Wavelet Analysis: Basic Concepts and Applications [1 ed.] 0367562189, 9780367562182

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Wavelet Analysis

Wavelet Analysis

Basic Concepts and Applications

Sabrine Arfaoui

University of Monastir, Tunisia University of Tabuk, Saudi Arabia

Anouar Ben Mabrouk

University of Kairouan, Tunisia University of Monastir, Tunisia University of Tabuk, Saudi Arabia

Carlo Cattani

University of Tuscia, Italy

First edition published 2021 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2021 Sabrine Arfaoui, Anouar Ben Mabrouk, and Carlo Cattani CRC Press is an imprint of Taylor & Francis Group, LLC The right of Sabrine Arfaoui, Anouar Ben Mabrouk, and Carlo Cattani to be identified as authors of this work has been asserted by them in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Arfaoui, Sabrine, author. | Ben Mabrouk, Anouar, author. | Cattani, Carlo, 1954- author. Title: Wavelet analysis : basic concepts and applications / Sabrine Arfaoui, University of Monastir, Anouar Ben Mabrouk, University of Kairouan, Carlo Cattani, University of Tuscia. Description: Boca Raton : Chapman & Hall/CRC Press, 2021. | Includes bibliographical references and index. Identifiers: LCCN 2020050757 (print) | LCCN 2020050758 (ebook) | ISBN 9780367562182 (hardback) | ISBN 9781003096924 (ebook) Subjects: LCSH: Wavelets (Mathematics) Classification: LCC QA403.3 .A74 2021 (print) | LCC QA403.3 (ebook) | DDC 515/.2433--dc23 LC record available at https://lccn.loc.gov/2020050757 LC ebook record available at https://lccn.loc.gov/2020050758

ISBN: 978-0-367-56218-2 (hbk) ISBN: 978-1-003-09692-4 (ebk) Typeset in Latin Modern font by KnowledgeWorks Global Ltd.

Contents List of Figures

ix

Preface

xi

Chapter

1  Introduction

1

Chapter

2  Wavelets on Euclidean Spaces

5

2.1

INTRODUCTION

5

2.2

WAVELETS ON R

6

2.2.1

Continuous wavelet transform

7

2.2.2

Discrete wavelet transform

10

2.3

MULTI-RESOLUTION ANALYSIS

11

2.4

WAVELET ALGORITHMS

13

2.5

WAVELET BASIS

16

2.6

MULTIDIMENSIONAL REAL WAVELETS

21

2.7

EXAMPLES OF WAVELET FUNCTIONS AND MRA

22

2.7.1

Haar wavelet

22

2.7.2

Faber–Schauder wavelet

24

2.7.3

Daubechies wavelets

25

2.7.4

Symlet wavelets

27

2.7.5

Spline wavelets

27

2.7.6

Anisotropic wavelets

29

2.7.7

Cauchy wavelets

30

2.8

Chapter

EXERCISES

3  Wavelets extended

31

35

3.1

AFFINE GROUP WAVELETS

35

3.2

MULTIRESOLUTION ANALYSIS ON THE INTERVAL

37 v

vi  Contents

3.3

3.4

Chapter

3.2.1

Monasse–Perrier construction

37

3.2.2

Bertoluzza–Falletta construction

37

3.2.3

Daubechies wavelets versus Bertoluzza–Faletta

39

WAVELETS ON THE SPHERE

40

3.3.1

Introduction

40

3.3.2

Existence of scaling functions

41

3.3.3

Multiresolution analysis on the sphere

43

3.3.4

Existence of the mother wavelet

44

EXERCISES

4  Clifford wavelets

47

51

4.1

INTRODUCTION

51

4.2

DIFFERENT CONSTRUCTIONS OF CLIFFORD ALGEBRAS

52

4.2.1

Clifford original construction

53

4.2.2

Quadratic form-based construction

53

4.2.3

A standard construction

54

4.3

GRADUATION IN CLIFFORD ALGEBRAS

56

4.4

SOME USEFUL OPERATIONS ON CLIFFORD ALGEBRAS

57

4.4.1

Products in Clifford algebras

57

4.4.2

Involutions on a Clifford algebra

58

4.5

CLIFFORD FUNCTIONAL ANALYSIS

60

4.6

EXISTENCE OF MONOGENIC EXTENSIONS

67

4.7

CLIFFORD-FOURIER TRANSFORM

70

4.8

CLIFFORD WAVELET ANALYSIS

76

4.8.1

Spin-group based Clifford wavelets

76

4.8.2

Monogenic polynomial-based Clifford wavelets

82

4.9

SOME EXPERIMENTATIONS

4.10 EXERCISES

Chapter

5  Quantum wavelets

92 96

99

5.1

INTRODUCTION

99

5.2

BESSEL FUNCTIONS

99

5.3

BESSEL WAVELETS

105

5.4

FRACTIONAL BESSEL WAVELETS

107

5.5

QUANTUM THEORY TOOLKIT

119

Contents  vii

5.6

SOME QUANTUM SPECIAL FUNCTIONS

123

5.7

QUANTUM WAVELETS

127

5.8

EXERCISES

134

Chapter

6  Wavelets in statistics

137

6.1

INTRODUCTION

137

6.2

WAVELET ANALYSIS OF TIME SERIES

138

6.2.1

Wavelet time series decomposition

138

6.2.2

The wavelet decomposition sample

140

6.3

WAVELET VARIANCE AND COVARIANCE

141

6.4

WAVELET DECIMATED AND STATIONARY TRANSFORMS

144

6.4.1

Decimated wavelet transform

144

6.4.2

Wavelet stationary transform

145

6.5

6.6

6.7

6.8

Chapter

WAVELET DENSITY ESTIMATION

145

6.5.1

Orthogonal series for density estimation

145

6.5.2

δ-series estimators of density

147

6.5.3

Linear estimators

148

6.5.4

Donoho estimator

150

6.5.5

Hall-Patil estimator

150

6.5.6

Positive density estimators

151

WAVELET THRESHOLDING

152

6.6.1

Linear case

152

6.6.2

General case

154

6.6.3

Local thresholding

155

6.6.4

Global thresholding

155

6.6.5

Block thresholding

156

6.6.6

Sequences thresholding

156

APPLICATION TO WAVELET DENSITY ESTIMATIONS

157

6.7.1

Gaussian law estimation

158

6.7.2

Claw density wavelet estimators

159

EXERCISES

7  Wavelets for partial differential equations

160

163

7.1

INTRODUCTION

163

7.2

WAVELET COLLOCATION METHOD

165

viii  Contents

7.3

WAVELET GALERKIN APPROACH

166

7.4

REDUCTION OF THE CONNECTION COEFFICIENTS NUMBER

171

7.5

TWO MAIN APPLICATIONS IN SOLVING PDEs

174

7.5.1

The Dirichlet Problem

174

7.5.2

The Neumann Problem

176

7.6

APPENDIX

179

7.7

EXERCISES

180

Chapter

8  Wavelets for fractal and multifractal functions

183

8.1

INTRODUCTION

183

8.2

HAUSDORFF MEASURE AND DIMENSION

184

8.3

WAVELETS FOR THE REGULARITY OF FUNCTIONS

186

8.4

THE MULTIFRACTAL FORMALISM

189

8.4.1

Frisch and Parisi multifractal formalism conjecture

189

8.4.2

Arneodo et al wavelet-based multifractal formalism

190

8.5

SELF-SIMILAR-TYPE FUNCTIONS

192

8.6

APPLICATION TO FINANCIAL INDEX MODELING

201

8.7

APPENDIX

205

8.8

EXERCISES

205

Bibliography

209

Index

237

List of Figures

2.1

Decomposition algorithm.

15

2.2

Inverse algorithm.

16

2.3

Haar scaling and wavelet functions.

24

2.4

Schauder scaling and wavelet functions.

26

2.5

Some Daubechies scaling functions and associated wavelets.

27

2.6

Some symlet scaling functions and associated wavelets.

28

4.1

Wedge product of vectors.

58

4.2

Some 2D Clifford wavelets.

93

4.3

ψ1,2 (x)-Clifford wavelet 1-level decomposition of Mohamed Amine’s photo.

94

4.4

The 3D ψ3,3 (x) Clifford wavelet brain processing at the level J = 2.

95

5.1

Graphs of Bessel functions Jv of the first kind for v = 0, 1, 2, 3.

102

5.2

Graphs of Bessel functions Yv of the second kind for v = 0, 1, 2, 3.

103

6.1

The normal reduced and centered density N (0, 1).

158

6.2

The gaussian density and its wavelet estimator.

159

6.3

Claw density.

159

6.4

Claw density wavelet estimator at the level J = 4.

160

8.1

Wavelet decomposition of the SP500 signal at the level J = 4.

203

8.2

SP500 spectrum of singularities.

204

8.3

Original SP500 and self-similar-type model.

204

ix

Preface Nowadays, wavelets are applied almost everywhere in science. Both pure fields, such as mathematics and theoretical physics, and applied ones, such as signal/image processing, finance and engineering, apply wavelets. Although the references and/or the documentation about wavelets and their applications are wide, it seems that with the advancement of technology and the appearance of many phenomena in nature and in life there still exist some places for more efforts and developments to understand the new problems, as the existing wavelet methods do not provide good understanding of them. The new COVID-19 pandemic may be one of the challenges that should be understood. On the other hand, especially for young researchers, existing references such as books in wavelet theory are somehow very restricted. The majority are written for specific communities. This is, in fact, not surprising and may be due to the necessity of developing such references to overcome the concerned problems in that time. Next, with the inclusion of wavelet theory in academic studies such as in master’s and PhD programs, the scientific and academic communities have had a great need to develop references in other forms. Students and generally researchers need sometimes self-containing references responding to their need, to avoid losing time in redeveloping existing results, which is a necessary step for both the generalization and the experiments. The present volume is composed of eight chapters. In the first introductory chapter, a literal introduction is developed discussing generally the topic. Chapter 2 is concerned with the presentation of the original developments of wavelet theory on the real Euclidean space. This is also a preliminary chapter that will be of great help for young researchers. Chapter 3 is more specialized and constitutes a continuation of the previous one, in which some extending cases of wavelet theory and applications have been provided. Chapter 4 is a very specialized part that is developed for the first time to our knowledge. It is concerned with the presentation of wavelet theory in a general functional framework based on Clifford algebras. This is very important as these algebras contain all the Euclidean structures and gather them in one structure to facilitate calculus. Readers will notice clearly that Clifford wavelet theory induces naturally the Euclidean ones such as real and complex numbers, circles and spheres. Chapter 5 is a continuation of the development of the theory in specialized fields such as quantum theory. Next, in Chapter 6, statistical application of wavelets has been reviewed. Topics such as density estimation, thresholding concepts, variance and covariance have been detailed. Chapter 7 is devoted to wavelets applied in solving partial differential equations. Recall that this field needs many assumptions on the functional bases applied, especially the explicit form of the basis elements and their xi

xii  Preface

regularities. The last chapter is devoted to the link and/or the use of wavelet theory in characterizing fractal and multifractal functions and their application. Each chapter contains a series of exercises and experimentations to help understand the theory and also to show the utility of wavelets. The present book stems, in fact, from lectures and papers on the topics developed, which have been gathered, re-developed, improved and sometimes completed with necessary missing developments. However, naturally it is not exhaustive and should be always criticized, sometimes corrected and improved by readers. So, we accept and wait for any comments and suggestions. We also want to stress the fact that we have provided in some chapters, especially those on preliminary concepts that may be useful to young researchers, some exercises and applications that are simple to handle with the aim to help the readers understand the theory. We apologize if there are simpler applications and details that may be more helpful to the readers but that have been left out from inclusion in this book. This, in fact, needs more time and may induce delays in the publication of the book. We hope that with the present form the readers become acquainted with the topics presented. The aim of this book is to provide a basic and self-contained introduction to the ideas underpinning wavelet theory and its diversified applications. Readers of our proposed book would include master’s degree students, PhD students and senior researchers. It may also serve scientists and research workers from industrial settings, where modeling real-world phenomena and data needs wavelets such as finance, medicine, engineering, transport, images and signals. Henceforth, the book will interest practitioners and theorists alike. For theorists, rigorous mathematical developments will be presented with necessary prerequisites that make the book selfcontaining. For the practitioner, often interested in model building and analysis, we provide the cornerstone ideas. As with any scientific production and reference, the present volume could not have been realized without the help of many persons. We thus owe thanks to many persons who have helped us in any direction such as encouragements, scientific discussions and documentation. We thank the Taylor & Francis Publishing Group for giving us the opportunity to write and publish the present work. We also would like to express our gratitude to our professors, teachers, colleagues, and universities. Without their help and efforts, no such work might be realized. We would also like to thank all the members of the publishing house, especially the editorial staff for the present volume, Callum Fraser and Mansi Kabra, for their hospitality, cooperation, collaboration and for the time they have spent on our project.

