a circle. The role of the outer spherical harmonics is taken over by the outer circular harmonics. Green’s functions with respect to the Beltrami operator on the sphere serve as monopoles. Analogously to the Euclidean case, limit formulas and jump relations of curve integrals are the main tools. C. Gerhards (2018) deals with boundary-value problems on the sphere, where some prework is available from contibutions given by S. Gemmrich et al. (2008), M.C.A. Kropinski, N. Nigam (2014). The solution of boundary-value problems by explicit representations of associated Green’s functions with respect to boundary circles and the application of the concept of the Runge-Walsh approximation for regular regions are described in C. Gerhards (2018).

Chapter 14

Multiscale Mollifier Potential Systems in Geoapplications

In what follows we discuss first order differential equations concerned with geoscientifically relevant applications, namely gravitational deflections of the vertical and geostrophic ocean flow. These geoapplications are known to be solvable by singular integral equations of the first kind, which – in the approach presented below – lead by regularization to mollifier wavelet systems.

14.1 Surface Mollifier Gradient Fields and Gravitational Deflections of the Vertical

We are interested in the first order differential equation ∇∗ 𝐹 (𝜉) = 𝑣(𝜉),

𝜉 ∈ Ω.

(14.1)

To this end we remember the fundamental theorem for ∇∗ on the unit sphere Ω: Theorem 14.1 Suppose that 𝐹 is of class 𝐶 (1) (Ω). Then, for all 𝜉 ∈ Ω, we have ∫ ∫ 1 𝐹 (𝜉) = 𝐹 (𝜂) 𝑑𝜔(𝜂) − ∇∗𝜂 𝐹 (𝜂) · ∇∗𝜂 𝐺 (Δ∗ ; 𝜉, 𝜂) 𝑑𝜔(𝜂), (14.2) 4𝜋 Ω Ω where 𝐺 (Δ∗ ; 𝜉, 𝜂) =

1 1 1 ln(1 − 𝜉 · 𝜂) + − ln(2), 4𝜋 4𝜋 4𝜋

and ∇∗𝜉 𝐺 (Δ∗ ; 𝜉, 𝜂) = −

1 𝜂 − (𝜉 · 𝜂)𝜉 , 4𝜋 1 − 𝜉 · 𝜂

−1 ≤ 𝜉 · 𝜂 < 1,

−1 ≤ 𝜉 · 𝜂 < 1.

© The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2022 W. Freeden and M. Schreiner, Spherical Functions of Mathematical Geosciences, Geosystems Mathematics, https://doi.org/10.1007/978-3-662-65692-1_14

(14.3)

(14.4) 669

670

14 Multiscale Mollifier Potential Systems in Geoapplications

As a consequence we are are led the following framework to solve the differential equation for ∇∗ on Ω: Corollary 14.2 Let 𝜐 : Ω → R3 be a continuously differentiable vector field on Ω with 𝜉 · 𝜐(𝜉) = 0, 𝐿 ∗𝜉 · 𝜐(𝜉) = 0, 𝜉 ∈ Ω. Then 𝐹 (𝜉) =

1 4𝜋

∫ Ω

1 (𝜉 − (𝜉 · 𝜂)𝜂) · 𝜐(𝜂) 𝑑𝜔(𝜂) 1−𝜉 ·𝜂

(14.5)

is the uniquely determined solution of the differential equation ∇∗ 𝐹 (𝜉) = 𝜐(𝜉),

𝜉 ∈ Ω,

(14.6)

𝐹 (𝜉) 𝑑𝑆(𝜉) = 0.

(14.7)

satisfying 𝐹 ∧ (0, 1) =

1 4𝜋

∫ Ω

∇∗ -Mollifier Wavelets. For real values 𝜌 > 0, we come back to the so-called 𝜌mollifier Green’s kernel function with respect to Δ∗

𝐺 𝜌 (Δ∗ ; 𝜉, 𝜂) =

 1 1   ln(1 − 𝜉 · 𝜂) + (1 − ln(2)), 1 − 𝜉 · 𝜂 > 𝜌,   4𝜋  4𝜋   1−𝜉 ·𝜂 1     4𝜋𝜌 + 4𝜋 (ln(𝜌) − ln(2)), 

(14.8)

1 − 𝜉 · 𝜂 ≤ 𝜌.

The mollifier Green’s kernel functions now enable us to reformulate the regularized fundamental theorem for ∇∗ (see W. Freeden, M. Schreiner (2006b)): Theorem 14.3 Suppose that 𝐹 is a continuously differentiable function on Ω. Then lim sup 𝐹 (𝜉) − 𝑆𝑇 𝜌 (𝜉) − 𝐹 ∧ (0, 1) = 0, (14.9) 𝜌→0 𝜉 ∈Ω

where we have used the abbreviations ∫ 𝑆𝑇 𝜌 (𝜉) = − ∇∗ 𝐹 (𝜂) · 𝑠 𝜌 (𝜉, 𝜂) 𝑑𝜔(𝜂) Ω

and 𝑠 𝜌 (𝜉, 𝜂) = ∇∗𝜂 𝐺 𝜌 (Δ∗ ; 𝜉, 𝜂)

(14.10)

14.1 Surface Mollifier Gradient Fields and Gravitational Deflections of the Vertical

=

1 1    − (𝜉 − (𝜉 · 𝜂)𝜂), 1 − 𝜉 · 𝜂 > 𝜌,   4𝜋 1 − 𝜉·𝜂     1   (𝜉 − (𝜉 · 𝜂)𝜂), −  4𝜋𝜌

671

(14.11)

1 − 𝜉 · 𝜂 ≤ 𝜌.

In what follows, scale discrete scaling vector fields and wavelets will be introduced to establish a numerical solution process: We start with the choice of a sequence that divides the continuous scale interval (0, 2] into discrete pieces. More explicitly, {𝜌 𝑗 } 𝑗 ∈N0 denotes a monotonically decreasing sequence of real numbers satisfying 𝜌0 = 2,

lim 𝜌 𝑗 = 0.

(14.12)

𝑗→∞

For example, one may choose 𝜌 𝑗 = 21− 𝑗 , 𝑗 ∈ N0 . Then we immediately obtain the following result: Corollary 14.4 For 𝐹 ∈ 𝐶 (1) (Ω) lim sup 𝐹 (𝜉) − 𝐹 ∧ (0, 1) − 𝑆𝑇 𝜌 𝑗 (𝜉) = 0,

(14.13)

𝑗→∞ 𝜉 ∈Ω

where

∫ 𝜌𝑗

𝑆𝑇 (𝜉) = −

∇∗ 𝐹 (𝜉) · 𝑠 𝜌 𝑗 (𝜉, 𝜂) 𝑑𝜔(𝜂).

(14.14)

Ω

Our procedure canonically leads to the following type of scale-discretized mollifier wavelets: 𝑤𝜌 𝑗 = 𝑠 𝜌 𝑗+1 − 𝑠 𝜌 𝑗 , (14.15) or, more explicitly,  0,  𝜌 𝑗 < 1 − 𝜉 · 𝜂,       1 1 1 𝑤𝜌 𝑗 (𝜉, 𝜂) = − 4 𝜋  𝜌 𝑗 − 1− 𝜉 ·𝜂 (𝜉 − (𝜂 · 𝜉)𝜂) , 𝜌 𝑗+1 < 1 − 𝜉 · 𝜂 ≤ 𝜌 𝑗 ,     − 41𝜋 𝜌 1𝑗+1 − 𝜌1𝑗 (𝜉 − (𝜂 · 𝜉)𝜂) , −𝜉 · 𝜂 ≤ 𝜌 𝑗+1 . 

(14.16)

Equation (14.15) is called the (scale) discretized mollifier scaling equation. Assume now that 𝐹 is a function of class 𝐶 (1) (Ω). Observing the discretized scaling equation (14.15) we obtain for 𝐽 ∈ N0 𝑆𝑇 𝜌𝐽 = 𝑆𝑇 𝜌0 +

𝐽 −1 ∑︁ 𝑗=0

with

𝑊𝑇 𝜌 𝑗 ,

(14.17)

672

14 Multiscale Mollifier Potential Systems in Geoapplications

∫

∇∗ 𝐹 (𝜂) · 𝑤𝜌 𝑗 (𝜉, 𝜂) 𝑑𝜔(𝜂),

𝑊𝑇 𝜌 𝑗 (𝜉) =

𝑗 = 0, . . . , 𝐽 − 1.

(14.18)

Ω

Therefore, we are able to formulate the following result: Corollary 14.5 Let {𝑠 𝜌 𝑗 } 𝑗 ∈N0 be a (scale) discretized scaling vector field. Then the multiscale representation of a function 𝐹 ∈ 𝐶 (1) (Ω) is given by ∞ 3 ∑︁ ∑︁ 1 © ª sup 𝐹 (𝜉) − ­ 𝑊𝑇 𝜌 𝑗 (𝜉) + 𝐹 ∧ (0, 1) + 𝐹 ∧ (1, 𝑚)𝑌1,𝑚 (𝜉) ® = 0. (14.19) 3 𝑚=1 𝜉 ∈Ω « 𝑗=0 ¬ The aforementioned result admits the following reformulation: 𝑆𝑇 𝜌𝐽 +

∞ ∑︁

𝑊𝑇 𝜌 𝑗 = 𝐹 − 𝐹 ∧ (0, 1) −

𝑗=𝐽

3 1 ∑︁ ∧ 𝐹 (1, 𝑚)𝑌1,𝑚 3 𝑚=1

(14.20)

for every 𝐽 ∈ N0 (in the sense of ∥ · ∥ 𝐶 (Ω) ). Again, the integrals 𝑆𝑇 𝜌 𝑗 , 𝑊𝑇 𝜌 𝑗 may be understood as lowpass and bandpass filters, respectively. Obviously,

𝑆𝑇 𝜌 𝑗+1 = 𝑆𝑇 𝜌 𝑗 + 𝑊𝑇 𝜌 𝑗 .

(14.21)

Equation (14.21) may be interpreted in the following way: The ( 𝑗 + 1)-scale lowpass filtered version of 𝐹 is obtained by the 𝑗-scale lowpass filtered version of 𝐹 added by the 𝑗-scale bandpass filtered version of 𝐹. Deflections of the Vertical. The Earth’s gravity potential 𝑊 = 𝑈 + 𝑇 is typically split into a normal gravity potential 𝑈 corresponding to a reference ellipsoid E (i.e., 𝑈 (𝑥) = const. for 𝑥 ∈ E) and a remaining disturbing potential 𝑇. The vertical deflection Θ(𝑥) measures the angular distance between the normal vector 𝜈 G (𝑥) at a point 𝑥 on the geoid G (i.e., 𝑊 (𝑥) = const. for 𝑥 ∈ G) and the corresponding ellipsoidal normal vector 𝜈 E (𝑥) with respect to E. Assuming that 𝜈 G − 𝜈 E and 𝜈 E are nearly orthogonal and that the deviation of the reference ellipsoid from a sphere is negligible, one can derive the following relation between the disturbing potential and the deflections of the vertical (cf. F.A. Vening Meinesz (1928)): ∇∗𝑇 (𝑅𝜉) = −

𝐺𝑀 Θ(𝑅𝜉), 𝑅

𝜉 ∈ Ω,

(14.22)

where 𝑅 is the Earth’s mean radius, 𝐺 the gravitational constant, and 𝑀 the Earth’s mass. For more details, the reader is referred to, e.g., W.A. Heiskanen, H. Moritz (1967), E. Groten (1979), B. Hofmann-Wellenhof, H. Moritz (2005), W. Freeden, M. Schreiner (2009).

14.1 Surface Mollifier Gradient Fields and Gravitational Deflections of the Vertical

673

As in classical Physical Geodesy of gravimetric determination of the disturbing potential and the geoid (see e.g., W.A. Heiskanen, H. Moritz (1967)) we assume that the zero-degree terms of the disturbing potential vanish, i.e., the difference of the mass of the Earth and the mass of the reference ellipsoid is supposed to be zero. Moreover, we assume that the center of the reference ellipsoid coincides with the center of gravity of the Earth so that the first-degree term is zero. In other words, we have ∫ 𝑇 (𝑅𝜉)𝑌𝑛,𝑚 (𝜉) 𝑑𝜔(𝜉) = 0, 𝑛 = 0, 1; 𝑚 = 1, . . . , 2𝑛 + 1. (14.23) Ω

Therefore, deflections of the vertical and disturbing potential are related by the following integral equation: Let Θ(𝑅·) : 𝜉 ↦→ Θ(𝑅𝜉), 𝜉 ∈ Ω be a continuous field on Ω. Then the geoidal undulations can be determined by the formula ∫ 𝐺𝑀 1 (𝜉 − (𝜉 · 𝜂)𝜂) · Θ(𝑅𝜉) 𝑑𝜔(𝜂). (14.24) 𝑇 (𝑅𝜉) = − 4𝜋𝑅 Ω 1 − 𝜉 · 𝜂 According to our approach the integral on the right-hand side of (14.24) can be written approximately by replacing the improper integral by proper scale-integrals involving the “mollifier Green’s function kernel” with respect to the Beltrami operator. In our notation, for 𝜉 ∈ Ω and 𝐽 ∈ N0 , we obtain

∫

𝐺𝑀 𝑇 (𝑅𝜉) = − 4𝜋𝑅 −

Θ(𝑅𝜂) · 𝑠 𝜌𝐽 (𝜉, 𝜂) 𝑑𝜔(𝜂)

𝐺𝑀 4𝜋𝑅

Ω ∞ ∫ ∑︁ 𝑗=𝐽

(14.25)

Θ(𝑅𝜂) · 𝑤𝜌 𝑗 (𝜉, 𝜂) 𝑑𝜔(𝜂).

Ω

Consequently we find 𝐺𝑀 𝑇 (𝑅𝜉) = − 4𝜋𝑅 −

𝐺𝑀 4𝜋𝑅

∫ Θ(𝑅𝜂) · 𝑠 𝜌𝐽 (𝜉, 𝜂) 𝑑𝜔(𝜂) Ω ∞ ∑︁ 𝑗=𝐽

∫ 1− 𝜉 · 𝜂 ≤𝜌 𝑗 𝜂 ∈Ω

Θ(𝑅𝜂) · 𝑤𝜌 𝑗 (𝜉, 𝜂) 𝑑𝜔(𝜂).

(14.26)

674

14 Multiscale Mollifier Potential Systems in Geoapplications

In other words, finer and finer detail information about the disturbing potential is obtained by wavelets with smaller and smaller local support. Gravitational Signatures of Mantle Plumes. Nowadays, the concept of mantle plumes is widely accepted in the geoscientific community. Mantle plumes are understood to be approximately cylindrical concentrated upflows of hot mantle material with a common diameter of about 100 km – 200 km. They are an upwelling of abnormally hot rock within the Earth’s mantle. As the heads of mantle plumes can partly melt when they reach shallow depths, they are thought to be the cause of volcanic centers known as hotspots. Hotspots have first been explained by J. Wilson (1963) as long-term sources of volcanism that are fixed relative to the tectonic plate overriding them. Following W.J. Morgan (1971), characteristic surface signatures of hotspots are due to the rise and melting of hot plumes from deep areas in the mantle. Special cases occur as chains of volcanic edifices whose age progresses with increasing distance to the plume, like the islands of Hawaii. They are the result of a pressure-release melting nearby the bottom of the lithosphere that produces magma rising to the surface and by plate motion relative to the plume. The term “hotspot” is used rather loosely. It is often applied to any long-lived volcanic center that is not part of the global network of mid-ocean ridges and island arcs, like Hawaii which deserves as a classical example. Anomalous regions of thick crust on ocean ridges are also considered to be hotspots, like Iceland.

Fig. 14.1: Illustration of the deflections of the vertical in the region of Hawaii in rad.

14.1 Surface Mollifier Gradient Fields and Gravitational Deflections of the Vertical

675

Hawaii: J.R.R. Ritter, U.R. Christensen (2007) believe that a stationary mantle plume located beneath the Hawaiian Islands created the Hawaii-Emperor seamount chain while the oceanic lithospere continuously passed over it. The Hawaii-Emperor chain consists of about 100 volcanic islands, atolls, and seamounts that spread nearly 6000km from the active volcanic island of Hawaii to the 75-80 Ma old Emperor seamounts nearby the Aleutian trench. With moving further south east along the island chain, the geological age decreases. The interesting area is the relatively young southeastern part of the chain, situated on the Hawaiian swell, a 1200 km broad anomalously shallow region of the oceanfloor, extending from the island of Hawaii to the Midway atoll. Here, a distinct gravity disturbance and geoid anomaly occur that has its maximum around the youngest island that coincides with the maximum topography and both decrease in northwest direction. The progressive decrease in terms of the geological age is believed to result from the continuous motion of the underlaying plate (cf. W.J. Morgan (1971), J. Wilson (1963)).

Fig. 14.2: The plume beneath Hawaii reconstructed from seismic tomography (in an illustration from T. Fehlinger (2009) following J.R.R. Ritter, U.R. Christensen (2007)).

With seismic tomography, several features of the Hawaiian mantle plume are gained (cf. J.R.R. Ritter, U.R. Christensen (2007) and the references specified therein). They result in a Low Velocity Zone (LVZ) beneath the lithosphere, starting at a depth of about 130 − 140km, beneath the central part of the island of Hawaii (and leading to

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14 Multiscale Mollifier Potential Systems in Geoapplications

the illustration in Fig. 14.2). So far plumes have just been identified as low seismic velocity anomalies in the upper mantle and the transition zone, by seismic wave tomography, which is a fairly new achievement. Because plumes are relatively thin according to their diameter, they are hard to detect in global tomography models. Hence, despite novel advantages, there is still no general agreement on the fundamental questions concerning mantle plumes, like their depth of origin, their morphology, their longevity and even their existence is still discussed controversial. This is due to the fact that many geophysical as well as geochemical observations can be explained by different plume models and even by models that do not include plumes at all (e.g., G. Foulger et al. (2005)). With our space-localized multiscale method of deriving gravitational signatures, more concretely the disturbing potential, from the deflections of the vertical, we add a new component in specifying essential features of plumes. The deflections of the vertical of the plume in the region of Hawaii are visualized in Fig. 14.1. From the bandpass filtered detail approximation we are able to conclude that the Hawaii plume has an oblique layer structure. As can be seen in the lower scale, which reflects the higher depths, the strongest signal is located in the ocean in a westward direction of Hawaii. With increasing scale, i.e., lower depths it moves more and more to the Big Island of Hawaii, i.e., in eastward direction.

14.1 Surface Mollifier Gradient Fields and Gravitational Deflections of the Vertical

677

Fig. 14.3: Bandpass filtered details of the disturbing potential in m2 s−2 in the region of Hawaii using the discretized space mollifier Green’s function with respect to the Beltrami operator for 𝑗 = 9, 11, 13 (left), 10, 12, 14 (right).

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14 Multiscale Mollifier Potential Systems in Geoapplications

Fig. 14.4: Illustration of the deflections of the vertical in the region of Iceland in rad.

Iceland: The plume beneath Iceland is a typical example of a ridge-centered mantle plume. An interaction between the North Atlantic ridge and the mantle plume is believed to be the reason for the existence of Iceland, resulting in melt production and crust generation since the continental break-up in the late Palaeocene and early Eocene. Nevertheless, there is still no agreement on the location of the plume before rifting started in the East. Controversial discussions, whether it was located under central or eastern Greenland about 62 Ma – 64 Ma ago are still in progress (cf. D. Schubert et al. (2001) and the references therein). Iceland itself represents the top of a nearly circular rise topography, with the maximum of about 2.8km above the surrounding seafloor in the south of the glacier “Vatnaj¨okull”. Beneath this glacier several active volcanoes are located, which are supposed to be feeded by a mantle plume. The surrounding oceanic crust consists of three different types involving a crust thickness that is more than three times as thick as average oceanic crusts. Seismic tomography provides evidence of the existence of a mantle plume beneath Iceland, resulting in low velocity zones in the upper mantle and the transition zone, but also hints for anomalies in the deeper mantle seem to exist. The low velocity anomalies have been detected in depths ranging from at least 400km up to about 150km. Above 150 km ambiguous seismic-velocity structures were obtained involving regions of low-velocities covered by regions of high-seismic-velocities.

14.1 Surface Mollifier Gradient Fields and Gravitational Deflections of the Vertical

679

For a deeper access into the theory of the Iceland plume the interested reader may be referred to J.R.R. Ritter, U.R. Christensen (2007) and the references therein.

Fig. 14.5: Bandpass filtered details of the disturbing potential in m2 s−2 in the region of Iceland using the discretized space mollifier Green’s function with respect to the Beltrami operator for 𝑗 = 10, 12, 14 (left) and 𝑗 = 11, 13, 15 (right).

From Fig. 14.5 it can be seen that the mantle plume in lower scales, i.e., in higher depths starts in the North of Iceland and with increasing scale, i.e., lower depths,

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14 Multiscale Mollifier Potential Systems in Geoapplications

it moves to the South. It is remarkable that from scale 13 on, the plume seems to divide into two sectors. Since it is known that the disturbing potential of the Earth is influenced by its topography, a view to a topographic map shows that the sector located more Eastern is (probably) caused by the Vatnaj¨okull glacier (being the biggest glacier in Europe). Altogether, our multi-scale method offers an additional component in characterizing mantle plumes with respect to their position and horizontal size. A serious description of their depth, however, cannot be realized. The detailed detection of the depths remains a great challenge for future work (for more details the reader is referred to the case studies due to T. Fehlinger (2009)), in which also gravimetric investigations should be taken into account. It seems that only a combined investigation of all available techniques is well-promising.

14.2 Surface Mollifier Curl Gradient Fields and Geostrophic Ocean Flow

Next we are interested in the first order differential equation 𝐿 ∗ 𝐹 (𝜉) = 𝑣(𝜉),

𝜉 ∈ Ω.