CHAPTER

1

Introduction

Wavelets were discovered in the eighteenth century, essentially in pertroleum extraction. They have induced a new type of analysis extending the Fourier one. Recall that Fourier analysis has been for long a time the essential mathematical tool, especially in harmonic analysis and related applications such as physics, engineering, signal/image processing and PDEs. Next, wavelets were introduced as new extending mathematical tools to generalize the Fourier one and to overcome, in some ways, the disadvantages of Fourier analysis. For a large community, especially non-mathematicians and non-physicists, a wavelet may be defined in the most simple sense as a wave function that decays rapidly and has a zero mean. Compared to the Fourier theory, wavelets are mathematical functions permitting themselves to cut up data into different components relative to the frequency spectrum and next focus on these components somehow independently, extracting their characteristics and lifting to the original data. One main advantage of wavelets is the fact that they are more able than Fourier modes to analyze discontinuities and/or singularities efficiently (see [20], [177], [218], [248], [279], [295], [305]). Wavelets have been also developed independently in the fields of mathematics, quantum physics, electrical engineering and seismic geology. Next, interchanges between these fields have yielded more understanding of the theory and more and more bases as well as applications such as image compression, turbulence, human vision, radar and earthquake. Nowadays, wavelets have become reputable and successful tools in quasi all domains. The particularity in a wavelet basis is that all its elements are deduced from one source function known as the mother wavelet. Next, such a mother gives rise to all the elements necessary to analyze objects by simple actions of translation, dilatation and rotation. The last parameter was introduced by Antoine and his collaborators ([11], [13]) to obtain some directional selectivity of the wavelet transform in higher dimensions. Indeed, unlike Fourier analysis, there are different ways to define multidimensional wavelets. Directional wavelets based on the rotation parameter just evoked. Another class is based on tensor products of one-dimensional wavelets. Also, some wavelets are related to manifolds, essentially spheres, where the idea is based on the geometric structure of the surface where the data lies. This gives rise to

1

2  Wavelet Analysis: Basic Concepts and Applications

the so-called isotropic and anisotropic wavelets (see [11], [12], [20], [205], [218], [219], [220], [229], [230], [248], [279], [295], [305], [357], [358], [363]). Wavelet theory provides for functional spaces and time series good bases, allowing their decomposition into spaces associated with different horizons known as the levels of decomposition. A wavelet basis is a family of functions obtained from one function known as the mother wavelet, by translations and dilations. Due to the power of their theory, wavelets have many applications in different domains such as mathematics, physics, electrical engineering and seismic geology. This tool permits the representation of L2 -functions in a basis well localized in time and in frequency. Wavelets are also associated with many special functions such as orthogonal polynomials and hypergeometric series. The most well known may be the Bessel functions that have been developed in both classic theory of Bessel functional analysis and the modified versions in fractional and quantum calculus. As its name indicates, Bessel wavelets are related to Bessel special function. Historically, special functions differ from elementary ones such as powers, roots, trigonometric and their inverses, mainly with the limitations that these latter classes are known for. Many fundamental problems such as orbital motion, simultaneous oscillatory chains and spherical body gravitational potential were not best described using elementary functions. This makes it necessary to extend elementary functions’ classes to more general ones that may describe well unresolved problems. Wavelets are also developed and applied in financial time series such as market indices and exchange rates. In [42], for example, a study of the largest transaction financial market was carried out. The exchange market gave some high-frequency data. Compared to other markets, such data can be available at long periods and with high frequency. The data were detected for very small periods, which means that the market is also liquid. Until 1990, economists were interested in intra-daily data because of which the detection of some behaviors did not appear in the daily analysis of data such as homogeneity. A well-known hypothesis in finance is the homogeneity of markets where all investigators have almost the same behavior. The idea of nonhomogeneous markets is more recent, and it suggests that investigators have different perceptions and different laws. For the exchange market, for example, investigators can differ in profiles, geographic localizations and also in institutional constraints. Another natural suggestion can be done about traders. Naturally, traders investigating at short time intervals allow some high-frequency behaviors in the change market. Long-time traders are interested in the general tendency and the volatility of the market along a microscopic greed. Short-time traders, however, are interested in fractional perceptions and so in macroscopic greed. This leads to the wavelet analysis of financial time series. Recently, other models have been introduced in modeling financial time series by means of fractals, which are in turn strongly related to wavelets. For example, in Olsen & Associates, operating the largest financial database, has noticed that the tick frequency has strongly increased in one decade, causing problems in studying the time series extracted from such a database. Such problems can be due to transmission delays, input errors and machine damages. So, some filtering procedure has to be done

Introduction  3

before using the data. The first point that one must take into account in filtering time series is their scaling behavior. Scaling laws were empirically observed by Olsen et al until 1990 (see [306]). A time series X(t) has a scaling law if its so-called partition function has the form N/t t X Sq (t) = |X(jt)|q ∼ tτq +1 N j=1 where N stands for the size of the series for some appropriate function τ . This estimation is well understood when merging wavelet tools into fractal models. (See Chapter 8.) In such models, Fourier analysis could not produce good results in estimating such behavior. Indeed, Fourier transform of time series is generally limited because a single analysis window cannot detect features in the signals that are either much longer or shorter than the window size. Moving-window Fourier transform (MWFT) slides a fixed-size analysis window along the time axis and is able to detect non-stationarities. The fixed-size window algorithm of MWFT limits the detection of cycles at wavelengths that are longer than the analysis windows, and non-stationarities in short wavelengths (i.e., high frequencies) are smoothed. Use of the wavelet transform solves this problem, because it uses narrow windows at high frequencies and wide windows at low frequencies. This book is devoted to developing the basic concepts of wavelet analysis necessary for young researchers doing their Master’s level in science and researchers doing doctoral studies in pure mathematical/physical sciences, as well as applied and interacted ones by providing the basic tools required, with simple and rigorous methods. It also aims to serve researchers at advanced levels by providing them the necessary tools that will allow them to understand and adapt wavelet theory to their needs such as supervision and development of research projects. The book provides some highly flexible methods and ideas that can be manipulated easily by undergraduate students, and thus may be of interest for Bachelors in science by providing them a clear idea on what wavelets are, and thus permitting them to decide in their scientific future. Organization of the book

Chapter 2 presents the notion of wavelets as analyzing functions and as mathematical tools for analyzing square integrable functions known in signal theory as finite variance and/or finite energy signals. The analysis passes through two essential types of transforms: the continuous wavelet transforms and the discrete wavelet transforms. Such transforms are applied to represent the analyzed signals by means of wavelet series in the time–frequency domain and thus use the modes generating such series to localize singularities. Furthermore, we review multi-resolution analysis as a basic construction tool related to wavelets. It allows to split the whole space of analyzed signals into sub-spaces known as approximation spaces and detail ones. Multi-resolution analysis is based on nested sub-spaces that are related to each other by means of specific algorithms such as the decomposition low-pass filter-based algorithm and the inverse high-pass one. These algorithms are efficient for understanding the behavior of a series and eventual predictions.

4  Wavelet Analysis: Basic Concepts and Applications

Chapter 3 is a mixture of different concepts. We aim precisely to develop it as a continuation of the previous one in order to show some extensions of wavelet theory and also some other point of view to its theoretical introduction, essentially affine group method, and also to show to the readers that although manifolds are parts of Euclidean spaces for many cases, the concept of introducing wavelets on them may differ. We considered the special case of spheres. Chapter 4 is a new developed academic reference devoted to Clifford wavelet analysis. It offers a general context of Euclidean wavelet analysis by a higher-dimensional analogue. The notion of monogenic function theory is reviewed: monogenic polynomials and their application to yield Clifford wavelets. Mathematical formulations of harmonic analysis such as Fourier–Plancherel and Parseval are established in the new context. Applications in image processing are also developed. Readers will notice clearly that Clifford wavelet theory induces naturally the Euclidean ones such as real and complex circles and spheres. Chapter 5 is a continuation of the development of the theory in specialized fields such as quantum theory. We precisely present in detail quantum and fractional Bessel functions and associated wavelet theory. Plancherel/Parseval as well as reconstruction formula has been investigated. Bessel wavelets are applied in various domains, especially partial differential equations, wave motion, diffusion, etc. Next, in Chapter 6, statistical application of wavelets is reviewed. Density estimation, thresholding concepts, and variance and covariance topics are discussed in detail. Recall that statistical and time series constitute a very delicate area of study due to specific characteristics. Most of the time-varying series are nonlinear, in particular, the financial and economic series, which present an intellectual challenge. Their behavior seems to change dramatically, and uncertainty is always present. To understand and to discover hidden characteristics and behavior of these series, wavelet theory has been proved to be useful and necessary compared to existing previous tools in statistics. Chapter 7 is devoted to wavelets applied in solving partial differential equations. Recall that this field needs many assumptions on the functional bases applied, especially the explicit form of the basis elements and their regularities. We propose to show the contribution of wavelet theory in PDEs solving. The idea is generally not complicated, and it consists essentially of developing the unknown solution of the PDE into its eventual wavelet series decomposition (and its derivatives included in the PDE) and then using the concept of wavelet basis to obtain algebraic and/or matrices/vector equations on the wavelet coefficients that should be resolved. The crucial point in this theory is the so-called connection coefficients of wavelets. These have been investigated by many authors. In the present work, we develop a somehow new procedure to compute them. Some applications are also provided at the end of the chapter. The last chapter is devoted to the link and/or the use of wavelet theory in characterizing fractal and multifractal functions and their applications. Concepts such as H¨older regularity, spectrum of singularity, multifractal formalism for function and self-similar-type functions based on wavelets are discussed with necessary developments.

CHAPTER

2

Wavelets on Euclidean Spaces

2.1

INTRODUCTION

Wavelets were discovered in the eighteenth century in petroleum exploration, and since their discovery, they have proven to be powerful tools in many fields from pure mathematics to physics to applied ones such as images, signals, medicine, finance and statistics. The study of their constructions and their properties, especially in functional/signals decompositions on functional wavelet bases, has indeed grown considerably. In pure mathematics, wavelets constitute a refinement of Fourier analysis as they compensate and/or resolve some anomalies in Fourier series. The first wavelet basis has been, in fact, used before the pure mathematical discovery of wavelet bases since the introduction of Haar system, which by the next has been proved indeed to be a possible wavelet basis reminiscent of the regularity. Such a system dates back to the beginning of the 20th century and was precisely discovered in the year 1909. It was introduced in order to construct a functional basis permitting the representation of all continuous functions by means of a uniformly convergent series. Recall that in Fourier series, there are, as usual, many kinds of convergence that may be investigated such as the point-wise convergence subject of the well-known Dirichlet theorem, the uniform convergence which needs more assumptions on the series and the function, the convergence in norm and Carleson’s or almost everywhere convergence. Each kind of convergence requires special assumptions on the function. Although a Fourier series converges in sense of Dirichlet, it does not imply that the graph of the partial Fourier series converges to that of the function. This phenomenon is known as the Gibbs phenomenon and is related to the presence of oscillations in the Fourier series near the discontinuity points of the function. This means that the uniform convergence is not sufficient. One of the challenging concepts in wavelet analysis is its ability to describe well the behavior of the analyzed function near its singularities and join or more precisely extend the notion of Littlewood-Palay decomposition. To overcome some drawbacks of Fourier analysis, mathematicians have introduced a bit of modification called the windowed Fourier transform by computing the 5

6  Wavelet Analysis: Basic Concepts and Applications

original Fourier transform of the analyzed signals on a special localized extra function called the window. However, some situations remain non resolved especially with the emergence of irregular signals or high-frequency variations. The major problem in the windowed Fourier extension is due to the use of fixed window, which may not be well adapted to other problems such as high fluctuations of non stationary signals. This led researchers to think about a stronger tool taking into account nonlinear algorithms, nonstationary signals, nonperiodical, volatile and/or fluctuated ones. It holds that wavelets, since their discovery, have permitted to overcome these obstacles. These powers are related simultaneously to many properties of wavelets. Indeed, wavelet decomposition of functions joins Littlewood-Paley decomposition in many cases. Wavelets provide simultaneous local analyses related to time–frequency. They can be adapted to study-specific operators, especially differential and stochastic ones. From the numerical and/or applied point of view, wavelets provide fast and accurate algorithms, multi-resolution analyses as well as recursive schemes. These are very important especially in big data analysis, image processing and also in numerical resolution of partial differential equations. In this chapter, we propose to review the basic concepts of wavelets as well as their basic properties.

2.2

WAVELETS ON R

The first wavelet bases and thus analyses have been constructed on the real line R and have been next extended to the general cases of the real/complex Euclidean spaces Rm -Cm using different methods such as the natural tensor product. Mathematically speaking, a wavelet or an analyzing wavelet on the Euclidean space Rm may be defined in a large way as a function with specific properties that may or may not be required necessarily as simultaneous assumptions. More precisely, wavelet analysis is based primarily on the following points: • An effective representation for standard functions, • Robustness to the specification models, • A reduction in the computation time, • Simplicity of the analysis, • An easy generalization and efficient, depending on the dimension, • A location in time and frequency. A wavelet is a function ψ ∈ L2 (R) that satisfies the following conditions: • Admissibility, Z

2 ˆ |ψ(ω)|

R+

dω = Cψ < ∞. |ω|

(2.1)

• Zero mean, Z

+∞

b ψ(0) =

ψ(u)du = 0. −∞

(2.2)

Wavelets on Euclidean Spaces  7

• Localization in time/frequency domains Z

+∞

b ψ(0) =

|ψ(u)|2 du = 1.

(2.3)

−∞

• Enough vanishing moments, p = 0, ..., m − 1,

Z

ψ(t)tp dt = 0.

(2.4)

R

To analyze a signal by wavelets, one passes as in Fourier analysis by the wavelet transform of the signal. A wavelet transform (WT) is a re representation of a time– frequency signal. It replaces the Fourier sine by a wavelet. Generally, there are two types of processing: the continuous wavelet transform (CWT) and the discrete wavelet transform. 2.2.1

Continuous wavelet transform

The CWT is based firstly on the introduction of a translation parameter u ∈ R and another parameter s > 0 known as the scale to the analyzing wavelet ψ, which plays the role of Fourier sine and cosine and will be subsequently called mother wavelet. The translation parameter determines the position or the time around which we want to assess the behavior of the signal, while the scale factor is used to assess the signal behavior around the position. That is, it allows us to estimate the frequency of the signal at that point. Let   1 x−u ψs,u (x) = √ ψ . (2.5) s s The CWT at the position u and the scale s is defined by Z

du,s (f ) =

∞

−∞

ψu,s (t)f (t)dt,

∀ u, s.

(2.6)

By varying the parameters s and u, we may cover completely all the time–frequency plane. This gives a full and redundant representation of the whole signal to be analyzed (see [295]). This transform is called continuous because of the nature of the parameters s and u that may operate at all levels and positions. The original signal S can be reproduced knowing its CWT by the following relationship: S(t) =

1 Cψ



Z Z

du,s (S)ψ R

x−u s



dsdu . s2

(2.7)

It remains to notice that CWT is suitable for continuous time signals and those representing varying singularities. The function ψ ∈ L2 (R) satisfies some conditions such as the admissibility condition and somehow describes Fourier-Plancherel identity and says that Z R+

2 ˆ |ψ(ω)|

dω = Cψ < ∞. |ω|

(2.8)

8  Wavelet Analysis: Basic Concepts and Applications

The function ψ has to also satisfy a number of vanishing moments, which is related in wavelet theory to its regularity order. It states that p = 0, ..., m − 1,

Z

ψ(t)tp dt = 0.

(2.9)

R

Sometimes, we say that ψ is C m on R. The time localization chart is a normalization form that is resumed in the identity Z

+∞

|ψ(u)|2 du = 1.

(2.10)

−∞

To analyze a signal by wavelets, one passes via the so-called wavelet transforms. A wavelet transform is a representation of the signal by means of an integral form similar to Fourier one in which the Fourier sine and/or cosine is replaced by the analyzing wavelet ψ. In Fourier transform, the complex exponential source function yields the copies eis. indexed by s ∈ R, which somehow represent frequencies. This transform is continuous in the sense that it is indexed on the whole line of indices s ∈ R. In wavelet theory, the situation is more unified. A CWT is also well known. Firstly, a frequency, scale or a dilation or compression parameter s > 0 and a second one related to time or position u ∈ R have to be fixed. The source function ψ known as the analyzing wavelet is next transformed to yield some copies (replacing the eis. ):   1 x−u ψs,u (x) = √ ψ . (2.11) s s The CWT of a real-valued function f defined on the real line at the position u and the scale s is defined by Z

ds,u (f ) =

∞

−∞

f (t)ψs,u (t)dt,

∀ u, s.

(2.12)

By varying the parameters s and u, we cover completely all the time–frequency plane. This gives a full and redundant representation of the whole signal to be analyzed (see [295]). This transform is called continuous because of the nature of the parameters s and u that may operate at all levels and positions. So, wavelets operate according two parameters: the parameter u, which permits translation of the graph of the source mother wavelet ψ, and the parameter s, which permits compression or dilation of the graph of ψ. Computing or evaluating the coefficients du,s means analyzing the function f with wavelets. Theorem 2.1 The wavelet transform ds,u (f ) possesses some properties such as 1.

The linearity, in the sense that ds,u (αf + βg) = αds,u (f ) + βds,u (g), ∀f, g.

Wavelets on Euclidean Spaces  9

2.

The translation-invariance, in the sense that ds,u (τt f ) = ds,u−t (f ), ∀f ; and ∀u, s, t, and where (τt f )(x) = f (x − t).

3.

The dilation-invariance, in the sense that 1 ds,u (fa ) = √ das,au (f ), ∀f ; and ∀u, s, a, a and where for a > 0, (fa )(x) = f (ax).

The proof of these properties is easy and the readers may refer to [163] for a review. In wavelet theory, as in Fourier analysis theory, the original function f can be reproduced via its CWT by an L2 -identity. Theorem 2.2 For all f ∈ L2 (R), we have the L2 -equality 1 f (x) = Cψ

Z Z

ds,u (f )ψ(

x − u dsdu ) 2 . s s

The proof of this result is based on the following lemma: Lemma 2.3 Under the hypothesis of Theorem (2.2), we have Z Z

ds,u (f )ds,u (g)

dsdu = Cψ s

Z

f (x)g(x)dx, ∀ f, g ∈ L2 (R).

Proof. We have 1 ds,u (f ) = f ∗ ψs (u) = √ s

Z

f (x)ψ(

x−u 1 −iuy
ˆ )dx = { fˆ(y)ψ(sy)e . s 2π

Consequently 1 ds,u (f )ds,u (g)du = 2π u

Z

Z

2 ˆ fb(y)ˆ g (y)|ψ(sy)| dy.

y

By application of Fubini’s rule, we get Z

Z

ds,u (f )ds,u (g) s>0

u

dsdu s

1 2 dsdy ˆ fˆ(y)ˆ g (y)|ψ(sy)| 2π s>0 y s Z 1 Cψ fˆ(y)ˆ g (y)dy 2π y Z

= =

Z

Z

= Cψ

f (y)g(y)dy. y

10  Wavelet Analysis: Basic Concepts and Applications

Proof of Theorem 2.2. By applying the Riesz rule, we get ||F (x) −

1 Cψ

Z

=

sup ||G||=1

Z

Z |b|≤B

1/A≤a≤A

1 F (x) − Cψ

Ca,b (F )ψ(

x − b dadb ) 2 ||L2 a a !

x − b dadb ) 2 G(x)dx. Ca,b (F )ψ( a a |b|≤B

Z

Z

1/A≤a≤A

Next, using Fubini’s rule, we observe that the last line is equal to 1 Cψ

Z

F (x)G(x)dx −

sup ||G||=1

=

1 ||G||=1 Cψ

Z

Z

1/A≤a≤A

|b|≤B

Z

sup

(a,b)∈[1/A,A]×[−B,B] /

Ca,b (F )Ca,b (G)

Ca,b (F )Ca,b (G)

dadb a



dadb a

which by Cauchy–Schwarz inequality is bounded by 1 Cψ

"Z (a,b)∈[1/A,A]×[−B,B] /

"

|Ca,b (F )|

2 dadb

#1/2

a 2 dadb

Z

sup / ||G||=1 (a,b)∈[1/A,A]×[−B,B]

|Ca,b (G)|

a

1/2

.

Now, Lemma 2.3 shows that the last quantity goes to 0 as A, B tends to +∞. 2.2.2

Discrete wavelet transform

To analyze statistical series or discrete time signals and avoid redundancy problems and integrals calculations appearing in the CWT, one makes use of the discrete wavelet transform. It is to restrict to discrete calculations grids for scale parameters and position instead of browsing the entire domain. The most used method is dyadic grid based on taking s = 2−j and u = k2−j . In this case, the wavelet copy ψu,s will be denoted by ψj,k and defined by ψj,k (x) = 2−j/2 ψ(2j x − k),

j, k ∈ Z.

The discrete wavelet transform will be defined by Z

dj,k =

∞

−∞

ψj,k (t)S(t)dt.

(2.13)

These are often called wavelet coefficients or detail coefficients of the signal S. It holds that the set (ψj,k )j,k∈Z constitutes an orthonormal basis of L2 (R) and it is called wavelet basis. A signal S with finite energy (in L2 (R)) is then decomposed according to this basis into a series S(t) =

∞ X X

dj,k ψj,k (t)

(2.14)

j=0 k

called the wavelet series of S which replaces the reconstruction formula for the CWT.

Wavelets on Euclidean Spaces  11

2.3

MULTI-RESOLUTION ANALYSIS

Multi-resolution analysis (MRA) is a functional framework for representing a series of approximations to different levels called resolutions. MRA is a family of vector spaces (Vj )j∈Z of the whole space of finite variance (energy) signals L2 (R) nested in the sense of a scaling law. Recall that L2 (R) = {S : R −→ R;

Z

|S(t)|2 dt < ∞}.

(2.15)

R

For each j ∈ Z, Vj is called the approximation space at the scale or the level j. More precisely, we have the following definition ([295]). Definition 2.4 A multiresolution analysis is a sequence of closed subsets (Vj )j∈Z of L2 (R) that satisfies the following points. a) ∀j ∈ Z; V0 ⊂ V1 ⊂ .... ⊂ Vj ⊂ Vj+1 . b) ∀j ∈ Z; f ∈ Vj ⇔ f (2.) ∈ Vj+1 c) There exists ϕ ∈ V0 such that {ϕ0,k = ϕ(. − k); k ∈ Z} is a Riesz basis of V0 . d)

\

Vj = {0}.

j∈R

e)

[

Vj = L2 (R).

j∈R

f) ∀j ∈ Z; f ∈ Vj ⇐⇒ f (x − k) ∈ Vj The property (a) reflects that the approximation of a signal at the resolution’s level j + 1 contains the necessary information to yield the approximation at the level j. Assertion (b) called dilatation’s property permits passing from a level of resolution to another. The space Vj+1 contains signals that are coarser than Vj . Assertion (c) means that a scaling function exists and permits the decomposition of the signal at the starting level 0. The property (d) means that at very low resolution level (2−j −→ 0 as j −→ +∞) we lose all the details of the signal. At a minimal resolution, we lose all the information about the signal. Assertion (e) implies that the signal may be approximated with elements in Vj . At a maximal resolution, we reconstruct all the whole signal. Finally, the property (f) of translation means that the space Vj is invariant under integer translation. Definition 2.5 The source function ϕ is called the scaling function of the MRA or also the father wavelet. It holds that this function generates all the subspaces Vj ’s of the MRA by acting as dilation/translation parameters. Indeed, the property (d) combined with   (f) in j/2 j Definition 2.4 implies that, for all j, ∈ Z, the set ϕj,k (x) = 2 ϕ(2 x − k) is an k orthogonal basis of Vj . From assertion (a), it holds that we may complete Vj in Vj+1

12  Wavelet Analysis: Basic Concepts and Applications

in the sense of the direct sum. Let Wj be the orthogonal supplementary of Vj in Vj+1 , that is Vj+1 = Vj ⊕⊥ Wj . (2.16) We will see that Wj plays a primordial role in representing the details of the analyzed signal. This is why it is called the detail space at the level j. Iterating the relation (2.16), we obtain an orthogonal decomposition for all J ∈ Z, VJ = V0 ⊕⊥

J,⊥ M

Wj .

(2.17)

j=0

By exploiting (c), this leads to L2 (R) =

M

Wj .

(2.18)

j∈Z

L2 (R) is decomposed into subspaces that are mutually orthogonal. Definition 2.6 The space Wj , j ∈ Z, is called detail space at the scale or the level j. In wavelet theory, the following result is proved. Theorem and Definition 2.7 There exists a function ψ ∈ W0 that satisfies • (ψ(t − k))k is an orthogonal basis of W0 . •





ψj,k (t) = 2j/2 ψ(2j t − k)

k

is an orthogonal basis of Wj .

The function ψ is called the wavelet function or the mother wavelet associated with the scaling function ϕ of the MRA. We will now explain the relationship between ϕ and ψ. It holds from (2.18) that any S ∈ L2 (R) is decomposed into components according to the subspaces Wj ’s in the sense that S=

+∞ X

S Wj ,

j=−∞

where the SWj for j ∈ Z designates the orthogonal projection of S on W . Recall  j next that Wj is generated by the orthogonal basis ψj,k (t) = 2j/2 ψ(2j t − k) for any k j ∈ Z. Consequently, X SWj = < S, ψj,k > ψj,k . k

Which leads to

+∞ X

S=

X

< S, ψj,k > ψj,k .

j=−∞ k

Observing now that the wavelet coefficients are already defined by (2.12) or (2.13) as Z

dj,k (S) =

+∞

−∞

S(t)ψj,k (t)dt =< S, ψj,k >

Wavelets on Euclidean Spaces  13

we obtain the wavelet series of S defined already by (2.14). Returning next to (2.16) or also (2.17), we may write that S=

−1 X X

< S, ψj,k > ψj,k +

j=−∞ k

+∞ XX

< S, ψj,k > ψj,k .

(2.19)

j=0 k

It results from the nesting property of the Vj ’s and the fact that Wj ⊂ Vj+1 that the first part in (2.19) is an element of V0 which is generated by the basis (ϕk (t) = ϕ(t − k))k . Consequently, we may also write that −1 X X

< S, ψj,k > ψj,k =

X

j=−∞ k

Ck ϕk .

k

Hence, the wavelet series decomposition of S becomes S(x) =

X

Ck ϕk (x) +

∞ X X

dj,k ψj,k (x).

(2.20)

j=0 k

k

In fact, it may truncate at any level J and apply the ϕJ,k instead of ϕk to obtain S(x) =

X

CJ,k ϕJ,k (x) +

∞ X X

dj,k ψj,k (x)

(2.21)

j=J k

k

which is known as the wavelet series decomposition of S at the level J. In fact, the coefficients that appears in this decomposition are evaluated via the relation Z

CJ,k =

+∞

−∞

S(t)ϕJ,k (t)dt

and are called the scaling or approximation coefficients of S. The first component in VJ reflects the global behavior or the tendency of the whole signal S and the second component relative to the dj,k ’s represents the details of S and thus reflects the dynamic behavior of the signal.

2.4

WAVELET ALGORITHMS

The strongest point in MRA and wavelet theory is that the scaling function and the analyzing wavelet lead each one to the other. Indeed, recall that ϕ belongs to V0 ⊂ V1 and the latter is generated by the basis (ϕ1,k )k . Hence, ϕ is expressed by means of (ϕ1,k )k . More precisely, we have the following result. Theorem and Definition 2.8 • The scaling function satisfies the so-called two-scale relation √ X ϕ(x) = 2 hk ϕ(2x − k) (2.22) k

where the coefficients hk are Z

hk = R

ϕ(x)ϕ(2x − k)dx.

14  Wavelet Analysis: Basic Concepts and Applications

• The mother wavelet ψ is expressed as √ X ψ(x) = 2 gk ϕ(2x − k) k

where the gk ’s are evaluated by gk = (−1)k h1−k . For more details, we refer to [177], [236], [295]. We will now prove that these relations allow obtaining all the decomposition of a signal from each other through specific algorithms. Indeed, consider a signal S and its approximation coefficients Cj,k and details dj,k . For j ∈ Z, we have Cj,k = hS, ϕj,k i. It follows from the two-scale relation (2.22) that ϕj,k =

X

hl ϕj+1,l+2k .

l

Hence, Cj,k =

X

hl Cj+1,2k+l .

l

This means that the approximation at level j is obtained from the level j + 1 by the intermediate of a filter. We have, in fact, the following definition. Definition 2.9 The sequence H = (hk )k is called the discrete low-pass filter. Analogously, we have for j, k fixed, ψj,k (x) =

X

gl ϕj+1,l+2k .

Hence, dj,k = hS, ψj,k i =

X

gl Cj+1,l+2k .

l

This means that the approximation at level j is obtained from the level j + 1 by the intermediate of a filter. We have here also the following definition. Definition 2.10 The sequence G = (gk )k is called the discrete high-pass filter. Figure 2.1 presents the decomposition algorithm due to Mallat [295]. It consists of a cascade algorithm permitting to obtain all the levels of resolution. We will now explain the inverse algorithm which permits to obtain the level j + 1 from the level j. Recall that the orthogonal projection of S on the approximation space Vj is given by (Hardle et al (1997) [236]) SVj =

X k

Cj,k ϕj,k +

X k

dj,k ψj,k

Wavelets on Euclidean Spaces  15

Cj+1

Figure 2.1

H

G

↓2

↓2

Cj

dj

H

G

↓2

↓2

Cj−1

dj−1

Decomposition algorithm.

and that Cj+1,k = < SVj , ϕj+1,k > X X dj,k < ψj,k , ϕj+1,k > Cj,k < ϕj,k , ϕj+1,k > + = k

k

or else Cj+1,n =

X

hn−2k Cj,m +

k

X

gn−2k dj,m .

k

This reconstruction is illustrated by means of Figure 2.2. Finally, we recall some properties related to the filters H and G ([295]). Proposition 2.11 The filters H and G satisfy 1.

X

hn hn+2j = 0;

∀j 6= 0.

n

2.

X

h2n = 1.

n

3. gn = (−1)n hl−n ; ∀ n. 4.

X n

hn gn+2j = 0, ∀ j ∈ N. (called mutual orthogonality).

(2.23)

16  Wavelet Analysis: Basic Concepts and Applications

Cj−1

dj−1

↑2

↑2

H

G

Cj

dj

↑2

↑2

H

G

Cj+1 Figure 2.2

2.5

Inverse algorithm.