(14.27)

To this end we remember the fundamental theorem for L∗ on the unit sphere Ω (see W. Freeden, M. Schreiner (2009)): Theorem 14.6 (Differential Equation for 𝐿 ∗ on Ω) Let 𝑣 : Ω → R3 be a continuously differentiable vector field on Ω with 𝜉 · 𝑣(𝜉) = 0, ∇∗𝜉 · 𝑣(𝜉) = 0, 𝜉 ∈ Ω. Then 𝐹 (𝜉) =

1 4𝜋

∫ Ω

1 (𝜂 ∧ 𝜉) · 𝑣(𝜂) 𝑑𝜔(𝜂) 1−𝜉 ·𝜂

(14.28)

is the uniquely determined solution of the differential equation 𝐿 ∗ 𝐹 (𝜉) = 𝑣(𝜉),

𝜉 ∈ Ω,

(14.29)

𝐹 (𝜉) 𝑑𝜔(𝜂) = 0.

(14.30)

satisfying 1 4𝜋

∫ Ω

𝐿 ∗ -Mollifier Wavelets. The mollifier Green’s kernel functions now enable us to reformulate the regularized fundamental theorem for 𝐿 ∗ :

14.2 Surface Mollifier Curl Gradient Fields and Geostrophic Ocean Flow

681

Theorem 14.7 Suppose that 𝐹 is a continuously differentiable function on Ω. Then lim sup 𝐹 (𝜉) − 𝑆𝑇 𝜌 (𝜉) − 𝐹 ∧ (0, 1) = 0, (14.31) 𝜌→0 𝜉 ∈Ω

where we have used the abbreviations ∫ 𝜌 𝑆𝑇 (𝜉) = − 𝐿 ∗ 𝐹 (𝜂) · 𝑠 𝜌 (𝜉, 𝜂) 𝑑𝜔(𝜂)

(14.32)

Ω

and 𝑠 𝜌 (𝜉, 𝜂) = 𝐿 ∗𝜂 𝐺 𝜌 (Δ∗ ; 𝜉, 𝜂) 1 1    − (𝜂 ∧ 𝜉), 1 − 𝜉 · 𝜂 > 𝜌,   4𝜋 1 − 𝜉 · 𝜂   =   1   (𝜂 ∧ 𝜉), 1 − 𝜉 · 𝜂 ≤ 𝜌. − 4𝜋𝜌 

(14.33)

Again, scale discrete scaling vector fields and wavelets can be introduced to establish a numerical solution process: We start with the choice of a sequence that divides the continuous scale interval (0, 2] into discrete pieces. More explicitly, {𝜌 𝑗 } 𝑗 ∈N0 denotes a monotonically decreasing sequence of real numbers satisfying 𝜌0 = 2,

lim 𝜌 𝑗 = 0.

(14.34)

𝑗→∞

For example, one may choose 𝜌 𝑗 = 21− 𝑗 , 𝑗 ∈ N0 . Our procedure leads to the following type of scale-discretized mollifier wavelets: 𝑤𝜌 𝑗 = 𝑠 𝜌 𝑗+1 − 𝑠 𝜌 𝑗 ,

(14.35)

or, more explicitly,  0,  𝜌 𝑗 < 1 − 𝜉 · 𝜂,       1 1 1 𝑤𝜌 𝑗 (𝜉, 𝜂) = − 4 𝜋  𝜌 𝑗 − 1− 𝜉 ·𝜂 (𝜂 ∧ 𝜉) , 𝜌 𝑗+1 < 1 − 𝜉 · 𝜂 ≤ 𝜌 𝑗 ,     − 41𝜋 𝜌 1𝑗+1 − 𝜌1𝑗 (𝜂 ∧ 𝜉) , −𝜉 · 𝜂 ≤ 𝜌 𝑗+1 .  Equation (14.35) is called the (scale) discretized mollifier scaling equation.

(14.36)

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14 Multiscale Mollifier Potential Systems in Geoapplications

+

+ lowpass filtering (scale 𝑗 = 1)

+

bandpass filtering (scale 𝑗 = 1)

(Frd, D.Michel,V. Michel ( 2005))

+ bandpass filtering (scale 𝑗 = 3)

bandpass filtering (scale 𝑗 = 2)

+

+ bandpass filtering (scale 𝑗 = 5)

bandpass filtering (scale 𝑗 = 4)

+

= bandpass filtering (scale 𝑗 = 6)

lowpass filtering (scale 𝑗 = 7)

(Frd, D.Michel,V. Michel ( 2005))

Fig. 14.6: Multiscale approximation of the ocean topography in cm (a rough lowpass filtering at scale 𝑗 = 1 is improved with several bandpass filters of scale 𝑗 = 1, ..., 6, where the last picture shows the multiscale approximation at scale 𝑗 = 7).

14.2 Surface Mollifier Curl Gradient Fields and Geostrophic Ocean Flow

Fig. 14.7: Ocean topography in cm of the gulf stream.

683

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14 Multiscale Mollifier Potential Systems in Geoapplications

Fig. 14.8: Geostrophic oceanic flow in cm s−1 of the gulf stream.

Ocean flow has a great influence on mass transport and heat exchange. By modeling oceanic currents we therefore gain, for instance, a better understanding of weather and climate. In what follows we devote our attention to the geostrophic oceanic circulation. In subregions Γ𝑅 ⊂ Ω𝑅 of the ocean with a sufficiently large horizontal extent, away from the top and bottom Ekman layers and coastal regions, the geostrophic balance holds true: the horizontal pressure gradients in the ocean balance the Coriolis force resulting from horizontal currents. The Coriolis force term in a point 𝑥 ∈ Γ𝑅 is given as the tangential contribution of −2𝑅𝜌 𝑤 ∧ 𝑣(𝑥), where 𝑣(𝑥) is the horizontal ocean flow velocity and 𝑤 = |𝑤|𝜀 3 the Earth’s rotation vector. 𝜌 denotes the density and is assumed to be constant. The pressure 𝑃(𝑥) in 𝑥 ∈ Γ𝑅 can be regarded as being proportional to the mean dynamic topography (MDT) 𝐻 (𝑥), which denotes the height of the sea surface relative to the geoid G and can be determined from altimetry measurements. More precisely, 𝑃(𝑥) = 𝜌𝐺𝐻 (𝑥), where 𝐺 denotes the gravitational constant. Using the geostrophic balance, we therefore obtain −2𝑅𝜌(𝑤 · 𝜉) 𝜉 ∧ 𝑣(𝑅𝜉) = 𝜌𝐺∇∗ 𝐻 (𝑅𝜉), or, equivalently,

𝜉 ∈ Γ,

(14.37)

14.2 Surface Mollifier Curl Gradient Fields and Geostrophic Ocean Flow

2𝑅 |𝑤|(𝜉 · 𝜀 3 )𝑣(𝑅𝜉) = 𝐿 ∗ 𝐻 (𝑅𝜉), 𝐺

𝜉 ∈ Γ.

685

(14.38)

For more details on the geophysical background, the reader is referred, e.g., to J. Pedlovsky (1979).

The data points used for our demonstration are extracted from the French CLS01 model (in combination with the EGM96 model). Clearly, ocean currents are subject to different influence factors, such as wind field, warming of the atmosphere, salinity of the water, etc, which are not accounted for in our modeling. Our approximation (see Figs. 14.6, 14.7, 14.8) must be understood in the sense of a geostrophic balance as proposed earlier. An analysis shows that its validity may be considered as given on spatial scales of an approximate expansion of a little more than 30km, and on time scales longer than approximately one week. Indeed, the geostrophic velocity field is perpendicular to the tangential gradient of the ocean topography (i.e., perpendicular to the tangential pressure gradient). This is a remarkable property. The water flows along curves of constant ocean topography (see Fig. 14.6 for a multiscale modeling). Despite the essentially restricting assumptions necessary for geostrophic modeling, we obtain instructive circulation models of the ocean surface current for the northern or southern hemisphere, respectively (however, difficulties for the computation of the flow arise from the fact that the Coriolis parameter vanishes on the equator). Point Vortex Motion. Vorticity describes the rotational motion of a fluid. In the ocean, for horizontal flows 𝑣 which extend over regions Γ𝑅 ⊂ Ω𝑅 at spatial scales of several tens or hundreds of kilometers, the following relation for the vorticity 𝜔 holds true: 𝜔(𝑅𝜉) = 𝐿 ∗ · 𝑣(𝑅𝜉),

𝜉 ∈ Γ.

(14.39)

The incompressible horizontal flow 𝑣 itself can be represented by a stream function Ψ via 𝑣 = 𝐿 ∗ Ψ, so that we obtain 𝜔(𝑅𝜉) = Δ∗ Ψ(𝑅𝜉),

𝜉 ∈ Γ.

(14.40)

The geostrophic flow is an example for such a current. For more geophysical background on vorticity, the reader is again referred to J. Pedlovsky (1979). Further multiscale developments and graphical illustrations can be found in C. Gerhards (2018).

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14 Multiscale Mollifier Potential Systems in Geoapplications

14.3 Bibliographical Notes

First it should be mentioned that our background of Earth’s gravitational field determination is taken from standard monographs (e.g., M. Hotine (1969), E. Groten (1979), W.A. Heiskanen, H. Moritz (1967), B. Hofmann-Wellenhof, H. Moritz (2005), W. Torge, J. M¨uller (2012)). Recent works concerned with Mathematical Geodesy are W. Freeden, M.Z. Nashed (2018a) and W. Freeden, R. Rummel (2020). An overview about Mathematical Geodesy is given by W. Freeden, M. Schreiner (2020a) (see also W. Freeden et al. (2018b), W. Freeden, F. Sans`o (2018), W. Freeden, B. Witte (2020)). The particular role of gravitation as a key observable for mass distribution even from space was already pointed out in ESA-reports of the last century (cf. ESA (1996), ESA (1998), ESA (1999), K.H. Ilk et al. (2004)). The locally oriented “zooming-in” techniques for determining disturbance potential, gravity anomalies, deflections of the vertical etc. are based on different approximation methods leading to a palette of innovative spherical base systems. Most of the aspects in constructive approximation are collected in W. Freeden et al. (1998), W. Freeden et al. (2018a), W. Freeden, M. Schreiner (2020b) (see also the references therein). Mollifier regularizations of fundamental solutions and the Green’s functions and its multiscale application to problems of modeling Earth’s graviational field and/or (geostrophic) ocean circulation are given in W. Freeden, M. Schreiner (2006b), W. Freeden, C. Mayer (2003), T. Fehlinger et al. (2007), T. Fehlinger et al. (2008), W. Freeden, K. Wolf (2008), T. Fehlinger et al. (2009), T. Fehlinger (2009),K. Wolf (2009), W. Freeden, M. Schreiner (2009), C. Gerhards (2011), W. Freeden, C. Gerhards (2013), W. Freeden, C. Blick (2013), C. Blick (2015), C. Blick et al. (2018), W. Freeden, M.Z. Nashed (2018b), W. Freeden (2021), C. Blick et al. (2021)).

Part VI

Conclusion

Chapter 15

Concluding Remarks

Today’s geosciences profit so much from the possibilities that result from highly advanced electronic measurement concepts, modern computer technology and, most of all, artificial satellites. In fact, the exceptional situation of getting simultaneous and complementary observations from a multiple of low-orbiting satellites opens new opportunities to contribute significantly to the understanding of our planet, its climate, its environment and about an expected shortage of natural resources. All of a sudden, key parameters for the study of the dynamics of our planet and the interaction of its solid part with ice, oceans, atmosphere etc. become accessible. In this context, new types of vector and tensor data measured on (almost) spherical reference surfaces such as the (spherical) Earth or (near-)circular orbits are very likely the greatest challenge to model temporal trends and changes. Consistent data structures help geoscientists to determine the Earth’s gravitational field from spaceborne gravity sensors, the oceanographers to see the oceans flow, people from geomagnetics to get insight in the spatio-time variation of the magnetic field, solid Earth physicists to better understand the dynamics of the Earth’s interior, meteorologists to simulate wind fields, etc. Clearly, (non-standard) observations and data having a different type, location, and distribution cannot be handled by traditional modeling and simulation techniques. This is the reason why adequate components of mathematical thinking, adapted formulations of theories and models, and economical and efficient numerical developments are indispensable. Up to now, the modeling of vector and tensor data is done on global scale by orthogonal expansions by means of polynomial structures such as (certain types of) vector and tensor spherical harmonics. Thus far, they can not keep pace with the prospects and the expectations of the “Earth system sciences”. Moreover, there is an increasing need for high-precision modeling on local areas. In this respect, zonal kernel functions, i.e., in the jargon of constructive approximation, radial basis functions, become more and more important because of their space localizing properties even in the vectorial and tensorial context. The current book shows that the addition theorem of the theory of spherical harmonics enables us to express all types of zonal kernel functions in terms of a one-dimensional function, viz. the Legendre polynomial. In other words, additive clustering of spherical harmonics © The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2022 W. Freeden and M. Schreiner, Spherical Functions of Mathematical Geosciences, Geosystems Mathematics, https://doi.org/10.1007/978-3-662-65692-1_15

689

690

15. Concluding Remarks

generates specific classes of space localizing zonal kernel functions, i.e., Legendre series expansions, ready for approximation within scalar, vectorial, and tensorial framework. Furthermore, our investigations demonstrate that the closer the Legendre series expansion is to the Dirac kernel, the more localized is the zonal kernel in space, and the more economical is its role in (spatial) local computation. In addition, the Funke-Hecke formula provides the natural tool for establishing convolutions of spherical fields against zonal kernels. As a consequence, by specifying scaling functions, i.e., sequences of zonal functions tending to the Dirac kernel, (spacelocalized) filtered versions of (square-integrable) spherical fields are obtainable by convolution leading to a “zooming-in” approximation within a multiscale procedure. Altogether, the vectorial and tensorial counterparts of the Legendre polynomial are the essential keystones in our work. They enable the transition from spherical harmonics via zonal kernels up to the Dirac kernel. In addition, the Funk-Hecke formula and its consequences in spherical convolutions opens new methological perspectives for global as well as local approximation in vectorial and tensorial physically motivated application (for specific purposes of constructive approximation see also W. Freeden et al. (2018a), W. Freeden, M. Schreiner (2020b)). Approximation by Spherical Harmonics. The context of spherical harmonics is a well-established tool, particularly for access to the inversion of problems under the assumption of a bandlimited Earth’s gravitational and/or magnetic model. Nowadays, spectral reference models, i.e., Fourier expansions in terms of spherical harmonics for the Earth’s gravitational and magnetic potential are widely known by tables of expansion coefficients as frequency determined constituents. However, the relative advantages of the classical Fourier (orthogonal) expansion method by means of spherical harmonics have to be realized not only in the frequency domain, but also in the space domain. It is characteristic for Fourier techniques that the spherical harmonics as polynomial trial functions admit no localization in space domain, while in the frequency domain, they always correspond to exactly one degree, i.e., frequency. Because of the ideal frequency localization and the simultaneous absence of space localization due to the uncertainty principle local changes of fields (signatures) in the space domain affect the whole table of orthogonal (Fourier) coefficients. This, in turn, causes global changes of the corresponding (truncated) Fourier series in the space domain. Nevertheless, frequency based approximation is often helpful for meaningful physical interpretations, for example, by relating the different observables of a geopotential to each other at a fixed frequency, e.g., the Meissl scheme in Physical Geodesy. In conclusion, Fourier expansion methods are well suited to resolve low and medium frequency phenomena, i.e., the “trends” of a signal, while their application to obtain high resolution in global or local models is critical. Approximation by Zonal Functions. Geoscientific modeling demands its own nature. Concerning modeling one is usually not interested in the separation into scalar Cartesian component functions involving product ingredients. Instead, inherent physical properties should be observed. For example, the deflections of the vertical form a vector-isotropic surface gradient field on the Earth’s surface, the equations for

15. Concluding Remarks

691

(geostrophic) ocean (surface) flow involving geoidal undulations (heights) imply a divergence-free vector-isotropic nature, satellite gradiometer data lead to tensorisotropic Hesse fields. As a consequence, in a geoscientifically reflected spherical framework, all these physical constraints result in a formulation by zonal functions and rotation-invariant pseudodifferential operators. In fact, rotational symmetry (isotropy) is an indispensable ingredient in the bridging transformation, that relates geoscientific quantities, i.e., the object parameters, to the observed and/or measured data sets, and vice versa. Zonal kernel function theory relies on the following principles: (i) Weighted Legendre kernels are the summands of zonal kernel functions. (ii) The Legendre kernel is ideally localized in frequency. The Dirac kernel is ideally localized in space. (iii) The only frequency- and spacelimited zonal kernel is the zero function. Zonal kernels exist as bandlimited and non-bandlimited functions. Every bandlimited kernel refers to a cluster of a finite number of polynomials i.e., spherical harmonics, hence, it corresponds to a certain band of frequencies. In contrast to a single polynomial which is localized in frequency but not in space, a bandlimited kernel such as the Shannon kernel already shows a certain amount of space localization. If we move from bandlimited to non-bandlimited kernels the frequency localization decreases and the space localization increases in accordance with the relationship provided by the uncertainty principle. Zonal kernel function approximation exists in spline and wavelets specifications naturally based on certain realizations in frequency and space localization. Obviously, if a certain accuracy should be guaranteed in kernel function approximation, adaptive sample point grids are required for the resulting extension of the spatial area determined by the kernels under investigation. If data sets on the sphere are localized in size, typically by spherical rectangles, approximation problems can be attacked by application of tensor product techniques in terms of polar coordinates originally designed for bivariate Euclidean space nomenclature. However, global problems like the determination of the gravitational field, magnetic field, tectonic movements, ocean circulation, climate change, hydrological and meteorological purposes, etc. involve essentially the entire surface of the sphere, so that modeling the data as arising in Euclidean two-space via latitude-longitude coordinate separation is no longer appropriate. Even more, since there is no differentiable mapping of the entire sphere to a bounded planar region, there is a need to develop approximations such as sampling methods over the sphere itself, thereby avoiding (artificially occurring) singularities. Looking at a numerically efficient and stable global model in today’s literature, a spherical (and usually not a physically more suitable but numerically harder to handle ellipsoidal) reference shape of the Earth has been taken into account in almost all practical approximations because of their

692

15. Concluding Remarks

conceptual simplicity and numerical computability (see W. Freeden et al. (2018a) for more details). Commonly, zonal functions on the sphere recognized as positive definite kernels may be interpreted as generating reproducing kernels of Sobolev spaces. This is the reason why spherical splines may be based on a variational approach (cf. A.Y. Bezhaev, V.A. Vasilenko (1993), W. Freeden (1981a), W. Freeden (1981b), W. Freeden (1990), W. Freeden et al. (1998), W. Freeden (2021), M.M.J.-A. Simeoni (2020), G. Wahba (1981), G. Wahba (1990)) that minimizes a weighted Sobolev norm of the interpolant, with a large class of spline manifestations provided by pseudodifferential operators being at the disposal of the user.

Approximation by Splines. Sobolev space framework involving rotation-invariant pseudodifferential operators (as originated by observables, e.g., in Physical Geodesy) shows some important benefits of spline interpolation as preparatory tool for spherical sampling. Accordingly we are confronted with the following situation:

(i) In analogy to the classical 1D-approach (see, e.g., T.N.E. Greville (1969)), interpolating/smoothing splines are canonically based on a variational approach that minimizes a weighted Sobolev norm of the interpolating/ smoothing spline, with a large class of spline manifestations provided by pseudodifferential operators being at the disposal of the user. Regularly distributed as well as scattered data systems can be handled. (ii) Artificial singularities caused by the use of (polar) coordinates in global approximation can be avoided totally. (iii) The rotational invariance of observables (such as gravity anomalies, gravity disturbances, and disturbing potentials in geodetic theory) is perfectly maintained. (iv) Measurement errors can be handled by an adapted interpolation/smoothing spline method. Error bounds can be derived that include computable constants rather than only being given in terms of the order of convergence of the maximum distance from any point in the domain to the nearest data point. (v) Spline spaces serve as canonical reference spaces for purposes of spherical sampling relative to finite as well as infinite scattered data distributions. (vi) Spherical splines provide approximations using kernel functions with a fixed “window”, i.e., preassigned frequency and space relation.