WAVELET BASIS

Providing functional spaces with bases is the most important task in functional analysis. It permits in some sense to reduce the problem of proving properties and/or demanding characteristics from a function—usually unknown—to doing it and/or demanding it from the elements of the basis. In finite dimensional spaces, this is somehow equivalent. In general and especially for almost all functional spaces where the dimension is usually infinite, we intend that the basis satisfy more properties to permit the reduction. In wavelet bases, for example, one of the important properties required is the localization and regularity of the mother wavelet ψ, which is obviously inherited by all the elements ψj,k . Wavelet bases may also be related to the concept of multiresolution analysis where an important property is also required and looks like the localization and consists of the existence of Riesz basis to define a multiresolution analysis. Recall that in a separable Hilbert space, a countable collection (ek )k∈Z is said to be a Riesz basis if all its finite linear combinations are dense in the whole Hilbert space and in addition if it satisfies an equivalence inequality of the form K1

X k

|λk |2 ≤ k

X k

λk ek k2 ≤ K1

X k

|λk |2

Wavelets on Euclidean Spaces  17

for some positive constants K1 ≤ K2 . The following result provides a characterization of the scaling function—father wavelet—of a multiresolution analysis to be effectively a good candidate. Proposition 2.12 Let ϕ ∈ L2 (R) and denote Γϕ (ξ) =

X

b + 2kπ)|2 . Then, the |ϕ(ξ

k∈Z

collection (ϕ(x − k))k∈Z is a Riesz basis for V0 if and only if 0 < inf Γϕ ≤ Γϕ (ξ) ≤ sup Γϕ < ∞, ∀ξ. Proof 2.1 Assume that the collection X (ϕ(x − k))k∈Z is a Riesz basis for V0 and let cϕ (ξ) = (λk )k∈Z ∈ `2 (R). Denote Fϕ (x) = λk ϕ(x − k). Its Fourier transform is F k∈Z

X

ikξ

λk e

b ϕ(ξ). We next have

k∈Z

cϕ k2 2 kF

Z

L (R)

=

|

X

R

= =

XZ

2 b λk eikξ ϕ(ξ)| dξ

k 2π(l+1)

|

l∈Z 2πl X Z 2π 0

=

0

2 b λk eikξ ϕ(ξ)| dξ

k

|

Zl∈Z2π X

|

X

X

b + 2πl)|2 dξ λk eikξ ϕ(ξ

k

X

λk eikξ |2 (

k

b + 2πl)|2 )dξ. |ϕ(ξ

l∈Z

As a result, kFϕ k2L2 (R)

1 1 c 2 kFϕ kL2 (R) = = 2π 2π

Z

2π

0

|

X

λk eikξ |2 Γϕ (ξ)dξ.

k

1 X Now, observing that the map Φ : λ = (λk ) → Φ(λ) = √ λk eikξ is an iso2π morphism from `2 (R) topL2 ([0, 2π]), to get the proposition, the map K defined on L2 ([0, 2π]) by K(f ) = f Γϕ should also be an isomorphism. This is the case if and 1 only if Γϕ and are both bounded (in L∞ ([0, 2π])). Γϕ Sometimes we need more regularity on the wavelet basis especially in the case of partial differential equations. Besides, we sometimes need compactly supported wavelets and/or wavelets with fast decay which are more adaptable and suitable for boundary and/or limit conditions due to partial differential equations. It is sometimes necessary to assume that the scaling function ϕ and its derivatives to an order r (usually the order of the MRA) are of fast decay. This permits to avoid the assumption on the overlap function Γϕ of ϕ to be upper bounded, as this characterization becomes already holding. Indeed, whenever ϕ is of fast decay, its Fourier transform ϕb is therefore infinitely continuously differentiable and is also of fast decay. It satisfies, in particular, b |ϕ(ξ)| ≤

K 1 + |ξ|

18  Wavelet Analysis: Basic Concepts and Applications

for some constant K. This leads to an upper-bounded overlap function Γϕ . As a consequence of these facts, we may assume that Γϕ ≡ 1. This results in a correspondence between `2 (R), which are r-order fast decay, and the elements of V0 , which are fast decay, as well as their derivatives to the order r. In some other applications, one may need more assumptions on the wavelet basis, especially in numerical analysis where the exponential decay behavior is more suitable than just fast decay. It holds, in fact, that whenever the sequence (λk )k is exponentially decaying in `2 (R), the associated function Fϕ is of exponential decay in V0 . More generally, we may prove the following result. Lemma 2.13 Let m(ω) =

X

λk eikω . Then, m is analytic in some annulus |Imω|
0 be such that |ϕ(x) − ϕ(x0 )| ≤  whenever |x − x0 | ≤ η. Therefore, X ϕ(x0 )ϕ(2j x − k) ϕ(x0 ) − Iϕ

X

≤ Iϕ | kj 2

k

−x0 |≤η

|

k 2j

k )| + |ϕ(x0 )| |ϕ(2j x − k)| 2j 



X

+

|ϕ(2j x − k)| |ϕ(

−x0 |>η j

+C + (2−3 2 ). η k Observe next whenever |x − x0 | ≤ then we necessarily get | j − x0 | > η, which 2 2 k η implies that | j − x| ≥ . Hence, 2 2 |ϕ(2j x − k)| ≤

C (1 +

|2j x

−

k|)4

≤

(1 +

|2j x

C . − k|)2 (2j η2 )2

As a consequence, X X 1 j −3 2j ϕ(x0 )ϕ(2 x − k) ≤ C ) ϕ(x0 ) − Iϕ η 2 + C + O(2 j 2 j (1 + |2 x − k|) (2 2 ) k k j

≤ C2−2j η −2 + C + O(2−3 2 ).

20  Wavelet Analysis: Basic Concepts and Applications

It follows consequently that for j large enough X j ϕ(x0 )ϕ(2 x − k) ≤ C. ϕ(x0 ) − Iϕ k

This yields that M 6= 0, and whenever |y − 2j x0 | ≤ 2j η2 , we get X 1 ϕ(y − k) − ≤ C. Iϕ k

X

Now observe that the function y 7−→

ϕ(y − k) is 1-periodic. Hence, for j large

k

enough, we get that such a function remains constant on the whole line R, which means that X 1 ϕ(y − k) = . M k This yields Z

+∞

ϕ(y)dy =

M= ∞

XZ

k+1

Z

1

(

ϕ(y)dy =

k∈Z k

0

X

ϕ(y − k))dy =

k∈Z

1 . M

Hence, M 2 = 1 and \ thus may be chosen to be equal to 1. We now prove that Vj = {0}. Let F be an element in the intersection. Hence, for j

all j, we may write F =

X

aj,k ϕj,k ,

k

where the aj,k are uniformly bounded. Moreover, kF k2 =

X

|aj,k |2 .

k

Therefore, for all j we get |F | ≤ C

X

j

|ϕj,k | ≤ C2 2

k

which means that F = 0. It remains now to show assertion (e) in Definition 2.4. To do this, we prove that any piecewise constant function may be approximated by elements of the union of the Vj ’s. So, let [a, b] be an interval of R and let for j ∈ Z the function X

Fj = k 2j

2j/2 ϕj,k .

∈[a,b]

We immediately observe that for all x we have X

|Fj (x)| ≤ C k 2j

∈[a,b]

1 C ≤ j . (1 + |2j x − k|2 ) 2 dist(x, [a, b])

Wavelets on Euclidean Spaces  21

On the other hand, X

|1 − Fj (x)| = | k 2j

c . 2j dist(x, [a, b]c )

ϕ(2j x − k)| ≤

∈[a,b] /

Now observe that Z

2

|χ[a,b] (x) − Fj (x)| dx =

Z j dist(x,[a,b]c )≥2− 2

|χ[a,b] (x) − Fj (x)|2 dx

Z

+ Zdist(x,[a,b])≥2

+

−

j 2

|χ[a,b] (x) − Fj (x)|2 dx j

dist(x,∂[a,b])≤2− 2

|χ[a,b] (x) − Fj (x)|2 dx.

Next, observing the estimations above we obtain Z

j

|χ[a,b] (x) − Fj (x)|2 dx ≤ C2−2j 2j + C2−2j 2j + C2− 2 .

Consequently, j

kχ[a,b] − Fj k22 ≤ C2− 2 . Letting j → ∞ we observe Fj goes to χ[a,b] in L2 .

2.6

MULTIDIMENSIONAL REAL WAVELETS

Many ideas have been exploited to introduce multidimensional wavelet analysis. Some are based on the adoption of multiresolution analysis on R to the multidimensional case. The first constructed bases were separable ones. Their construction focus on an analogy with the Haar one. Recall that in one-dimensional case, this basis is defined by ( ψj,k (x) = 2j/2 ψ(2j x − k) ; j, k ∈ Z ψ = ξ[0,1/2[ − ξ[1/2,1[ Generally, let (Vj1 ) be a multiresolution analysis of L2 (R) with a scaling function ϕ and a wavelet ψ, and let the orthogonal projection on Vj be denoted by Pj . The main idea in the multiresolution analysis is the ability to construct an orthonormal wavelet basis {ψj,k ; j, k ∈ Z}, ψj,k (x) = 2j/2 ψ(2j x − k), such that Pj+1 f = Pj f +

X

< f, ψj,k > ψj,k ∀ f ∈, L2 (R).

k

The adoption of these one-dimensional MRA and wavelets will be the starting point to construct the multidimensional case. Indeed, let (Vj1 ) be a multiresolution analysis of L2 (R) with a scaling function ϕ, a wavelet ψ and an orthogonal projections Pj1 . Consider then orthogonal projection Pjd in L2 (Rd ) defined as the tensor product of d copies of Pj1 Pjd = Pj1 ⊗ Pj1 ⊗ ... ⊗ Pj1 .

22  Wavelet Analysis: Basic Concepts and Applications

Denote Vjd = Pjd (L2 (Rd )). We have Vjd = Vj1 ⊗ Vj1 ⊗ ... ⊗ Vj1 . The closure in L2 (Rd ) of Vjd has an orthonormal basis (

ϕdj,k = 2jd/2 ϕd (2j x − k) ; j ∈ R , k ∈ Rd ϕd = ϕ ⊗ ϕ ⊗ ... ⊗ ϕ

The detail spaces Wjd will be defined by Wjd = ⊕ε6=(0,..,0) Vjε1 ⊗ Vjε2 ⊗ ... ⊗ Vjεd . This yields an orthonormal basis of L2 (Rd ) associated with Pjd (

ε ψj,k (x) = 2jd/2 ψ ε1 (2j x1 − k1 )...ψ εd (2j xd − kd ) ; j, ki ∈ Z, ψ 1 = ψ , ψ 0 = ϕ = ξ[0,1[ .

This last formula looks better than the one defined by tensor products and yields simple separable wavelets. In contrast, nonseparable wavelets remain difficult to be used and to construct. However, in analysis and in nature, one can speak about propagations in privileged directions. One plans to study their behavior by means of well-adapted wavelets. An important example of directional phenomena is supplied by spirals, such as the domain between the two curves of equations (in polar coordinates) r = θ−α

and

r = (θ + π)−α .

Another example that bears similarities with spirals is the set Cα =

[ n

1 1 , . α (2n + 1) (2n)α 

If the aim is a pointwise analysis, without particular emphasis on directions, the topic will be more economical. However, if the signal to be analyzed has a preferred direction, then one needs a wavelet with good angular selectivity.

2.7

EXAMPLES OF WAVELET FUNCTIONS AND MRA

In this section, we propose to present some examples of wavelets and associated MRA. The readers can be referred to [163], [248], [289], [295] for more details and examples of original wavelet analysis on the real line and Euclidean spaces in general. 2.7.1

Haar wavelet

The example of Haar is the simplest example in the theory of wavelets and MRA. It is based on the Haar scaling function explicitly given by ϕ = χ[0,1[ and characterized by the possibility of explicit computations of the transforms and coefficients. The basic approximation space is given by (

V0 =

f ∈ L2 (R) ; f =

) X k∈Z

ak ϕk ; (ak )k ⊂ R such that

X k∈Z

a2k < ∞ .

Wavelets on Euclidean Spaces  23

For k ∈ Z, we denote ϕk by the function ϕk (t) = ϕ0,k (t) = χ[0,1[ (t − k) = χ[k,k+1[ (t). Consequently, observing that the ϕk ’s have disjoints supports, any element S ∈ V0 may be written in the form f (t) = ak ; t ∈ [k, k + 1[. hence, V0 is the subspace of signals that are constant on intervals of the form [k, k+1[, k ∈ Z. By exploiting the assertion (d) in Definition 2.4, we obtain for any j ∈ Z, (

Vj =

) 2

f ∈ L (R) ; f =

X

aj,k ϕj,k ; (aj,k )k ⊂ R such that

k∈Z

X

a2j,k

0 be in N. The spline wavelets are defined inductively by ϕ0 = ϕ and ϕr = ϕr−1 ∗ ϕ, ∀r ≥ 1. Definition 2.15 The function ϕr is said to be the spline wavelet of order r. Obviously, the starting multiresolution space V0 will be the closure in L2 (R) of spann(ϕr (. − k))k∈Z and for j ∈ W the j-level approximation space Vj may be obtained by scaling the elements of V0 by 2j . In other words, F (.) ∈ Vj whenever F ( 2.j ) ∈ V0 and vice versa. We will show in the next part that spline wavelets may be seen as piecewise polynomial functions on intervals of the form [k, k + 1[, k ∈ Z with degrees at most the order r of the spline. More precisely, we have the following result. Proposition 2.16 Let Pr be the space of all (r − 1) continuously differentiable functions with their restriction on each interval [k, k + 1[, k ∈ Z, a polynomial of degree at most r. Then, ϕr ∈ Pr .

Wavelets on Euclidean Spaces  29

Proof 2.4 We proceed by recurrence on r. The function ϕ0 is constant. Therefore, it may be seen as a polynomial of degree 0. Assume next that on each interval [k, k + 1[, k ∈ Z, the function ϕr is a polynomial of degree at most r. Observing next that ϕr+1 = ϕr ∗ ϕ, it suffices to show that xr ∗ ϕ ∈ Pr . Indeed, for all x ∈ [0, 1] we explicitly have Z 1 r X (−1)r−l l r r (x ∗ ϕ)(t) = x (y − t) dy = Crl r+1−l 0 l=0 which is q polynomial of degree r. In fact, we may prove more properties of these wavelets such as the minimality of supports. We now investigate briefly the important assumption in the construction of multiresolution analysis stating that the collection ϕr (x − k), k ∈ Z, constitutes a Riesz basis of the space that they span. We thus have to show that there exist constants K1 , K2 with 0 < K1 < K2 < ∞ such that almost everywhere we have K1 ≤

X

cr (ω + 2kπ)|2 ≤ K2 . |ϕ

Indeed, it is straightforward that cr (ω) = ϕ

eiω − 1 iω

!r+1

.

X

cr (ω + 2kπ)|2 is periodic with period Notice now that the overlap function Γϕ = |ϕ 2π and it is upper bounded on [−π, π]. On the other hand, the function Θ(ω) = exp iω − 1 | | is lower bounded away from 0 on [−π, π]. So is the function Γϕ . As a iω consequence, the function ϕr yields a multiresolution on R.

2.7.6

Anisotropic wavelets

The first anisotropic wavelet is due to Morlet ([279]). Its basic function is 2 /2

ψ(X) = eiK.X e−kXk

which is in fact Gaussian modulated according to the direction of K. It is not, in any rigor, a wavelet; additional terms must be added to it for reason of oscillation. Note that it is not isotropic but has a privileged direction given by K. It is used to analyze images having anisotropic characteristics. A second example of anisotropic wavelets is the so-called Mexican hat ([279]). It is based on the function ψ(X) = (2 − XAX)e−XAX/2 where A is an unspecified symmetric matrix of M2 (R). It differs from that of Morlet due the fact that the anisotropy is in its module. If A = λI, the Mexican hat is of a radial symmetry. However, if the spectrum of the matrix A is not a singleton, the wavelet will be anisotropic.