15. Concluding Remarks

693

(vii) The accuracy of spline approximation can be controlled by a decreasing sampling width. (viii) Global spherical spline interpolation in terms of zonal, i.e., radial basis functions, has its roots in physically motivated problems of minimizing a “(linearized) curvature energy” variational problem consistent to data points. Numerical experiences with the linear system of equations have shown that the system tends to be ill-conditioned unless the number of data points is not too large. Clearly, oscillation phenomena in spline interpolation may occur for larger data gaps. Spherical splines in their manifestations (interpolation, smoothing, best approximation) were independently developed in W. Freeden (1978), W. Freeden (1981c), and G. Wahba (1981). A spline survey on the unit sphere Ω including aspects in spherical sampling is given, e.g., in W. Freeden (1999), W. Freeden (2015), W. Freeden, C. Gerhards (2013), W. Freeden, M. Gutting (2013), W. Freeden, M. Schreiner (2009), W. Freeden et al. (1998), F.J. Narcowich, J.D. Ward (2002), F. Narcowich et al. (2006), G. Wahba (1990) (see also the references therein). For a certain class of exponential kernels the numerical computation may be organized very efficiently and economically by fast multipole methods in combination with near/far field procedures (for more details the reader is referred to M. Gutting (2008, 2012, 2014, 2018)). For generalizations to non-spherical manifolds and the use in boundary-value problems of elliptic partial differential equations the reader is, e.g., referred to M.J. Fengler et al. (2006b), W. Freeden (1981b), W. Freeden (1982a), W. Freeden (1982b), W. Freeden (1987a), W. Freeden (1987b), W. Freeden (1990), W. Freeden (1999), W. Freeden, M. Gutting (2013), W. Freeden, V. Michel (2004), W. Freeden, V. Michel (2005), W. Freeden, R. Reuter (1990b), W. Freeden et al. (1990), L. Shure et al. (1982), and the references therein. Applications in flow problems of meteorology and ocean circulation are given, e.g., in M.J. Fengler, W. Freeden (2005), W. Freeden (2015), W. Freeden et al. (2005). A multivariate manifestation of spline interpolation in Paley-Wiener spaces with relations to Shannon-type sampling can be found in W. Freeden, M.Z. Nashed (2015), W. Freeden et al. (2018a,b), W. Freeden, M.Z. Nashed (2020b). Wavelets. The wavelet approach is a more flexible approximation method than the Fourier expansion approach in terms of spherical harmonics or the variational spline method using zonal kernels with fixed window. Due to the fact that variable “window kernel functions”, i.e., zooming-in, are being applied, a substantially better representation of the high frequency “short-wavelength” part of a function is achievable (global to scale-dependent local approximation). The zooming-in procedure allows higher global resolutions and, therefore, makes a better exposure of the strong correlation between the function (signal) under consideration and the local phenomena that should be modeled. Furthermore, the multiscale analysis can be used to modify and improve the standard approach in the sense that a local approximation can be

694

15. Concluding Remarks

established within a global orthogonal (Fourier) series and/or spline concept (see W. Freeden et al. (1998)). In essence, the characterization of spherical wavelets contains three basic attributes: (i) Basis Property. Wavelets are building blocks for the approximation of arbitrary functions (signals). In mathematical terms this formulation expresses that the set of wavelets forms a “frame” (see, e.g., I. Daubechies (1992) and the references therein for details in classical one-dimensional theory). (ii) Decorrelation. Wavelets possess the ability to decorrelate the signal. This means, that the representation of the signal via wavelet coefficients occurs in a “more constituting” form than in the original form, reflecting any certain amount of space and frequency information. The decorrelation enables the extraction of specific information contained in a signal through a particular locally reflected number of coefficients. Signals usually show a correlation in the frequency domain as well as in the space domain. Obviously, since data points in a local neighborhood are more strongly correlated as those data points far-off from each other, signal characteristics often appear in certain frequency bands. In order to analyze and reconstruct such signals, we need “auxiliary functions” providing localized information in the space as well as in the frequency domain. Within a “zooming-in process”, the amount of frequency as well as space contribution can be specified in a quantitative way. A so-called scaling function forms a compromise in which a certain balanced amount of frequency and space localization in the sense of the uncertainty principle is realized. As a consequence, each scaling function on the sphere depends on two variables, namely a rotational parameter (defining the position) and a dilational (scaling) parameter, which controls the amount of the space localization to be available at the price of the frequency localization, and vice versa. Associated to each scaling function is a wavelet function, which in its simplest manifestation may be understood as the difference of two successive scaling functions. Filtering (convolution) with a scaling function takes the part of a lowpass filter, while convolution with the corresponding wavelet function provides a bandpass filtering. A multiscale approximation of a signal is the successive realization of an efficient (approximate identity realizing) evaluation process by use of scaling and wavelet functions which show more and more space localization at the cost of frequency localization. The wavelet transform within a multiscale approximation lays the foundation for the decorrelation of a signal by specifying the transitional areas between different bands in the signature. (iii) Efficient Algorithms. Wavelet transform provides efficient algorithms because of the space localizing character. The successive decomposition of the signal by use of wavelets at different scales offers the advantage for efficient and economic numerical calculation. The detail information stored in the wavelet coefficients leads to a reconstruction from a rough to a fine resolution and to a decomposition

15. Concluding Remarks

695

from a fine to a rough resolution in form of tree algorithms. In particular, the decomposition algorithm is an excellent tool for the post-processing of a signal into “constituting blocks” by decorrelation, e.g., the specification of signature bands corresponding to certain signal specifics. A particular type is formed by mollifier regularization wavelets in inverse problems (see W. Freeden, M. Schreiner (2006b), W. Freeden et al. (2020), W. Freeden (2021), C. Blick et al. (2021)), which show a local support. As a consequence, spherical wavelets may be regarded as constituting multiscale building blocks, which provide a fast and efficient way to decorrelate a given signal data set. The properties (basis property, decorrelation, and efficient algorithms) are common features of all wavelets, so that these attributes form the key for a variety of applications particularly for signal reconstruction and decomposition, thresholding, data compression, denoising by use of, e.g., multiscale signal-to-noise ratio, etc. The essential power of spherical wavelets is based on the “zooming-in” property, i.e., scale dependent varying amounts of both frequency and space localization. This multiscale structure is the reason why spherical wavelets can be used as mathematical means for breaking up a complicated structure of a function into many simple pieces at different scales and positions. It should be pointed out that several wavelet approaches involving spherical wavelets have been established, all of them providing multiscale approximation, but not all of them showing a structural “breakthrough” in form of a multiresolution of the whole reference space by nested detail spaces. In all cases, however, wavelet modeling is provided by a two-parameter family reflecting the different levels of localization and scale resolution. Roughly speaking, the wavelet transform is a space localized replacement of the Fourier transform, providing space-varying frequency distribution in banded form. Wavelets provide sampling by only using a small fraction of the original information of a function. Typically the decorrelation is achieved by wavelets which have a characteristic local band (localization in frequency). Different types of wavelets can be established from certain constructions of space/frequency localization. It turns out that the decay towards long and short wavelengths (i.e., in information theoretic jargon, bandpass filtering) can be assured without any difficulty. Moreover, vanishing moments of wavelets enable us to combine polynomial (orthogonal) expansion (responsible for the long-wavelength part of a function) with wavelet expansions responsible for the medium-to-short-wavelength contributions (see, e.g., W. Freeden, U. Windheuser (1997), W. Freeden et al. (1998), W. Freeden et al. (2018a)). Because of the rotation-invariant nature of a large number of geodetic and geophysical quantities, resulting sampling methods of zonal spline and wavelet nature have much to offer. This is the reason why the authors decided to add to the plane involved matter some of the significant sphere intrinsic features and results of spherical sampling in a unifying concept in the second half of the book (note that vector- and tensor-isotropic sampling follows by obvious arguments from the context provided in this monograph based on settings developed in W. Freeden, C. Gerhards (2013), W. Freeden, M. Gutting (2013), W. Freeden, M. Schreiner (2009)). Furthermore,

696

15. Concluding Remarks

the fundamental idea of handling inverse problems of satellite technology (such as satellite-to-satellite tracking, satellite gravity gradiometry) is to understand regularization of inverse (non-bounded) pseudodifferential operators by a multiresolution analysis using certain kernel function expressions as regularizing wavelets.

List of Symbols

In the following definitions, the first digit refers to the the chapter in which the notation occurs and the second to the section within the chapter. Basic Nomenclature N0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . set of non-negative integers: N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . set of positive integers: Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . set of rational numbers: Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . set of integers: R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . set of real numbers: C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . set of complex numbers: ∪ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . union ∩ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . intersection ⊂ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . is contained in ⋐ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . is totally contained in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three-dimensional Euclidean space: 𝑥, 𝑦, 𝑧 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . elements of R3 : 𝑥 · 𝑦 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . scalar product of vectors: 𝑥 ∧ 𝑦 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vector product of vectors: 𝑥 ⊗ 𝑦 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dyadic product of vectors: |𝑥| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclidean norm of 𝑥: 𝜀 𝑖 , 𝑖 = 1, 2, 3 . . . . . . . . . . . . . . . . . . . . . . . . . canonical orthonormal basis in R3 : 𝛿𝑖 𝑗 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kronecker symbol: i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . identity tensor: t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . orthogonal matrix: t𝑇 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . transpose of the matrix t: det t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . determinant of the matrix t: tr t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . trace of tensor t: s ⊗ t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tensor product of tensors: s · t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . scalar product of tensors: © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2022 W. Freeden and M. Schreiner, Spherical Functions of Mathematical Geosciences, Geosystems Mathematics, https://doi.org/10.1007/978-3-662-65692-1

2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 697

698

List of Symbols

G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . point set in R3 : 𝜕G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . boundary of G: G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . closure of G: G 𝑐 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . complement to G: 𝐹 |𝑀 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . restriction of 𝐹 to 𝑀: ∇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gradient: 𝐿 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . curl gradient: ∇·, div . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . divergence: 𝐿·, curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . curl: Δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laplace operator:

2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2

Spherical Nomenclature Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . unit sphere in R3 around the origin: Ω𝑅 . . . . . . . . . . . . . . . . . . . . . . . . . sphere in R3 with radius 𝑅 around the origin: Ωint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inner space of Ω: Ωext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . outer space of Ω: Ωint 𝑅 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inner space of Ω 𝑅 : Ωext 𝑅 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . outer space of Ω 𝑅 : 𝑡, 𝜑 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . polar coordinates: 𝜉, 𝜂, 𝜁 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . elements of Ω: 𝜀𝑟 , 𝜀 𝜑 , 𝜀 𝑡 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . orthonormal triad on Ω: 𝑑𝜔 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . surface element: 𝑑𝑥, 𝑑𝑦, 𝑑𝑧 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . volume element: 𝑑𝜎 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . line element: 𝑂 (3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . group of real orthogonal matrices: 𝑆𝑂 (3) . . . . . . . . . . . . . . . . . . . group of real orthogonal matrices with det t = 1: 𝑅𝑡 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t-transform: C (𝑘 ) , L 𝑝 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . classes of scalar-valued function: c (𝑘 ) , l 𝑝 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . classes of vector-valued functions: c (𝑘 ) , l 𝑝 . . . . . . . . . . . . . . . . . . . . . . . classes of (rank-2) tensor-valued functions: C (𝑘 ) , L 𝑝 . . . . . . . . . . . . . . . . . . . . . classes of (rank-4) tensor-valued functions: 𝐹, 𝐺 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . scalar-valued functions: 𝑓 , 𝑔 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vector-valued functions: f, g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rank-2 tensor-valued functions: F, G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rank-4 tensor-valued functions: ∇∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . surface gradient: 𝐿 ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . surface curl gradient: ∇∗ ·, div∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . surface divergence: 𝐿 ∗ ·, curl∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . surface curl: Δ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . scalar Beltrami operator:

2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.5 2.4 2.4 2.4 2.7 2.7 2.7 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.5 2.5 2.5 2.5 2.5

List of Symbols

𝚫∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vectorial Beltrami operaror: ▲ ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tensorial Beltrami operator: 𝑓nor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . normal surface vector field: 𝑓tan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tangential surface vector field: fnor,nor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . left normal/right normal surface tensor field: fnor,tan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . left normal/right tangential surface tensor field: ftan,nor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . left tangential /right normal surface tensor field: ftan,tan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . left tangential/right tangential surface tensor field:

699

6.8 7.8 2.6 2.6 7.2 7.2 7.2 7.2

Spherical Harmonics (with respect to 𝑜 (𝑖) , 𝑂 (𝑖) ; o (𝑖,𝑘 ) , 𝑂 (𝑖,𝑘 ) -operators) 𝐻𝑛, 𝑗 . . . . . . . . . . . . . scalar homogeneous harmonics polynomials of degree 𝑛 and order 𝑗: 4.3 𝑌𝑛, 𝑗 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . spherical harmonic of degree 𝑛 and order 𝑗: 4.4 𝑃𝑛 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Legendre polynomial of degree 𝑛: 4.4 𝑃𝑛,𝑚 . . . . . . . . . . . . . . . . . . . . . . . . . . . associated Legendre function of degree 𝑛 and order 𝑚: 4.12 𝑜 (𝑖) , 𝑂 (𝑖) . . . . . . . . . . . . . . . . . . . . . . . . . . . adjoint operators (vectorial context): 5.3 (𝑖) 𝑦 𝑛, 𝑗 . . . . . . . . . . . . . . . . . . . . . vector spherical harmonics of degree 𝑛, order 𝑗, and type 𝑖 (with respect to 𝑜 (𝑖) , 𝑂 (𝑖) ): 5.3 (𝑖) 𝑝 𝑛 (·, ·) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Legendre vector kernel of degree 𝑛 and type 𝑖 (with respect to 𝑜 (𝑖) , 𝑂 (𝑖) ): 6.6 𝑝 𝑛 (·, ·) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Legendre vector kernel of degree 𝑛 (with respect to 𝑜 (𝑖) , 𝑂 (𝑖) ) : 6.6 (𝑖,𝑘 ) 𝑣p (·, ·) . . . . . . . . . . . (vectorial) Legendre rank-2 tensor kernel of degree 𝑛 𝑛 and type (𝑖, 𝑘) (with respect to 𝑜 (𝑖) , 𝑂 (𝑖) ): 6.8 𝑣 p (·, ·) . . . . . . . . . . . . . (vectorial) Legendre rank-2 tensor kernel of degree 𝑛 𝑛 (with respect to 𝑜 (𝑖) , 𝑂 (𝑖) ): 6.8 (𝑖,𝑘 ) (𝑖,𝑘 ) o ,𝑂 . . . . . . . . . . . . . . . . . . . . . . . . adjoint operators (tensorial context): 6.4 (𝑖,𝑘 ) y𝑛, . . . . . . . . . . . . . . . . . . . . tensor spherical harmonics of degree 𝑛, order 𝑗, 𝑗 and type (𝑖, 𝑘) (with respect to 𝑜 (𝑖,𝑘 ) , 𝑂 (𝑖,𝑘 ) ): 6.4 𝑡 p (𝑖,𝑘 ) (·, ·) . . . . . . . . . . . (tensorial) Legendre rank-2 tensor kernel of degree 𝑛 𝑛 and type (𝑖, 𝑘) (with respect to 𝑜 (𝑖,𝑘 ) , 𝑂 (𝑖,𝑘 ) ): 7.11 𝑡 p (·, ·) . . . . . . . . . . . . . .(tensorial) Legendre rank-2 tensor kernel of degree 𝑛 𝑛 (with respect to 𝑜 (𝑖,𝑘 ) , 𝑂 (𝑖,𝑘 ) ): 7.11 (𝑖,𝑘,𝑙,𝑚) P𝑛 (·, ·) . . . . . . . . (tensorial) Legendre rank-4 tensor kernel of degree 𝑛

700

List of Symbols

and type (𝑖, 𝑘, 𝑙, 𝑚) (with respect to 𝑜 (𝑖,𝑘 ) , 𝑂 (𝑖,𝑘 ) ): P𝑛 (·, ·) . . . . . . . . . . . . . . (tensorial) Legendre rank-4 tensor kernel of degree 𝑛 (with respect to 𝑜 (𝑖,𝑘 ) , 𝑂 (𝑖,𝑘 ) ):

7.9 7.9

Spherical Harmonics (with respect to 𝑜˜ (𝑖) , 𝑂˜ (𝑖) ; o˜ (𝑖,𝑘 ) , 𝑂˜ (𝑖,𝑘 ) -Operators) 𝐻𝑛, 𝑗 . . . . . . . . . . . . . scalar homogeneous harmonics polynomials of degree 𝑛 and order 𝑗: 𝑌𝑛, 𝑗 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . spherical harmonic of degree 𝑛 and order 𝑗: 𝑃𝑛 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Legendre polynomial of degree 𝑛: 𝑃𝑛,𝑚 . . . . . . . . . . . . . . . . . . . . . . . . . . . associated Legendre function of degree 𝑛 and order 𝑚: 𝑜˜ 𝑛(𝑖) , 𝑂˜ 𝑛(𝑖) . . . . . . . . . . . . . . . . . . . . . . . . . . . adjoint operators (vectorial context): (𝑖) e 𝑦 𝑛, 𝑗 . . . . . . . . . . . . . . . . . . . . . vector spherical harmonics of degree 𝑛, order 𝑗, and type 𝑖(with respect to o˜ (𝑖) , 𝑂˜ (𝑖) ): (𝑖) 𝑝e𝑛 (·, ·) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Legendre vector kernel of degree 𝑛 and type 𝑖 (with respect to o˜ (𝑖) , 𝑂˜ (𝑖) ): 𝑝e𝑛 (·, ·) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Legendre vector kernel of degree 𝑛 (with respect to o˜ (𝑖) , 𝑂˜ (𝑖) ): (𝑖,𝑘 ) 𝑣e p𝑛 (·, ·) . . . . . . . . . . . (vectorial) Legendre rank-2 tensor kernel of degree 𝑛 and type (𝑖, 𝑘) (with respect to o˜ (𝑖) , 𝑂˜ (𝑖) ): 𝑣e p𝑛 (·, ·) . . . . . . . . . . . . . (vectorial) Legendre rank-2 tensor kernel of degree 𝑛 and type (𝑖, 𝑘) (with respect to o˜ (𝑖) , 𝑂˜ (𝑖) ): (𝑖,𝑘 ) (𝑖,𝑘 ) o˜ , 𝑂˜ . . . . . . . . . . . . . . . . . . . . . . . . adjoint operators (tensorial context): (𝑖,𝑘 ) e y𝑛, 𝑗 . . . . . . . . . . . . . . . . . . . . tensor spherical harmonics of degree 𝑛, order 𝑗, and type (𝑖, 𝑘)(with respect to o˜ (𝑖,𝑘 ) , 𝐾 (𝑖,𝑘 ) ): (𝑖,𝑘 ) 𝑡e p𝑛 (·, ·) . . . . . . . . . . . (tensorial) Legendre rank-2 tensor kernel of degree 𝑛 and type (𝑖, 𝑘) (with respect to o˜ (𝑖,𝑘 ) , 𝑂˜ (𝑖,𝑘 ) ): e p𝑛 (·, ·) . . . . . . . . . . . . . . (tensorial) Legendre rank-2 tensor kernel of degree 𝑛 (with respect to o˜ (𝑖,𝑘 ) , 𝑂˜ (𝑖,𝑘 ) ): (𝑖,𝑘,𝑙,𝑚) e P𝑛 (·, ·) . . . . . . . . (tensorial) Legendre rank-4 tensor kernel of degree 𝑛 and type (𝑖, 𝑘, 𝑙, 𝑚) (with respect to o˜ (𝑖,𝑘 ) , 𝑂˜ (𝑖,𝑘 ) ): e P𝑛 (·, ·) . . . . . . . . . . . . . . (tensorial) Legendre rank-4 tensor kernel of degree 𝑛 (with respect to o˜ (𝑖,𝑘 ) , 𝑂˜ (𝑖,𝑘 ) ):

4.3 4.4 4.4 4.12 6.12 5.3 6.12 6.12 6.12 6.12 6.12 7.13 7.11 7.11 7.9 7.9

List of Symbols

701

Spin-Weighted Spherical Harmonics Ω0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . unit sphere without poles : 𝑜ˇ𝑖 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . auxiliary spherical function : 𝑜ˆ𝑖 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . auxiliary spherical function : sw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . spin-weight : ð 𝑁 , ð 𝑁 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . spin-weighted differential operators : Δ∗, 𝑁 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . spin-weighted Beltrami operator : 𝑋 𝑘 (Γ) . . . . . . . . . . . . . . . . . . . . class of spin-weighted differentiable functions: 𝐷 𝑛𝑗, 𝑁 (𝛼, 𝛽, 𝛾) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wigner 𝐷-function: 𝑁 𝑃 𝑛 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . spin-weighted Legendre polynomial: Harm𝑛𝑁 (Ω0 ) . . . . . . . . . . . . . . . . . . . . . . . . set of (∗, 𝑁)-harmonic polynomials :

8.1 8.1 8.1 8.1 8.1 8.1 8.1 8.3 8.3 8.3

Spherical Harmonic Spaces Harm𝑛 . . . . . . . . . . . . . . . . . . . space of scalar spherical harmonics of degree 𝑛: harm𝑛(𝑖) . . . . . . . . . . . . . . . . . . space of vector spherical harmonics of degree 𝑛 and type (𝑖) (with respect to 𝑜 (𝑖) , 𝑂 (𝑖) ): harm𝑛 . . . . . . . . . . . . . . . . . . . .space of vector spherical harmonics of degree 𝑛 (with respect to 𝑜 (𝑖) , 𝑂 (𝑖) ): (𝑖,𝑘 ) harm𝑛 . . . . . . . . . . . . . . . . space of tensor spherical harmonics of degree 𝑛 and type (𝑖, 𝑘) (with respect to 𝑜 (𝑖,𝑘 ) , 𝑂 (𝑖,𝑘 ) ): harm𝑛 . . . . . . . . . . . . . . . . . . . space of tensor spherical harmonics of degree 𝑛 (with respect to 𝑜 (𝑖,𝑘 ) , 𝑂 (𝑖,𝑘 ) ): (𝑖) ] 𝑛 . . . . . . . . . . . . . . . . . . space of vector spherical harmonics of degree 𝑛 harm and type (𝑖) (with respect to 𝑜˜ (𝑖) , 𝑂˜ (𝑖) ): ] 𝑛 . . . . . . . . . . . . . . . . . . . .space of vector spherical harmonics of degree 𝑛 harm (with respect to o˜ (𝑖) , 𝑂˜ (𝑖) ): (𝑖,𝑘 ) ]𝑛 harm . . . . . . . . . . . . . . . . space of tensor spherical harmonics of degree 𝑛 and type (𝑖, 𝑘) (with respect to o˜ (𝑖,𝑘 ) , 𝑂˜ (𝑖,𝑘 ) ): ] 𝑛 . . . . . . . . . . . . . . . . . . . space of tensor spherical harmonics of degree 𝑛 harm (with respect to o˜ (𝑖,𝑘 ) , 𝑂˜ (𝑖,𝑘 ) ):