30  Wavelet Analysis: Basic Concepts and Applications

2.7.7

Cauchy wavelets

Cauchy wavelets are one step in the direction of introducing spherical wavelets as they aim to take into account the angular behavior of the analyzed functions. In one-dimensional case, Cauchy wavelets are defined via their Fourier transform (

ψbm (ω) =

0 for ω < 0 m −ω ω e for ω ≥ 0.

with m > 0. In one dimension, the positive half-line is a convex cone. Thus, a natural generalization to two dimensions will be a wavelet whose support in spatial frequency space is contained in a convex cone with apex at the origin. Let C ≡ C(α, β) be the convex cone determined by the unit vectors eα , eβ , where α < β, β − α < π and for all θ, eθ ≡ (cos θ, sin θ). The axis of the cone is ξαβ = e α+β . In other words, 2



k ∈ R2 , α ≤ arg(k) ≤ β

C(α, β) =







k ∈ R2 , k.ξαβ ≥ eα .ξαβ = eβ .> 0 .

= The dual cone to C(α, β) is 



e C(α, β) = k ∈ R2 , k.k 0 > 0, ∀ k 0 ∈ C(α, β) . e Remark that C(α, β) may be also seen as b e b , β). C(α, β) = C(α

where b=β− α

π b π , β =α+ 2 2

and eα .eαb = eβ .eβb = 0. Thus, the axis of Ce is ξαβ . The two-dimensional Cauchy wavelet is defined via its Fourier transform ( C,η ψblm (k)

=

(k.eαe)l (k.eβe)m e−k.η , k ∈ C(α, β), 0 , otherwise.

where η ∈ Ce and l, m ∈ N∗ . Note that such a wavelet is also supported by the cone C. It satisfies the admissibility condition AψC,η ≡ (2π)2 lm

Z

C,η |ψblm (k)|2

d2 k < ∞. |k|2

The following result of Antoine et al is proved in [13] and yields an explicit form for the two-dimensional Cauchy wavelet.

Wavelets on Euclidean Spaces  31

Proposition 2.17 For even η ∈ Ce and l, m ∈ N∗ , the two-dimensional Cauchy C,η wavelet ψlm (x) with support in C belongs to L2 (R2 , dx) and is given by C,η ψlm (x) =

il+m+2 [sin(β − α)]l+m+1 l!m! . 2π [(x + iη).eα ]l+1 [(x + iη).eβ ]m+l

We can, with analogous techniques, define multidimensional Cauchy wavelets. See [13] and the references therein for more details.

2.8 EXERCISES Exercise 1.

√ Denote φ = χ[0,1[ and φjk (t) = 2j φ(2j t − k), t ∈ R. Denote further E = L2 ([0, 1[) the vector space of square-integrable functions on [0, 1[. Next for j fix V j , the vector k k+1 space of constant functions, on { j ; j } for k ∈ {0; 2j − 1}. 2 2 a. Show that (φjk )k∈{0,2j −1} is an orthonormal basis of V j . < (φjk ); (φjh ) >= 0∀(k, h) ∈ {0, 2j − 1} × {0, 2j − 1}. and that the elements φjk are unitary. b. Sketch the graphs of (φ1k )k∈{0,1 } and (φ2k )k∈{0,3 }. Exercise 2.

Consider the same assumptions as in Exercise 1 above. a. Find two functions (ψk1 )k∈{0,1} constituting an orthonormal basis of a vector space W 1 such that V 2 = V 1 ⊕ W 1 . b. Express the functions (ψk1 )k∈{0,1 } and (φ1k )k∈{0,1 } by means of (φ2k )k∈{0,3 }. c. Express conversely the functions (φ2k )k∈{0,3 } in terms of (φ1k )k∈{0,1 } and (ψk1 )k∈{0,1 }. d. Express the function ψk1 = ψ(2t − k). Exercise 3.

Consider the same assumptions as in Exercises 1 and 2 above. a. Provide the 2j Haar wavelets (ψk1 )k∈{0,1 } which constitute an orthonormal basis of W j = V j+1 V j . b. Write the (φjk ) and (ψkj ) by means of (ψkj+1 ). j j c. Write the (φj+1 k ) by means of (φk ) and (ψk ).

d. Write the approximation coefficients ajk and the detail coefficients (djk ) by means of the (aj+1 k ).

32  Wavelet Analysis: Basic Concepts and Applications j j e. Write the coefficients (aj+1 k ) by means of (ak ) and dk .

f. Compare the relation between the coefficients in V j ⊕ W j to the one in V j+1 . Exercise 4.

Consider the discrete signal S = [2 4 8 12 14 0 2 1] and the Haar multiresolution analysis on R. a. Decomposer S in V 0 ⊕2j=0 W j b. Sketch the graphs of S relatively to V 2 , V 1 and V 0 . b. Sketch the graphs of S relatively to the details W 2 , W l and W 0 Exercise 5. b Let ψ : R → C be wavelet in the Schwarz class such that ψ(0) = 0). Denote Z

|xψ(x)|dx.

C= R

Denote also for any function f : R → C, s > 0 and u ∈ R, 1 Wf (s, u) = √ s

x−u f (x)dx. ψ s R

Z





1. Let f : R → C be 1-Lipschitzian. Prove that for all s and u we have 3

|Wf (s, u)| ≤ Cs 2 . 2. a) Show that there exists φ : R → C such that ψ = φ0 and ψ(x) → 0 as |x| → ±∞. b) Let a ∈ R be fixed and consider f = 1[a,+∞] . Compute Wf by means of φ. 3. Let f : R → C be piecewise continuous with finite number of discontinuities − x1 < ... < xn and denote for k ≤ n αk = f (x+ k ) − f (xk ). Assume further that f is 1-Lipschitzian on any interval not containing any of the xk ’s. Sketch the graph of Wf (s, .) for s small enough. Exercise 6.

Let f, ψ : R → C be in the Schwarz class such that ψ(0) = 0) and denote Z

+∞

Z

ψs (t − u)Wf (u, s)du

F (t) = 0

R

dt . s2

Show that F = Cψ f , where Cψ is a constant depending on ψ and not on f . Exercise 7. b Let ψ be a wavelet such that ψ(0) = 0, where Z b ψ(w) = R

ψ(t)e−izt dt, ∀w ∈ R.

Wavelets on Euclidean Spaces  33

Show the following Heisenberg uncertainty inequality: 2 b kψk22 ≤ √ ktψ(t)k2 kξ ψ(ξ)k 2. 2π (Hint. We may use the derivative (tψ(t))0 ). Exercise 8.

Let f ∈ L2 (R) be such that fb(ξ) = 0 for ξ < 0. For z ∈ C such that I, (z) ≥ 0, let 1 fp (z) = π

+∞

Z

(iξ)p fb(ξ)eizξ dξ.

0

a) Prove that fp is the derivative of f at the order p whenever f is C p . b) Prove that fp is analytic on the upper half complex plane Im(z) > 0. c) Prove that for z = x + iy, y > 0, we have fp (z) = y −p−1/2 dy,x (f ), calculated with an analytic wavelet ψ that you will specify. Exercise 9.

Let for N ∈ N, f : [0, N ] → R be continuous and (Vm )m≥0 be a multiresolution analysis on [0, N ] with a scaling function ϕ. 1) Verify that for all ε > 0 there exists m ≥ 0 such that for all elements t1 , t2 ∈ [0, N ] with |t1 − t2 | < 2−m we have |f (t1 ) − f (t2 )| < ε. 2) Denote next tm,n+1/2 = (n + 21 )2−m and fε (t) =

N −1 2X

f (tm,n+1/2 )ϕm,n (t).

n=0

Prove that |f (t) − fε (t)| < ε, ∀t ∈ [n2−m , (n + 1)2−m ]. 3) Show that

√ kf − fε k2 ≤ ε N .

Exercise 10.

Part A. Consider the Haar system on R with its scaling function ϕ = χ[0,1[ and wavelet function ψ(x) = χ[0,1/2[ − χ[1/2,1[ . Let next f (t) = (t2 + 1)χ[−1,1[ and its projections on Vj and Wj be, respectively, fj =

X k

aj,k ϕj,k

and dj =

X

dj,k ψj,k .

k

a) Evaluate f0 (t), f1 (t), f2 (t), d0 (t) and d1 (t). b) Draw the graphs of f (t), f0 (t), f1 (t), f2 (t), d0 (t) and d1 (t).

34  Wavelet Analysis: Basic Concepts and Applications

Part B. Denote for (x, y) ∈ R2 φ(x, y) = ϕ(x)ϕ(y), Ψ0 (x, y) = ϕ(x)ψ(y), Ψ1 (x, y) = ψ(x)ϕ(y), Ψ2 (x, y) = ψ(x)ψ(y). a) Prove that the system (Φ, Ψi , i = 0, 1, 2) constitutes a scaling function and a wavelet on R2 . b) Express the approximation and detail spaces Vj and Wj associated. c) Let f (x, y) = cos(x) sin(x+y) on the cube C = [−π, π]×[−π, π] and its projections on Vj and Wj be, respectively, fj =

X k

aj,k Φj,k

and dij =

X

dj,k Ψij,k , i = 0, 1, 2.

k

a) Evaluate f0 (x, y), f1 (x, y), f2 (x, y), d0 (x, y) and d1 (x, y). b) Draw the graphs of f (x, y),f0 (x, y), f1 (x, y), f2 (x, y), d0 (x, y) and d1 (x, y).

CHAPTER

3

Wavelets extended

3.1

AFFINE GROUP WAVELETS

Wavelet transform of functions may be understood as a group action on the space of square integrable functions L2 (R) for suitable group. In this chapter, we aim to present the eventual link between wavelet transforms and the real affine group. To make the section self-contained and easy for readers from different fields, we firstly recall the basic definition of groups. Definition 3.1 Let G be a nonempty set. G is said to be a group if there exists an operation ∗ defined on G for which 1. x ∗ y ∈ G for all x, y ∈ G (known as closure property). 2. (z ∗ y) ∗ z = z ∗ (y ∗ 2) for all x, y, z ∈ G (known as associative property). 3. There is an element e ∈ G such that e ∗ x = x ∗ e = x for all x ∈ G (e is known as the identity element). e ∈ G such that x ∗ x e=x e ∗ x = e (x e is 4. For each x ∈ G, there is an element x called the inverse of x).

If, in addition, x ∗ y = y ∗ x for all x, y ∈ G, the group G is said to be commutative. A subset H of a group G is a subgroup of G if it is a group under the same operation as G. The group G may have additional structure. For instance, if G is a topological space and the operation ∗ is continuous on G × G with respect to this topology, G will be a topological group. Furthermore, if G has a differentiable structure, G will be a differentiable manifold, and is called a Lie group. A differentiable manifold, for our purpose, is a set that looks like Euclidean space. For example, Rn is one since it is Euclidean space. On Euclidean spaces, we can choose different parametrizations; however, the change of coordinates must be differentiable. Rectangular coordinates and polar coordinates are familiar examples of such parametrizations of Rn . We recall that change of coordinates must also be invertible locally, that is, the Jacobian of the transformation should be invertible. 35

36  Wavelet Analysis: Basic Concepts and Applications

In our case, we will focus on the so-called linear or affine group on Rn . These are groups that can be realized as groups of linear transformations of a vector space or, equivalently, as matrix groups. We will see that group representation theory may help explain the essential properties of wavelet transforms. Consider the connected affine group G+ , also called the ax + b group, consisting of transformations of R of the type x 7−→ gx = ax + b, x ∈ R, where a > 0, b ∈ R and where we denoted g = (b, a) ∈ G+ with the multiplication law gl g2 = (b1 + a1 b2 , a1 a2 ). dadb . a2 A nontrivial unitary irreducible representations of G on Hilbert space L2 (R, dx) will be 1 x−b (U (a, b)f )(x) = √ f ( ), f ∈ L2 (R, dx), (a, b) ∈ G. a a It holds that an invariant measure on G is dµ (a, b) =

For the sequel, we define the following operators on the set of functions defined on the Euclidean space R: • The translation: Tb f (x) = f (x − b), a ∈ R. √ • The dilation: Da f (x) = af (ax), a > 0. • The modulation: Ea f (x) = e2iπax f (x), a ∈ R. • The multiplication: Ma f (x) = e2iπa f (x), a ∈ R. These operators are easily seen to be unitary. Using these operators, we immediately deduce that (U (a, b)f ) = Da−1 Tb f. As a consequence, the wavelet transform of f at the scale a > 0 and the position b ∈ R may be expressed by means of the representation U as Cf (a, b) = hf, U (a, b)ψi = hf, Da−1 Tb ψi. This inverse transform will be expressed as f (•) =

1 hCf (a, b), U (a, b)ψ(•)idµ (a,b) , Aψ

where Aψ is the admissibility constant expressed as Aψ =

Z R

2 b |ψ(ξ)| . |ξ|

The results presented in this section may be extended in a usual way to multidimensional case.

Wavelets extended  37

3.2

MULTIRESOLUTION ANALYSIS ON THE INTERVAL

Different ways have been proposed to adapt the AMR of L2 (R) to L2 (]0, 1[). [68, 69, 70, 71, 72]. Most of these constructions are based on the idea of considering the scaling functions ϕj,k with supports in [0, 1] and adding suitable linear combinations of elements ϕj,k whose support crosses the left (respect. the right) boundary of the interval. 3.2.1 Monasse–Perrier c onstruction

Monasse and Perrier [304] have proceeded as follows: for all integer j ≥ j0 = log(4N ), ]0,1[ the space Vj is spanned by the following functions. • 2j/2 ϕ˜0k (2j x)|]0,1[ , k = 0, · · · , M, • ϕj,k K = N, · · · , 2j − N, • 2j/2 ϕ˜1k (2j (x − 1))|]0,1[ , k = 0, · · · , M, where the functions ϕ˜0k and ϕ˜1k are defined as linear combinations of ϕj,k with supports containing 0 or 1 and with suitable polynomial coefficients such as • ϕ˜0k (x) =

N −1 X

Pk0 (l)ϕ(x − l),

−N +1

• ϕ˜1k (x) =

N −1 X

Pk1 (l)ϕ(x − l).

−N +1 0 1 P00 , P10 , ..., PM and P01 , P11 , ..., PM are two bases of the space of polynomials of degree at most M . The functions ϕj,k , k = N, ..., 2j − N , whose supports are contained in [0,1], are called interior scaling functions. The functions ϕ˜0k and ϕ˜1k , k = 0, ..., M are called edge scaling functions.

3.2.2 Bertoluzza–Falletta construction

Recently, Bertoluzza and Faletta [70] have proposed a new construction of AMR on the interval by considering the following spaces Vjloc = Span < ϕj,k , k ∈ Z >L2loc (R) , where ϕ is the scaling function of an AMR in L2 (R) with compact support. Let Vj∗ be the subspace of Vjloc of functions with coefficients having polynomial behavior on some neighborhood of 0 or 1. The degrees of polynomials depend possibly on j. More precisely, they have proceeded as follows. For j ≥ 0, let Nj = 2j − 2N + 2M + 3, Mj = min(M, Nj − 1) and (

Vj∗ =

) X

k∈Z

fkj ϕj,k ; ∃Pl , Pr

∈

PMj , fkj

= Pl (k), k ≤ N −

1, fkj

= Pr (k), k ≥ 2j − N + 1 .

38  Wavelet Analysis: Basic Concepts and Applications

Vj∗ is well defined for all j ≥ ˆj0 = [log2 (2(N − M ) − 1)]. Compared to j0 of Monasse– Perrier, the integer ˆj0 depends directly on N − M . For example, if we consider the compactly supported Daubechies wavelets, then N − M = 1 and Vj∗ are defined for any j ≥ 0. Besides, the polynomials Pl and Pr of Vj∗ are not necessarily independent. In fact, for all j ≤ [log2 (2(N − 1) − M )], the polynomials are the same. Finally, the parameters Nj and Mj are, respectively, the dimensions of Vj∗ and the degree of polynomials exactly reproduced in Vj∗ . Proposition 3.2 The sequence (Vj∗ )j≥j0 satisfies ∗ 1/ Vj∗ ⊂ Vj+1 . 2/ PMj ⊂ Vj∗ . We will recall the important points in this construction. For j ≥ j0 , let Ij = {N − M − 1, N − M, ..., 2j − N + M + 1}. For a given sequence a = (ak )k∈Ij , we consider N −M −1+Mj

X

Pl (a, x) =

am LMj,m (x),

m=N −M −1 2j −N +M +1 X

Pr (a, x) =

am LMj,m (x),

m=2j −N +M +1−M

where

j

N −M −1+Mj

LMj,m (x) =

x−i , m − i m=N −M −16=i Y

2j −NY +M +1

LMj,m (x) =

m=2j −N +M +1−Mj6=i

x−i . m−i

The linear extension operator Ej : S(Ij ) −→ S(Z) is defined for all a = (ak )k∈Ij in 



S(Ij ) by Ej (a) = Ej,k (a)

Ej,k (a) =

, where k∈Z

   Pl (a, k)

, k ≤N −M −2 ak , N − M − 1 ≤ k ≤ 2j − N + M + 1   Pr (a, k) , k ≥ 2j − N + M + 2.

S(A) for A ⊆ Z stands for the set of real sequences indexed by A. Let εj be the X surjective operator which corresponds to f = fj,k ϕj,k of Vjloc , the element in Vj∗ k

defined by εj (f ) =

X k

where f j = (fj,k )k∈Z .

Ej,k (f j )ϕj,k |]0, 1[,

Wavelets extended  39

Proposition 3.3 For all j ≥ j0 , we have ]0,1[

Vj∗ |]0, 1[≡ Vj ]0,1[

For all j ≥ ˆj0 , Vj

.

is defined as the restriction of Vj∗ on ]0, 1[.

Proof 3.1 It follows immediately from a simple comparison of the dimensions of the two spaces. 3.2.3

Daubechies wavelets versus Bertoluzza–Faletta

In this section, we propose to apply the method of Bertoluzza and Faletta to Daubechies wavelets in order to obtain a wavelet basis on ]0, 1[. Let N be the larger integer such that |supp(ϕ)| ≥ 2N − 1. Without loss of generality, we can suppose that this support coincides with the interval [−N + 1, N ].The two-scaler relation takes the form N X

ϕ=

hk ϕ1,k

k=−N +1

and the wavelet ψ has the form N X

ψ=

hk ϕ1,k .

k=−N +1

Recall that in this link we have M = N − 1, Nj = 2j + 1, Mj = N − 1, ˆj0 = 0, Ij = {1, 1, ..., 2j }. The polynomial Pl (c, .) of degree M = N − 1 interpolates (c) at the nodes 0, ..., N − 1 and Pr (c, .) of the same degree M = N − 1 interpolates (c) at the nodes 2j − N + 1, ..., 2j . More precisely, we have Pl (c, x) =

N −1 X

cm L0M,m (x),

m=0

where L0M,m is the Lagrange polynomial of degree M . L0M,m (x) =

M Y

x−i . m−i i=0,i6=m j

2 X

Pr (c, x) =

cm L1M,m (x),

m=2j −N +1

where L1M,m is the Lagrange polynomial of degree M = N − 1, taking the value 1 at x = m and 0 at x = i 6= m ∈ {2j − N + 1, ..., 2j } j

L1M,m (x)

=

2 Y

x−i . m−i i=2j −N +16=m

40  Wavelet Analysis: Basic Concepts and Applications

The operator Ej : S(Ij ) → S(Z) becomes for all c = (ck )k in S(Ij ) 

   Pl (c, k)



Ej (c)

k

, k ≤ −1 ck , 0 ≤ k ≤ 2j =   P (c, k) , k ≥ 2j + 1 r

Via this operator, we define εj : Vjloc → Vj∗ , which associates with each element f=

X

fkj ϕj,k ∈ Vjloc , the image εj (f ) =

k

X

fkj ϕj,k .

k∈Ij

Thanks to the previous facts, the following definition will be consistent. ]0,1[

Definition 3.4 For all j ≥ 0, we define Vj to be the restriction of Vj∗ to the unit interval. That is, ]0,1[ Vj = Vj∗ |]0, 1[. ]0,1[

To construct a wavelet basis for Vj , we simply apply the operator εj to the subset of {ϕj,k } which cross the interval ]0, 1[. So that, we obtain for all k ∈ Ij , ϕj,k = |]0, 1[). In [70], it has been proven that for all f ∈ Vj∗ , f|]0,1[ =

XZ

f ϕj,k ϕ¯j,k .

R

k∈Ij

3.3 WAVELETS ON THE SPHERE 3.3.1

Introduction

Spherical wavelets are adapted for understanding complicated functions defined or supported by the sphere. The classical are essentially done by convolving the function against rotated and dilated versions of one fixed function ψ. Gegenbauer polynomials are also applied to yield first examples of spherical wavelets ([205]). Let S n+1 be the unit sphere in Rn+2 and σ the Lebesgue mean/2 sure on S n+1 . Let Pk be the kth Gegenbauer polynomial of order n/2 and define for G ∈ L1 ([−1, 1]) (4π)n/2 Γ(k + 1)Γ(n/2) b G(k) = Γ(k + n)

Z

1

−1

n/2

G(t)Pk (t)(1 − t2 )(n−1)/2 dt.

Let ψr ∈ L1 ([−1, 1]), r ≥ 0. The family { ψr }r is said to be spherical wavelet of order p if  ˆ ˆ ˆ  ψr (0) = ψr (1) = ... = ψr (p) = 0  Z ∞  (ψˆr (k))2 dr = 1 for k ≥ p + 1.   0

Wavelets extended  41

The associated spherical wavelet transform is defined on L2 (S n+1 ) by Z

Cψ F (r, η) =

S n+1

F (ξ)ψr (ξ.η)dσ(ξ).

To introduce a special wavelet analysis on the sphere related to zonals, we recall firstly some useful topics. Let F ∈ L2 [−1, 1] and Ln be the Legendre polynomial of degree n. The coefficients Z

Fb (n) = 2π < F, Ln >= 2π

1

−1

F (x)Ln (x) dx,

n∈N

are called the Legendre coefficients or the Legendre transforms of F . It is proved in harmonic Fourier analysis that F may be expressed via a series form F =

∞ X

Fb (n)

n=0

2n + 1 Ln 4π

(3.1)

called the Legendre series of F . 3.3.2

Existence of scaling functions

Definition 3.5 A family {φj }j∈N ⊂ L2 [−1, 1] is called a spherical scaling function system if the following assertions hold. 1. For all n, j ∈ N, we have φbj (n) ≤ φbj+1 (n). 2. lim φbj (n) = 1 for all n ∈ N. j−→∞

3. φbj (n) ≥ 0 for all n, j ∈ N. Above, φbj (n) is the Legendre transform of φbj . We will now investigate a way of construction of a scaling function (see [218], [219], [220]). Definition 3.6 A continuous function γ : R+ 7−→ R is said to be admissible if it satisfies the admissibility condition ∞ X 2n + 1 n=0

4π

!2

sup

|γ(x)|

< +∞.

x∈[n,n+1]

γ is called an admissible generator of the function ψ given by ψ=

∞ X 2n + 1 n=0

4π

γ(n)Ln .

We immediately obtain the following characteristics [357]. Proposition 3.7 The following assertions are true:

42  Wavelet Analysis: Basic Concepts and Applications

1. If γ is an admissible generator, then its generated function ψ ∈ L2 ([−1, 1]). b 2. For all n ∈ N, ψ(n) = γ(n).

Proof. 1. Since the Legendre polynomials form an orthogonal basis for L2 [−1, 1] with 4π < Ln , Ln >L2 [−1,1] = , 2n + 1 the admissibility condition imposed on γ yields that |ψ||2L2 [−1,1] =

∞ X 2n + 1 n=0

≤

4π

∞ X 2n + 1 n=0

4π

(γ0 (n))2 !2

sup

|γ0 (x)|

x∈[n,n+1]

< +∞. 2. It is an immediate result from (3.1). We now investigate the idea to construct a whole family of admissible functions starting from one source admissible function. Definition 3.8 The dilation operator is defined for γ : [0, ∞) → R and a > 0 by Da γ(x) = γ(ax), ∀x ∈ [0, ∞). For a = 2−j , j ∈ Z, we denote γj = Dj γ = D2−j γ. Definition 3.9 An admissible function ϕ : [0, ∞) → R is said to be a generator of a scaling function if it is decreasing, continuous at 0 and satisfies ϕ(0) = 1. The system {φj }j∈N ⊂ L2 [−1, 1], defined by φj =

∞ X 2n + 1 n=0

4π

ϕj (n)Ln ,

is said to be the corresponding spherical scaling function associated with ϕ. cj (n))n is stationary with zero stationIt holds sometimes that for all j, the sequence (φ ary value. In this case, the system {φj }j∈N ⊂ L2 [−1, 1] is called bandlimited. It holds that for a bandlimited scaling functions, each φj is a one-dimensional polynomial, and for all F ∈ L2 (S 2 ), φj ∗ F is a polynomial on S 2 . The following theorem affirms that scaling functions permit to approximate L2 -ones with polynomial approximates (see [357]).

Theorem 1 Let {φj }j∈N be a scaling function and F ∈ L2 (S 2 ). Then (k)

lim ||F − φj ∗ F ||L2 (S 2 ) = 0, ∀k ∈ N.

j→∞

Wavelets extended  43

Here, for a function Φ ∈ L2 , we designate by Φ(k) the k-times self-convolution of Φ with itself. The last approximation is called spherical approximate identity. Proof. Observe firstly that +∞ X 2n+1 X

(k)

φj ∗ F =

b (n)Fb (n, j)Y . Φ J n,j

n=0 j=1

Thus, F−

(k) φj

∗F =

+∞ X 2n+1 X

b (n))Fb (n, j)Y , (1 − Φ J n,j

n=0 j=1

which, by applying Parseval identity, yields that +∞ X 2n+1 X

(k)

kF φj ∗ F k22 =

b (n))2 (Fb (n, j))2 . (1 − Φ J

n=0 j=1

Now, observing that the last series is J-uniformly convergent and the fact that b (n)) = 0, ∀n, lim (1 − Φ J

J→+∞

it results that (k)

lim ||F − φj ∗ F ||L2 (S 2 ) = 0.

j→∞

3.3.3

Multiresolution analysis on the sphere

In this section, we will show that such scaling functions are suitable candidates to approximate functions in L2 as it is needed in wavelet theory in general and thus they are suitable sources to define multi-resolution analysis and/or a wavelet analysis on the sphere. The next theorem shows the role of spherical scaling functions in the construction of multi-resolution analysis on the sphere. Theorem 2 Let for j ∈ Z, 

Vj =

(2) φj

2

2



∗ F | F ∈ L (S )

where {φj }j∈N ⊂ L2 [−1, 1] is a scaling function. Then, the sequence (Vj )j defines a multi-resolution analysis on the sphere. That is, 1. Vj ⊂ Vj+1 ⊂ L2 (S 2 ), ∀j ∈ N. 2.

S∞

j=0

Vj = L2 (S 2 ).

For j ∈ Z, the spaces Vj represent the so-called scale or approximation space at the level j. Proof. 1. As Φ ∈ L2 and also F , the convolution Φ ∗ F is also L2 . Consider next, for J ∈ Z, the function γJ (n) =

b (n) Φ J b Φ J+1 (n)

!2

Fb (n, j) if ΦJ+1 (n) 6= 0

44  Wavelet Analysis: Basic Concepts and Applications

and 0 else, and define the function G by G=

+∞ X 2n+1 X

γj (n)Yn,j .

n=0 j=1

b It is straightforward that G ∈ L2 and that G(n, j) = γj (n). Furthermore, (2)

φJ+1 ∗ G = = =

+∞ X 2n+1 X n=0 j=1 +∞ X 2n+1 X

b b Φ J+1 (n)G(n, j)Yn,j b (n)Fb (n, j)Y Φ J n,j

n=0 j=1 (2) φJ ∗ F.

Hence, (2)

(2)

φJ ∗ F = φJ+1 ∗ G ∈ VJ+1 . Consequently, VJ ⊂ VJ+1 . 2. The density property is an immediate consequence of the spherical approximate identity proved in Theorem 1. 3.3.4

Existence of the mother wavelet

Based on this multi-resolution analysis of L2 (S 2 ), we will be able to introduce spherical wavelets. Definition 3.10 Let Φ = {φj }j∈N in L2 ([−1, 1]) be a scaling function and Ψ = 2 e = {ψ˜ } {ψj }j∈N∪{−1} and Ψ j j∈N∪{−1} be in L ([−1, 1]) satisfying the so-called refinement equation 2 2 cj (n)ψ˜j (n) = (φd c ψ j+1 (n)) − (φj (n)) , ∀n, j ∈ [0, +∞).

c

Then, e are called, respectively, (spherical) primal wavelet and (spherical) dual a. Ψ and Ψ wavelet relatively to Φ.

b. The functions ψ0 and ψ˜0 are called the primal mother wavelet and the dual mother wavelets, respectively. Here, we set ψ−1 = ψ˜−1 = φ0 . The following result due to Volker in [357] shows the existence of primal and dual wavelets. Theorem 3 Let ϕ0 be a generator of a scaling function and ψ0 and ψ˜0 be admissible function such that x ψ0 ψ˜0 (x) = (ϕ0 ( ))2 − (ϕ0 (x))2 , ∀x ∈ R+ . 2 Then, ψ0 and ψ˜0 are generators of primal and dual mother wavelets, respectively.