4.1 5.3 5.3 7.13 7.13 6.12 6.12 7.13 7.13

702

List of Symbols

Zonal Kernel Functions 𝐾 (𝜉 · 𝜂) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . scalar zonal kernel function: 𝑘 (𝑖) (·, ·) . . . . . . . . . . . . . . . . . . . . . . . . (vectorial) zonal kernel function of type 𝑖 (with respect to the Legendre {𝑝 𝑛(𝑖) (·, ·)}-system ): 𝑘 (·, ·) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (vectorial) zonal kernel function (with respect to the Legendre {𝑝 𝑛 (·, ·)}-system): 𝑣 k (𝑖) (·, ·) . . . . . . . . . . . . . . . . . . . (vectorial) rank-2 tensor zonal kernel function of type 𝑖 (with respect to the Legendre { 𝑣 p𝑛(𝑖,𝑖) (·, ·)}-system) : 𝑣 k(·, ·) . . . . . . . . . . . . . . . . . . . . . (vectorial) rank-2 tensor zonal kernel function (with respect to the Legendre { 𝑣 p𝑛 (·, ·)}-system ) : 𝑡 k (𝑖,𝑘 ) (·, ·) . . . . . . . . . . . . . . . . . (tensorial) rank-2 tensor zonal kernel function of type (𝑖, 𝑘) (with respect to the Legendre {𝑡 p𝑛(𝑖,𝑘 ) (·, ·)}-system) : 𝑡 k(·, ·) . . . . . . . . . . . . . . . . . . . . . (tensorial) rank-2 tensor zonal kernel function (with respect to the Legendre {𝑡 p𝑛 (·, ·)}-system) : K (𝑖,𝑘 ) (·, ·) . . . . . . . . . . . . . . . . . (tensorial) rank-4 tensor zonal kernel function of type (𝑖, 𝑘) (with respect to the Legendre {P𝑛(𝑖,𝑘,𝑖,𝑘 ) (·, ·)}-system): K(·, ·) . . . . . . . . . . . . . . . . . . . . . (tensorial) rank-4 tensor zonal kernel function (with respect to the Legendre {P𝑛 (·, ·)}-system): e 𝑘 (𝑖) (·, ·) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (vectorial) zonal kernel function of type 𝑖 (with respect to the Legendre { 𝑝e𝑛(𝑖) (·, ·)}-system): e 𝑘 (·, ·) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (vectorial) zonal kernel function (with respect to the Legendre { 𝑝e𝑛 (·, ·)}-system): 𝑣e k (𝑖) (·, ·) . . . . . . . . . . . . . . . . . . . (vectorial) rank-2 tensor zonal kernel function of type 𝑖 (with respect to the Legendre { 𝑣e p𝑛(𝑖,𝑖) (·, ·)}-system): 𝑣e k(·, ·) . . . . . . . . . . . . . . . . . . . . . (vectorial) rank-2 tensor zonal kernel function (with respect to the Legendre { 𝑣e p𝑛 (·, ·)}-system) : 𝑡e (𝑖,𝑘 ) k (·, ·) . . . . . . . . . . . . . . . . . (tensorial) rank-2 tensor zonal kernel function of type (𝑖, 𝑘) (with respect to the Legendre {𝑡 e p𝑛(𝑖,𝑘 ) (·, ·)}-system): 𝑡e k(·, ·) . . . . . . . . . . . . . . . . . . . . . (tensorial) rank-2 tensor zonal kernel function (with respect to the Legendre {e p𝑛 (·, ·)}-system) : e (𝑖,𝑘 ) (·, ·) . . . . . . . . . . . . . . . . . (tensorial) rank-4 tensor zonal kernel function K of type (𝑖, 𝑘) (with respect to the Legendre {e P𝑛(𝑖,𝑘,𝑖,𝑘 ) (·, ·)}-system): e ·) . . . . . . . . . . . . . . . . . . . . . (tensorial) rank-4 tensor zonal kernel function K(·, (with respect to the Legendre {e P𝑛 (·, ·)}-system):

9.3 10.3 10.3 10.3 10.3 11.2 11.2 11.2 11.2 10.3 10.3 10.3 10.3 11.2 11.2 11.2 11.2

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Index

Abel-Poisson kernel, 128, 479, 584, 602 Abel-Poisson means, 129 Abel-Poisson summability, 127 addition theorem Chebyshev polynomial, 74 homogeneous harmonic polynomials, 95 scalar spherical harmonics, 109 spin-weighted spherical harmonics, 434 tensor spherical harmonics, 364 vector spherical harmonics, 290, 316 addition theorem of inner harmonics, 571 addition theorem of outer harmonics, 573 amplitude spectrum, 236 approximate identity, 466 associated Legendre (spherical) harmonic complex, 165, 410 real, 164 associated Legendre harmonic, 163 associated Legendre polynomial, 154 auxiliary function 𝑜ˇ 1 , 396 𝑜ˇ 2 , 396 𝑜ˆ 1 , 396 𝑜ˆ 2 , 396 bandlimited, 462, 516 kernel, 490 Beltrami operator scalar, 50 spin-weighted, 398, 423 tensorial, 360 vectorial, 288, 290 Bernstein summability, 123, 263, 346 Bernstein kernel, 123, 486 explicit representation, 123 oscillation, 486

properties, 123, 486 biharmonic field, 631 Bjerhammar ball, 648 boundary, 36 boundary-value problems Berltrami operator, 661 Laplace operator, 578 boundary-value problems (regular curve), 660 boundary-value problems (regular surface), 578 boundary-value problems of elasticity (spherical boundary), 643, 644 boundary-value problems of potential theory (regular surface), 590, 594, 595 Cauchy-Navier equation, 630 Chebyshev polynomial, 73 circular harmonic, 78 circular harmonics inner, 79 outer, 79 classification of zonal kernels, 491 Clebsch projection, 93 closure, 36 scalar spherical harmonics, 131 vector spherical harmonics, 270 tensor spherical harmonics, 353 compactly contained, 36 completeness scalar spherical harmonics, 132 vector spherical harmonics, 270 spin weighted spherical harmonics, 443 spin-weighted spherical harmonics, 443 tensor spherical harmonics, 353 consoidal, 261 convolution, 464

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2022 W. Freeden and M. Schreiner, Spherical Functions of Mathematical Geosciences, Geosystems Mathematics, https://doi.org/10.1007/978-3-662-65692-1

723

724 scalar context, 468 tensorial context, 546 vectorial context, 533 coordinates local, 47 polar, 44 curl, 38 surface, 49 curl gradient, 37 decomposition Hardy-Hodge, 252 Helmholtz, 251 decomposition theorem of Hom𝑛 , 90 deflection of the vertical, 669 differential equation of the Beltrami operator, 217 of the iterated Beltrami operator, 234 of the surface curl gradient, 216 of the surface gradient, 216 surface gradient, 670, 680 surface curl gradient, 681 differential operators, 51 spin-weighted, 398 differential operators in R3 , 37 Dirac kernel, 481 Dirac kernel, 246 Dirac rank-2 tensor family (vectorial) with respect to { 𝑣 p𝑛 }, 536 (vectorial) with respect to { 𝑣 p𝑛(𝑖,𝑖) }, 536 Dirac rank-4 tensor family, 548 (tensorial) with respect to {P𝑛 }, 548 (tensorial) with respect to {P𝑛(𝑖,𝑘,𝑖,𝑘) }, 548 Dirac vector family (vectorial) with respect to { 𝑝𝑛 }, 536 (vectorial) with respect to { 𝑝𝑛(𝑖) }, 536 Dirichlet Runge basis, 606 disc, 71 distribution, 236 divergence, 38 surface, 49 divergence theorem, 41 double layer potential, 576 dual operators o˜ (𝑖,𝑘) , 𝑂˜ (𝑖,𝑘) , 385 𝑜˜ (𝑖) , 𝑂˜ (𝑖) , 315 𝑜 (𝑖) , 𝑂 (𝑖) , 255 o (𝑖,𝑘) , 𝑂 (𝑖,𝑘) , 337 dyadic product in R3 , 32 dyadic tensor product, 35 Euler angles, 185 exact computation

Index homogeneous harmonic polynomials, 98 scalar homogeneous harmonic polynomials, 105 vector homogeneous harmonic polynomials, 274 field scalar-valued, 36 tensor-valued, 36 vector-valued, 36 first Green’s theorem, 54 Fourier expansion Legendre, 11, 463 spherical harmonics, 7 frame, 694 function continuous, 236 generalized, 236 Lebesgue, 236 scalar-valued, 37 Schwartz, 236 tensor-valued, 37 vector-valued, 37 function spaces, 45 fundamental solution for Laplace operator, 119 Laplace equation, 554 stereographic, 80 fundamental system, 664 relative to Hom𝑛 , 87 fundamental systems relative to Harm𝑛 , 147 Gauss kernel, 481 theorem, 53 Gauss theorem, 41 generalized Fourier series (potential theory), 606, 608 generalized function, 236 geostrophic flow, 669 gradient suface, 48 Green’s function Legendre expansion, 201 pseudodifferential operator, 246 regularized version, 214 uniqueness, 194, 209 with respect to iterated Beltrami operators, 229 with respect to the Beltrami operator, 194 with respect to the modified Beltrami operator, 208 Green’s surface identity

Index third, 213 Green’s theorem first, 54 second, 55 Haar kernel, 481 smoothed, 481 Haar scaling function, 483 harmonic, 446, 569 inner circular, 78 outer circular, 78 spin-weighted, 446 harmonics inner, 78 outer, 78 Harnack’s convergence theorem, 569 Helmholtz decomposition theorem tensorial fields, 359 vector fields, 259 Hobson’s formula, 92 homogeneous “pre-Maxwell equations”, 61 homogeneous harmonic polynomials scalar context, 84 tensor contex, 354 vector context, 271 identity tensor, 254 inner cicular harmonics, 78 inner harmonics, 78 inner product in R3 , 32 inner space, 52 inner space of Ω, 44 inner space of Ω𝑅 , 43 integral theorem with respect to iterated Beltrami operators, 230 with respect to the Beltrami operator, 212 invariant with respect to reflections, 63 with respect to rotations, 63 inverse Fourier transform, 8 inverse gravimetry, 558, 568 inverse magnetometry, 558 irreducibility ℎ𝑎𝑟 𝑚𝑛 , 281 Harm𝑛 , 145 harm𝑛 , 373 irreducible space, 64 iterated convolution, 468 iterated kernel, 465 jump relations, 575, 576