Wavelets extended  45

Proof. We will prove precisely that the dilated copies {ψj }j∈N∪{−1} and {ψ˜j }j∈N∪{−1} defined via their Legendre coefficients by dilating ψ0 and ψ˜0 (x) as cj (n) = ψj (n) = ψ0 (2−j n), ∀n, j ∈ N, ψ c ψ˜j (n) = ψ˜j (n) = ψ˜0 (2−j n), ∀n, j ∈ N

and d ˜ ψd −1 (n) = ψ−1 (n) = ϕ0 (n), ∀n ∈ N

are a primal and dual wavelets, respectively. Indeed, considering these dilated copies, we obtain for all n, j ∈ N, cj (n)ψ˜j (n) = ψ0 (2−j n)ψ˜0 (2−j n) ψ = (ϕ0 (2−j−1 n))2 − (ϕ0 (2−j n))2 2 2 c = (φd j+1 (n)) − (φj (n)) . c

A fundamental property of spherical wavelets is the scale-step property proved hereafter and which prepares to introduce the detail spaces. e = {ψ˜ } Theorem 4 Let Ψ = {ψj }j∈N∪{−1} and Ψ j j∈N∪{−1} be a primal and a dual wavelet corresponding to the scaling function {φj }j∈N ⊂ L2 [−1, 1]. The following assertions hold for all F ∈ L2 (S 2 ). (2)

(2)

i. φJ2 ∗ F = φJ1 ∗ F +

JX 2 −1

ψ˜j ∗ ψj ∗ F , ∀J1 < J2 ∈ N.

j=J1

ii. F =

(2) φJ

∗F +

∞ X

ψ˜j ∗ ψj ∗ F , ∀J ∈ N.

j=J

Proof. i. We will evaluate the last right-hand series term in the assertion. Using the definition of primal and dual wavelets, we obtain ψ˜j ∗ ψj ∗ F

=

+∞ X 2n+1 X

ψbj (n)ψej (n)Fb (n, s)Yn,s b

n=0 s=1

=

+∞ X 2n+1 X h

i

(φbj+1 (n))2 − (φbj (n))2 Fb (n, s)Yn,s

n=0 j=1 (2)

(2)

= φj+1 ∗ F − φj ∗ F. As a result, JX 2 −1

(2) (2) ψ˜j ∗ ψj ∗ F = φJ2 ∗ F − φJ1 ∗ F.

j=J1

ii. It is an immediate consequence of assertion i.

46  Wavelet Analysis: Basic Concepts and Applications

Theorem 5 Let, for j ∈ Z, Wj = {ψ˜j ∗ ψj ∗ F/F ∈ L2 (S 2 )}. Then, for all J ∈ Z, VJ+1 = VJ + WJ . Proof. The inclusion VJ ⊂ VJ−1 + WJ−1 is somehow easy and it is a consequence of Theorem 4. We will prove the opposite inclusion. So, let F1 ∈ VJ and F2 ∈ WJ . We seek a function F ∈ L2 for which we have (2)

ΦJ+1 ∗ F = F1 + F2 . Since F1 ∈ VJ and F2 ∈ WJ , there exist G1 and G2 in L2 such that (2)

F1 = ΦJ ∗ G1

e ∗Ψ ∗G . F2 = Ψ J J 2

and

Now, consider the function γ defined by γ(n, j) =

2 2 b b (n))2 G b b b 1 (n, j) + ((Φ (Φ J J+1 (n)) − (ΦJ (n)) )G2 (n, j)

!2

b Φ J+1 (n)

whenever ΦJ+1 (n) 6= 0 and 0 else, and define the function F by +∞ X 2n+1 X

F =

γ(n, j)Yn,j .

n=0 j=1

It is straightforward that F ∈ L2 and that Fb (n, j) = γ(n, j). Furthermore, (2) ΦJ+1

∗F

=

+∞,∗ X 2n+1 X

2b b (Φ J+1 (n)) F (n, j)Yn,j

n=0 j=1

=

+∞ X 2n+1 X

b (n))2 G b 1 (n, j)Yn,j (Φ J

n=0 j=1

+

+∞ X 2n+1 X

2 2 b b b ((Φ J+1 (n)) − (ΦJ (n)) )G2 (n, j)Yn,j

n=0 j=1

=

+∞ X 2n+1 X

b (n))2 G b 1 (n, j)Yn,j (Φ J

n=0 j=1

+

+∞ X 2n+1 X

e (n)Ψ b (n)G b 2 (n, j)Yn,j Ψ J J b

n=0 j=1

=

(2) φJ

e ∗Ψ ∗G ∗ G1 + Ψ J J 2

= F1 + F2 . Consequently, F1 + F2 ∈ VJ+1 .

Wavelets extended  47

Definition 3.11 For j ∈ Z, the space Wj is called the detail space at the level j and the mapping (SW T )j : L2 (S 2 ) → L2 (S 2 ) F 7−→ ψj ∗ F is called the spherical wavelet transform at the scale j. Based on this definition and the results above, any function F ∈ L2 (S 2 ) will be represented by means of an L2 -convergent series F =

∞ X

ψ˜j ∗ (SW T )j (F ).

(3.2)

j=−1

3.4

EXERCISES

Exercise 1.

Let Φ1 be a scaling function generating a multiresolution analysis on L2 ([0, π]) and Γ be a scaling function generating a multiresolution analysis on L2 (R). 1. Prove that the function Φ2 (x) =

X

Γ(x−k) is a scaling function that generates

k≥0

a multiresolution analysis on L2 ([0, 2π]). 2. Prove, for j, k1 , k2 ∈ Z, Φj,k1 ,k2 (θ, ϕ) = Φj,k1 (θ)Φj,k2 (ϕ) generates a multiresolution analysis on L2 (S2 ), where S2 is the unit sphere in R3 . Exercise 2.

Let Γ be a C 1 -curve in R2 with a parametrization γ(t) = (u(t), v(t)), t ∈ R, where u and v are C 1 on R. Assume that γ is a homeomorphism between R and Γ. 1) Prove that the map V defined on the cylinder by V (cos θ, sin θ, z) = (u(z) cos θ, u(z) sin θ, v(z)) transforms the cylinder homeomorphically to a surface of revolution G about the z-sxis. 2) Prove that the surface element dσG on the cylinder is transformed to dσG = w(z)dθdz, 1

where w(z) = |u(z)|[u0 (z)2 + v 0 (z)2 ] 2 . 3) Prove that V induces a unitary map Vˆ : L2 (E, dzdθ) → L2 (G, dσG ) by 1

(Vˆf ) (u(z) cos θ, u(z) sin θ, v(z))) = (w(z))− 2 f ((cos θ, sin θ, z)) .

48  Wavelet Analysis: Basic Concepts and Applications

Exercise 3.

Consider on the unit sphere S2 of R3 the map V : E → S 2 defined by V (cos θ, sin θ, z) = (cosh−1 (z) cos θ, sinh−1 (z) sin θ, tanh(z)). Show that V induces a unitary map V˜ : L2 (E, dzdθ) → L2 (S2 , cosh−2 (z)dzdθ) by 



(V˜ f ) cosh−1 (z) cos θ, sinh−1 (z) sin θ, tanh(z) = cosh(z).f (cos θ, sin θ, z). ˆ × RZ) × T, where π = RZ Let now G = R × RZ and H0 = (R × RZ) × (R 3 and T is the torus in R . 3) Show that the map U˜ defined by cosh−1 (z − x) cos(θ − ϕ) cosh−1 (z) cos θ     (U˜ ((x, ϕ); (w, k) : η)f )  cosh−1 (z) sin θ  = Φf  cosh−1 (z − x) sin(θ − ϕ)  tanh(z − x) tanh(z) 



where Φ = Φ((x, ϕ); (w, k) : η; z)) = η[





cosh(z) ]e2πi(w.z+kθ) , −1 cosh (z − x)

is a representation of HG on L2 (S 2 , cosh−2 (z)dzdθ). Exercise 4. )

(

z2 x2 + y 2 + = 1 be a two-dimensional ellipsoid For α, γ > 0, let εα,γ = (x, y, z); α2 γ2 of revolution and consider the map V : E → εα,γ defined by 



V : (cos θ, sin θ, z) 7→ α cosh−1 (z) cos(θ), α cosh−1 (z) sin(θ), γ tanh(z) . Show that V induces a unitary map V˜ : L2 (E, dzdθ) → L2 (εα,γ , ρεα,γ (z)dzdθ) by cos θ α cosh−1 (z) cos θ 1     (V˜ f )  α cosh−1 (z) sin θ  = q f  sin θ  ρεα,γ (z) z γ tanh(z) 

where we have ρεα,γ (z) = .



α cosh3 z

q



α2 sinh2 z + γ 2



Wavelets extended  49

Exercise 5.

Let, for (x, y) ∈ R2 , r =

p

x2 + y 2 and (x, y) = rω with

ω = (cos θ, sin θ); θ ∈ (−π, π). Consider the functions ψ0 (x, y) = C0 (1 + r2 )3/2 e−r

2 /2

,

−r2 /2

ψ1 (x, y) = C1 (r2 − 2)(1 + r2 )3/2 e 4

(x, y),

2 3/2 −r2 /2

6

ψ2 (x, y) = C2 (−4 + 6r − 2r )(1 + r )

e

,

and ψ3 (x) = C3 (1 − 16r2 + 21r4 + 15r6 + r8 )(1 + r2 )3/2 e−r

2 /2

(x, y),

where the Cj ’s (j = 0, 1, 2, 3) are constants. 1) Compute the constants Cj ’s (j = 0, 1, 2, 3) in such a way we get normalized functions on the circle S 1 . 2) Compute the Fourier transform of the ψj ’s, j = 1, 2, 3, 4. 3) Deduce possible wavelets on the circle S 1 . Exercise 6.

Let, for (x, y, z) ∈ R3 , ρ =

p

x2 + y 2 + z 2 and (x, y, z) = ρω with

ω = e1 cos θ sin φ + e2 sin θ sin φ + e3 cos φ; θ ∈ (−π, π) , φ ∈ (0, π). Consider the functions ψ1 (ρ, θ, φ) = C1 (1 + ρ2 )3/2 e−ρ −ρ2 /2

ψ2 (ρ, θ, φ) = C2 (1 + ρ2 )3/2 e −ρ2 /2

ψ3 (ρ, θ, φ) = C3 (1 + ρ2 )3/2 e

2 /2

,

(ρ2 − 2)ρω,

(−6 − ρ2 + 7ρ4 − 2ρ6 )ρω,

and ψ4 (ρ, θ, φ) = C4 (1 + ρ2 )3/2 e−ρ

2 /2

(1 − 16ρ2 + 18ρ4 + 15ρ6 + ρ8 )ρω,

where the Cj ’s (j = 0, 1, 2, 3) are constants. 1) Compute the constants Cj ’s (j = 1, 2, 3, 4) in such a way we get normalized functions on the sphere S 2 . 2) Compute the Fourier transform of the ψj ’s, j = 1, 2, 3, 4. 3) Deduce possible wavelets on the sphere S 2 . Exercise 7.

Denote T = R/Z and consider f ∈ L1 (T) such that f (x) = sin(2π(γx + θ)), where γ, θ ∈ R are fixed and γ ∈ Z \ {0}. Let the generalized Haar wavelet be ψ = 1[− 1 ,0) + 1[0, 1 ) . 2

Prove that Wψ f (a, b) = for all (a, b) ∈ R+ × R.

2

πγa 2 sin2 ( ) cos(2π(γb + θ)), γπ 2

(3.3)

CHAPTER

4

Clifford wavelets

4.1

INTRODUCTION

The English mathematician William Kingdon Clifford introduced in 1878 a general set of algebras, which he called geometric algebras and which are now attached to his name as Clifford algebras. The interest in multivariate analysis using Clifford algebras started to grow in the last century. Since then, great works on Clifford analysis referring different classes of special functions have appeared. Clifford analysis also offers some general context of Fourier one in signal/image processing by applying real, complex and quaternion numbers. In matrix spaces, Clifford algebras may be constructed using Pauli and Dirac matrices, which makes them suitable physical space such as Minkowski space-time and thus a unifying language for both mathematics and physics (see [83], [84]). Clifford analysis offers a function theory, which is a higher-dimensional analogue of the theory of holomorphic functions in the complex plane, centered around the notion of monogenic functions. Indeed, monogenic function theory is considered generalization of the holomorphic function theory in the complex plane to higher dimensions and a refinement of the harmonic analysis based on the Laplace operator’s factorizations. The construction of spherical monogenic functions has been studied for decades with different methods. Recently, orthogonal monogenic bases are developed for spheroidal reference domains. Many works have related spheroidal monogenic functions to wavelets in the context of Clifford analysis. Therefore, classical wavelet theory has been constructed in the framework of Clifford analysis (see [75], [79], [80], [82], [83], [86]). Clifford analysis, in its most basic form, is a refinement of harmonic analysis in higher-dimensional Euclidean spaces. By introducing the so-called Dirac operator, researchers introduced the notion of monogenic functions extending holomorphic ones. In this context, different concepts of real and complex analysis have been extended to the Clifford case, such as Fourier transform extended to Clifford-Fourier transform and derivation of functions. For example, Clifford-Fourier transform is related to or expressed in terms of an exponential operator. For the even-dimensional case, it yields a kernel based on Bessel functions. Similar to the classical Fourier transform, here also the new kernel satisfies a system of differential equations (see [79], [80], [82], [83], [86], [190]). 51

52  Wavelet Analysis: Basic Concepts and Applications

In [88], the authors survey the historical development of quaternion and CliffordFourier transforms and wavelets. Basic concepts have been revisited, and mathematical formulations have been enlightened. Hypercomplex Fourier transforms and wavelets have been revisited with overviews on quaternion Fourier transforms, Clifford-Fourier transforms and quaternion and Clifford wavelets. Clifford algebra is characterized by additional facts as it provides a simpler model of mathematical objects compared to vector algebra. It also permits a simplification in the notations of mathematical expressions such as plane and volume segments in two and three as well as higher dimensions by using a coordinate-free representation. Such representation is characterized by one important feature resumed in the fact that the motion of an object may be described with respect to a coordinate frame defined on the object itself. This means that it permits use of a self-coordinate system related to the object in hand (see [75], [187], [286]). Compared to spheroidal functions, wavelets are characterized by scale invariance of approximation spaces. Clifford algebra is one mathematical object that owns this characteristic. Recall that multiplication of real numbers scales their magnitudes according to their position in or out from the origin. However, multiplication of the imaginary part of a complex number performs a rotation: it is a multiplication that goes round and round instead of in and out. So, a multiplication of spherical elements by each other results in an element of the sphere. Again, repeated multiplication of the imaginary part results in orthogonal components. Thus, we need a coordinate system that results always in the object, a concept that we will see again and again in the algebra. In other words, Clifford algebra generalizes to higher dimensions by the same exact principles applied at lower dimensions, by providing an algebraic entity for scalars, vectors, bivectors and trivectors, and there is no limit to the number of dimensions it can be extended to. More details on Clifford algebra—its origins, history and developments—may be found in [2], [181], [182], [183], [184], [187], [318], [286]. From the applied point of view, Clifford algebras/analysis has been the object of many applications, especially in image processing. For example, in [92, 96, 250] algorithms and methods based on Clifford algebras have been developed for the aim of segmentation and color alterations. Clifford algebras have also been shown to be good framework for the geometry of images. See [44, 337]. In [223], a Marr wavelet model has been applied on samples of intensity values for each pixel of image to estimate the probability density function of the pixel intensity. See also [360, 223, 342]. In [90] and [91], the authors introduced new definition for general geometric Fourier transform covering some Clifford cases and showed possible applications in image processing.