725 Kelvin transform, 573 kernel Abel-Poisson, 479 bandlimited, 462 Dirac, 481 Gauss, 481 Haar, 481 Legendre, 478 locally supported, 462 spacelimited, 462 symbol, 12, 464 zonal, 11 Kronecker delta, 33 Lam`e parameters, 629 Laplace operator (Laplacian), 39, 50 Laplace representation of Legendre polynomial, 120 Legendre kernel, 478 polynomial, 12 symbol, 12 Legendre function spin-weighted , 433 Legendre harmonic, 137 Legendre polynomial, 95, 112 associated, 154 constituing properties, 112 estimates, 113 generating series expansion, 118 Laplace representation, 120 orthogonality, 112 recurrence formulas, 116 Rodriguez formula, 115 spin-weighted , 433 Legendre rank-2 tensor kernel (tensorial) with respect to the o˜ (𝑖,𝑘) , 𝑂˜ (𝑖,𝑘) system, 387 (tensorial) with respect to the o (𝑖,𝑘) , 𝑂 (𝑖,𝑘) system, 377 (vectorial) with respect to 𝑜 (𝑖) , 𝑂 (𝑖) , 292 (vectorial) with respect to 𝑣 p˜ 𝑛 , 315 (vectorial) with respect to 𝑣 p˜ 𝑛(𝑖,𝑖) , 315 (vectorial) with respect to 𝑣 p𝑛 , 292 (vectorial) with respect to 𝑣 p𝑛(𝑖,𝑖) , 292 (vectorial) with respect to the 𝑜˜ (𝑖) , 𝑂˜ (𝑖) system, 315 (vectorial) with respect to the 𝑜 (𝑖) , 𝑂 (𝑖) system, 315 Legendre rank-4 tensor kernel (tensorial) with respect to the o˜ (𝑖,𝑘) , 𝑂˜ (𝑖,𝑘) system, 385 (tensorial) with respect to the o (𝑖,𝑘) , 𝑂 (𝑖,𝑘) system, 364

726 Levi-Civit`a alternating symbol, 32 limit formulas, 575 limit relations, 576 Lipschitz continuity, 37, 45 localization Abel-Poisson kernel, 479 Bernstein kernel, 486 Dirac kernel, 480 Gaussian kernel, 480 Haar kernel, 481 Legendre kernel, 478 scheme, 491 Shannon kernel, 483 spherical harmonics, 478 localization in frequency, 471 localization in space, 469 locally supported, 462 locally uniform approximation in elasticity, 595 locally uniform approximation in potential theory, 595 logarithmic kernel, 603 lowpass filter, 672 maximum principle, 561, 654 Maxwell’s representation formula, 119 mean locally supported, 483 modulus of continuity, 37, 45 mollifier, 196 mollifier Green’s kernel function, 670 mollifier regularization, 555, 654 monomials, 84 moving triad { 𝜀 1𝜉 , 𝜀 2𝜉 , 𝜀 3𝜉 }, 65 multipoles, 147 Navier equation, 630 Neumann problem, 590 Neumann Runge basis, 606 Newton integral, 565 Newton law, 553 norm estimate, 45, 47 normal, 328 normal vector field, 253 numerical integration, 222 ocean flow, 669 ocean topography, 685 oceanic flow, 684 operator 𝑅t - operator , 63 Beltrami, 50 gradient, 37 Laplace, 39, 50 Nabla, 37

Index spin lowering, 402 spin raising, 402 operator ð, 401 operator ð, 401 operators 𝑂 (𝑖,𝑘) , 339 o (𝑖,𝑘) , 337 𝑜 (𝑖) , 255 𝑂 (𝑖) , 255 orthogonal expansion spherical harmonics, 7 orthogonal invariance, 62 scalar context, 65 tensor context, 67 vector context, 66 orthogonally invariant, 63 orthonormal basis { 𝜀 1 , 𝜀 2 , 𝜀 3 }, 32 { 𝜀 𝑟 , 𝜀 𝜑 , 𝜀 𝑡 }, 48 orthonormal system L2 (Ω)-orthonormal system, 97 l2 (Ω)-orthonormal system, 258 l2 (Ω)-orthonormal system, 378 oscillations Bernstein kernel, 487 Shannon kernel, 484, 487 outer cicular harmonics, 78 outer harmonics, 78, 572 outer harmonics expansions, 648 outer space, 52 outer space of Ω, 44 outer space of Ω𝑅 , 43 Parseval identity spin-weighted spherical harmonics, 445 partial derivatives, 37 point set boundary, 36 closure, 36 Poisson problem Beltrami operator, 655 Laplace operator, 554 Poisson’s ratio, 630 polar coordinates, 44 potential function, 60 potential of the double layer, 576 potential of the single layer, 575 projection operator 𝑝nor , 253 projection operator 𝑝tan , 253 pseudodifferential operator, 236 amplifying, 245 exponentially growing, 245 Green’s function, 246 order 𝑡, 242

Index rotation invariant, 236 slowly growing, 241 smoothing, 245 symbol, 242 radial basis function, 463 recursion relations spin-weighted spherical harmonics, 422 reducible, 63 reflection, 62 region, 36 regular, 54 regular, 41 regular region, 41, 56 Euclidean, 40 spherical, 52 reproducing kernel in Harm𝑛 , 110 restriction, 37 right normal/left normal, 328 right normal/left tangential, 328 right tangential/left normal, 328 right tangential/left tangential, 328 Rodriguez formula, 115 Rodriguez rule, 115 rotation, 62, 185 rotational invariant, 62 Runge region, 648 Runge-Walsh theorem, 648 Runge-Walsh theorem (classical formulation), 596 Runge-Walsh theorem (potential theory), 596 scalar function, 36 scalar Legendre kernel, 113 scalar product in R3 , 32 scalar spherical harmonics degree, order, 109 definition, 108 degree and order variances, 149 eigenfunctions, 143 Fourier expansion, 123 Funk-Hecke formula, 141 irreducibility, 145 orthogonal expansions, 129 orthogonality, 108 Parseval identity, 135 restrictions of homogeneous harmonic polynomials, 108 scalar spherical harmonics addition theorem, 109 scalar zonal function, 46 scalar zonal kernel function, 467 scalars, 31 scaling

727 equation, 671, 681 scaling function, 694 Schwartz function, 236 Schwartz space, 236 Second Green’s Surface Identity, 399 second Green’s theorem, 55 semigroup of contraction operators, 498 sequence summable, 238 Shannon kernel, 484 oscillation, 484 properties, 484 single layer potential, 575 singularity kernel, 603 smoothed Haar kernel, 481 Sobolev Lemma, 238 Sobolev space, 236, 238 space inner, 52 outer, 52 space X 𝑘 (Γ), 399 spacelimited, 462, 501 special orthogonal group 𝑆𝑂 (3), 62 spherical harmonics, 6 spherical differential operators, 51 spherical harmonics scalar, 7 spherical nomenclature, 44 spherical notation, 44 spherical potential function, 60 spherical stream function, 60 spherical vector field, 56, 57 consoidal, 261 normal, 57 spheroidal, 261 tangential, 57 toroidal, 261 spheroidal, 261 spin lowering, 402 spin raising, 402 spin weight, 397 spin weights, 31 spin-weighted associated Legendre polynomial, 433 spin-weighted Beltrami operator, 398, 423 spin-weighted differential operators, 398 spin-weighted Legendre function, 433 spin-weighted Legendre polynomial, 433 spin-weighted spherical harmonics completeness, 443 Parseval identity, 445 spline, 693 spline admissible, 246

728 spline function classical, 223 spline integration formula, 248 square-summable, 464 stereographic projection, 75 Stokes theorem, 53 stream function, 60 summability Abel-Poisson, 123, 127 Bernstein, 123 summable, 238 surface curl, 49 surface curl gradient, 48 surface divergence, 49 surface identity tensor field, 254 surface rotation tensor field, 254 symbol kernel, 12, 464 Legendre, 12, 464 pseudodifferential operator, 242 symmetric gradient, 38 tangential, 254, 328 tangential vector field, 253 tensor rank 𝑘, 35 rank four, 36 rank two, 33 tensor field right normal/left normal, 329 right normal/left tangential, 329 right tangential/left normal, 329 right tangential/left tangential, 329 tensor function, 36 tensor product in R3 , 32 tensor spherical harmonic Funk - Hecke formulas, 372 tensor spherical harmonics addition theorem, 363 definition, 335 degree, order, type, 336 eigenfunctions, 360 Fourier expansion, 378 orthogonality, 339 with respect to the o˜ (𝑖,𝑘) , 𝑂˜ (𝑖,𝑘) - system, 379, 382 with respect to the o (𝑖,𝑘) , 𝑂 (𝑖,𝑘) -system, 341 tensorial Beltrami operator, 360 tensors, 31 toroidal, 261 totally contained, 36 transformed field

Index t-transformed, 62 translation operator, 121 uncertainty principle, 9, 469, 473, 477, 490 unit circle, 71 unit matrix i, 68 unit sphere without poles, 397 up function, 509 upward contonuation, 574 vector calculus differential, 48 vector field normal, 253 tangential, 253 vector function, 36 vector product in R3 , 32 vector spherical harmonics 𝑜˜ (𝑖) , 𝑂˜ (𝑖) - system, 309 𝑜 (𝑖) , 𝑂 (𝑖) - system, 256 Fourier expansion, 273 Funk - Hecke formulas, 297 addition theorem, 290, 296 degree and order variances, 305 degree, order, type, 258 eigenfunctions, 290 exact generation, 275 Hardy-Hodge definition, 309 Helmholtz definition, 256 irreducibility, 280 orthogonality, 256 vector spherical harmonics addition theorem, 283 vectorial Beltrami operator, 288, 290 vectors, 31 volume potential, 554 vorticity, 685 wavelet, 671, 681, 694 function, 694 wavelets, 499 well-posedness of boundary-value problem of potential theory, 594 Wigner 𝐷-function, 186, 425 Wigner matrices, 184 zeros of Legendre polynomial, 115 zonal function, 462 kernel, 462, 465 zonal function Legendre kernel, 109 zonal functions

Index frequency localization, 472 space localization, 472 zonal kernel Abel - Poisson, 512 Bernstein, 126 Gauss-Weierstraß, 513 Haar, 501 Shannon, 516 smoothed Shannon, 516 up function, 509 zonal kernel functions scalar context, 466 tensorial scheme, 539 vectorial context, 524 vectorial scheme, 524 zonal rank-2 tensor kernel, 544 (tensorial) with respect to { 𝑡 p˜ 𝑛 }, 544 (tensorial) with respect to { 𝑡 p˜ 𝑛(𝑖,𝑖) }, 544 (tensorial) with respect to { 𝑡 p𝑛 }, 544

729 (tensorial) with respect to { 𝑡 p𝑛(𝑖,𝑖) }, 544 (vectorial) with respect to { 𝑣 p˜ 𝑛 }, 530 (vectorial) with respect to { 𝑣 p˜ 𝑛(𝑖,𝑖) }, 530 (vectorial) with respect to { 𝑣 p𝑛 }, 525 (vectorial) with respect to { 𝑣 p𝑛(𝑖,𝑖) }, 525 zonal rank-4 tensor kernel, 540 (tensorial) with respect to { P˜ 𝑛 }, 540 (tensorial) with respect to { P˜ 𝑛(𝑖,𝑘,𝑖,𝑘) }, 540 (tensorial) with respect to {P𝑛 }, 540 (tensorial) with respect to {P𝑛(𝑖,𝑘,𝑖,𝑘) }, 540 zonal scaling functions scalar context, 491 tensor context, 548 vector context, 535 zonal vector kernel function (vectorial) with respect to { 𝑝˜ 𝑛 }, 533 (vectorial) with respect to { 𝑝˜ 𝑛(𝑖) }), 533 (vectorial) with respect to { 𝑝𝑛 }, 531 (vectorial) with respect to { 𝑝𝑛(𝑖) }), 531