4.2

DIFFERENT CONSTRUCTIONS OF CLIFFORD ALGEBRAS

Clifford analysis appeared as a generalization of the complex analysis and Hamiltonians. It extended complex calculus to some type of finite-dimensional associative algebra known as Clifford algebra endowed with suitable operations as well as inner products and norms. It is now applied widely in a variety of fields including geometry

Clifford wavelets  53

and theoretical physics. Clifford analysis offers a functional theory extending the theory of holomorphic functions of one complex variable. 4.2.1

Clifford original construction

The original idea of construction due to Clifford is based on the exterior Grassmann V algebra Rm associated with the linear space Rm . Consider an orthonormal basis {e , e , · · · , en } of Rm such as the canonical basis. The basis of the Clifford algebra V 1 m2 R is composed of all possible k-exterior products of the elements {e1 , e2 , · · · , en }, . 0 ≤ k ≤ n. We obtain e = 1, e = e ∧ e .. ∧ e , 1 ≤ k ≤ n by assuming that ∅

i1 i2 ...ik

i1

i2

ik

ei ∧ ej = −ej ∧ ei for i 6= j and ei ∧ ei = 1, 1 ≤ i ≤ n. By this way, we obtain an associative non-commutative algebra on R with dimension 2n denoted by Rm . Besides, Clifford himself has used a quite different way in [168] by putting ei ∧ei = −1 and thus obtaind what he called the anti-Euclidean space R0,n and the associated algebra. 4.2.2

Quadratic form-based construction

Clifford algebras have been introduced by using quadratic forms such as the construction developed in [331]. Let (V, Q) be an n-dimensional quadratic space and B be the bilinear form associated with the quadratic form Q. Consider next an associative algebra A with the following rules on addition and multiplication: x2 = Q(x) and xy + yx = 2B(x, y). Let next {e1 , e2 , · · · , en } be an orthonormal basis of V (orthonormality relatively to the bilinear form B). We may obtain a structure of Clifford algebra for A. More precisely, the Clifford algebra may be introduced as follows (see, for example, [186, 288]). Let V be an n-dimensional real vector space and Q : V −→ R a quadratic form on V . Let also A be a real associative algebra with identity 1A such that i. R and V are linear subspaces of A. ii. For all v ∈ V , we have v 2 = vv = B(v, v) = Q(v). iii. The algebra A is generated as a real algebra by 1 and V . Then, the algebra A is the Clifford algebra associated with the quadratic space (V, Q). Already with the quadratic forms, we may observe Clifford algebras as a quotient result of suitable vector space and suitable quadratic form. This was the subject of [158] and resembles somehow the original construction due to Clifford and is based on exterior product. The construction here will instead apply the tensor product. Let V be a vector space over the real field R with dim(V ) = n and Q : V −→ R, a quadratic form. Consider next the tensor algebra of V denoted by T (V ) =

∞ M

⊗k V .

k=0

Consider next the two-sided ideal I(Q) generated by the elements of the form

54  Wavelet Analysis: Basic Concepts and Applications

x ⊗ x − Q(x)1V , where x ∈ V and 1V is the identity element for the multiplication in V . Then, the quotient Cl(V, Q) = T (V )/I(Q) is called the Clifford algebra associated with (V, Q). 4.2.3

A standard construction

We now present a standard construction of Clifford algebra issued from the Euclidean space Rm (or Cm ), which is the simplest one used in the majority of works dealing with this subject. See [288]. Consider an orthogonal basis of Rm (or Cm ) such as the canonical one and let A be the 2n -dimensional linear space generated by the set {1, ei1 ···ik = ei1 ei2 · · · eik |1 ≤ i1 < i2 < · · · < ik ≤ n and 1 ≤ k ≤ n}, where a multiplication is defined by ei ej + ej ei = 2δij , where δij stands for the Kronecker symbol. We get here a Clifford algebra, which will be denoted by Rn (or Cn for the complex case). More precisely, we have in the literature the following definition. Definition 4.1 The Clifford algebra associated with Rm , endowed with the usual Euclidean metric, is the extension of Rm to a unitary, associative algebra denoted by Rm over the reals, for which 1. x2 = −|x|2 for any x ∈ Rm , 2. Rn is generated (as an algebra) by Rm , 3. Rn is not generated (as an algebra) by any proper subspace of Rm . Theorem 6 The Clifford algebra Rm is uniquely determined up to an algebra isomorphism. The proof is based on the following preliminary result. Lemma 4.2 Denote for I = (i1 , i2 , . . . , il ), 0 ≤ l ≤ n, eI = (−1)l eil eil−1 . . . ei1 and for a = (a1 , a2 , . . . , an ) ∈ Rm , aI = ail ai2 . . . ail . We have 0

2−n

(

X

eI aeI =

I

a∅ if n is even, a∅ + a(1,2,...,n) e(1,2,...,n) if n is odd.

Proof. Denote |.| the cardinality function. It is straightforward that for all I, J ⊆ {1, 2, . . . , n}, eI eJ = (−1)|I||J|−|I∩J| eJ eI . 0

As a result, whenever x =

X

xJ eJ ∈ Rm , we get

J 0

X I

0

eI xeI =

X I,J

0

xJ eI eJ eI =

(−1)|I||J|−|I∩J| xJ eJ .

X I,J

Clifford wavelets  55

Furthermore, for a fixed J ⊆ {1, 2, ..., n}, 0

X

(−1)|I||J|−|I∩J| =

|J| n−|J| X X

0

X

(−1)(i+j)|J|−i

i=0 j=0 |I|=i+j,|I∩J|=i

I

=

|J| n−|J| X X

0

j i (−1)j|J| (−1)i|J|−1 C|J| Cn−|J|

X

i=0 j=0 |I|=i+j,|I∩J|=i

= (1 + (−1)|J| )n−|J| (1 + (−1)|J|−1 )|J| = 0 unless either n is odd and J = {1, 2, ..., n}, or J = ∅. In these cases, we end up with 2n . Proof of Theorem 6. It suffices to show that the set (eI )0≤|I|=l≤n is linearly inde0

pendent. In other words, whenever the sum

X

aI eI is zero, necessarily aI = 0 for all

I

I. Using the minimality of Rn , we observe that 1 and e1,2,...,n are linearly independent. Hence, a∅ = 0. Next, by replacing a with el a, for arbitrary I, the result follows. In the sequel, it will also be useful to embed Rm+1 into Rm by identifying (x0 , x) ∈ m+1 R = R ⊕ Rm with x0 .e0 + x ∈ Rm (note that, by Assertion 1 in Definition 4.1, 1∈ / Rm ), and call these elements Clifford vectors. • ∀x ∈ Rm \ {0}, it has a multiplicative in-

Proposition and Definition 4.3 verse given by

x−1 =

x . |x|2

• The multiplicative group generated by all Clifford invertible vectors in Rm is called the Clifford group. Definition 4.4

• For a ∈ Rn , the real part of a is defined by Re(a) = a∅ .

• Rm is endowed with the natural Euclidean metric |a| = Lemma 4.5

q

Re(aa) =

q

Re(aa).

• For all x, y ∈ Rm , we have |xy| ≤ 2n−1 |x||y|

• For all x, y ∈ Rm , the quality |xy| = |x||y| yields that at least one of x, y belongs to the Clifford group of Rm .

56  Wavelet Analysis: Basic Concepts and Applications

4.3

GRADUATION IN CLIFFORD ALGEBRAS

We will focus, from now on, on the real Clifford algebra Rm (respectively, its complexification Cm if necessary) associated with the real (respectively, complex) Euclidean space Rm (respectively, Cm ). The concept of graduation in Clifford algebra is an important task as it permits observation of the whole algebra in the form of a direct sum of subalgebras where the operations, especially multiplication, are easier and simpler. P Recall that an element a ∈ Rm is written in the form a = A aA eA , where A is an arbitrary ordered multi-index set A = {(i1 i2 · · · ik )} with 1 ≤ i1 < i2 < · · · < ik ≤ n and where k = |A| is the cardinality of A. For |A| = 0, we set e∅ = 1. This way the element a may be written as a=

n X X

aA eA .

k=0 k=|A|

The subspace Rkm = spanR {eA , |A| = k} is known as subspace of grade k. The subspace of grade 0 is the field R whose elements are called scalars, the subspace of grade 1 is the vector space Rm composed of vectors, the subspace of grade 2 is composed of the so-called bivectors. This may be continued for any grade k ≤ n. This last grade gives the so-called pseudo-scalars. We obtain the decomposition Rm = R0m ⊕ R1m ⊕ .... ⊕ Rnm An element a ∈ Rn will be written as a = a∅ 1 + (a1 e1 + .... + an en ) vectors

scalars

+ (a12 e12 + a13 e13 + ....aij eij + ..... + an−1n en−1 en ) bivectors

+ ..... + (a123...n e123....n ). pseudo−scalar

Also, the element a may be written as a = nk=0 [a]k , where [a]k is the projection of a on Rkm . It is straightforward that the projector [.]k : Rm −→ Rkm for k = 0, 1, · · · , n satisfies P

i. [[a]k ]k = [a]k , ∀a ∈ Rm . ii. [λa]k = λ [a]k = [a]k λ, ∀a ∈ Rm and λ ∈ R. iii. [a + b]k = [a]k + [b]k , ∀a, b ∈ Rm and λ ∈ R. To finish with this review, we recall finally that the Clifford algebra Rm is Z2 -grader in the sense that it may be split as Rm =

M k even

Rkm ⊕

M k odd

Rkm .

Clifford wavelets  57

We usually denote R+ m =

M

Rkm

k even

and R− m =

M

Rkm .

k odd

Any element a may be written consequently as a = [a]+ + [a]− , where [a]± ∈ R± m. Moreover, we have the inclusions + R+ + m Rm − − } ⊂ Rm Rm Rm

and

− R+ − m Rm − + } ⊂ Rm . Rm Rm

Definition 4.6 The center of the Clifford algebra Rm is Z(Rm ) := {a ∈ Rm , ab = ba, ∀b ∈ Rm }. We may observe easily that (

R R ⊕ Re123···n

Z(Rm ) =

4.4

for n even, for n odd.

SOME USEFUL OPERATIONS ON CLIFFORD ALGEBRAS

4.4.1

Products in Clifford algebras

Although there are different constructions of Clifford algebras, they intersect in many characteristics such as the different products. Indeed, we will see in the present section that tensor, exterior and Clifford products yield each other. Let x =

n X

ei xi and y =

i=1

n X

ei yi be two elements in Rm . We recall that the

i=1

exterior is defined by x∧y =

X

ei ej (xi yj − xj yi ).

iRm

j=1

These two products combined yield the Clifford product xy = x · y + x ∧ y. Otherwise, we observe conversely that 1 1 x · y = (xy + yx) and x ∧ y = (xy − yx). 2 2

58  Wavelet Analysis: Basic Concepts and Applications

Figure 4.1

4.4.2

Wedge product of vectors.

Involutions on a Clifford algebra

There are many types of involutions that may be defined on Clifford algebras. The first one, known as the main involution or grade involution, is usually denoted by the tilde symbol e and constitutes an extension of vector reflection relatively to the origin in Rm to the whole algebra Rn . It is defined on the basis elements eA by |A| ef A = (−1) eA ,

where as usual A is a multi-index. As a consequence, we get e = λ and x e = −x, ∀λ ∈ R, ∀x ∈ Rm . λ

More generally, we have e = [a]+ − [a]− , ∀a ∈ Rm . i. a e e = a, ∀a ∈ Rm . ii. a

^ e+e iii. (a + b) = a b, ∀a, b ∈ Rm . g =a ee iv. (ab) b, ∀a, b ∈ Rm .

The second involution type operation is known as the reversion, denoted by the symbol ∗ , and is defined on the basis elements by e∗A = (−1) As a consequence, we get i. λ∗ = λ, ∀λ ∈ R. ii. x∗ = x, ∀x ∈ Rm .

|A|(|A|−1) 2

eA .

Clifford wavelets  59

iii. a∗ = [a]+ + [a]− , ∀a ∈ Rm . ∗

iv. a∗ = a, ∀a ∈ Rm . v. (a + b)∗ = a∗ + b∗ , ∀a, b ∈ Rm . vi. (ab)∗ = b∗ a∗ , ∀a, b ∈ Rm . Sometimes we also need to introduce a Clifford conjugation, which consists of a composition of the main involution and the reversion. It is defined on the basis elements as |A|(|A|+1) eA = (−1) 2 eA . As a result, we obtain i. λ = λ, ∀λ ∈ R. ii. x = −x, ∀x ∈ Rm . iii. a = a, ∀a ∈ Rm . iv. (a + b) = a + b, ∀a, b ∈ Rm . v. (ab) = ba, ∀a, b ∈ Rm . We achieve now the involutions with the complex case Cn , where we deal with the complex Clifford conjugation. Recall here that an element z ∈ Cm can be written in the form z = a + ib, where a, b ∈ Rm . The complex conjugation will be z † = a − ib. We may show easily that (uv)† = u† v †

and (αu + βv)† = αc u† + β c v † ,

where u, v ∈ Cm and α, β ∈ C and where the symbol .c stands for the complex conjugate in C. To finish with the operations on the Clifford algebra, we recall now the norm on Clifford algebra Rm or its complexification Cm . We define an inner product on Rm as h i ha, bi = ab = [ba]0 . 0

The associated norm will be defined for elements a = |a|2 = aa = aa =

X A

It is easy to check that i. |a + b| ≤ |a| + |b|, ∀a, b ∈ Rm . ii. |ab| = 6 |a||b| in general. iii. |ab| ≤ 2n |a||b|, ∀a, b ∈ Rm . iv. |ax| = |a||x|, ∀x ∈ Rm .

P

a2A .

A

aA eA ∈ Rm by

60  Wavelet Analysis: Basic Concepts and Applications

4.5

CLIFFORD FUNCTIONAL ANALYSIS

In this section, we propose to review the basic concepts of functional analysis on Clifford algebras. Recall that Clifford analysis constitutes in its basic form an extension of harmonic analysis in higher-dimensional Euclidean spaces. Different concepts of real and complex analysis will be reviewed. In this context, we will use for many times the Euclidean space Rm+1 as a subalgebra in the Clifford algebra Rm . The most important concept in Clifford algebra-valued functions is the concept of monogenicity. One of the definitions starts by introducing a concept of differentiability for a pair of functions. Definition 4.7 Let f, g be two locally bounded, Clifford-valued functions defined in an open domain Ω ⊆ Rm+1 . We introduce the Clifford derivative of the ordered pair (f, g) at a point X ∈ Ω by R ∂Q

D(f |g)(x) = lim

f η g dσ Q dV

R

Q↓x

for any domain Q such that x ∈ A ⊂ Ω and where dσ and dV are, respectively, the Lebesgue measures on ∂Q and Q. η is the outward normal vector to ∂Q. More specifically, the pair (f, g) is Clifford differentiable at x if there exists an element c ∈ Rm+1 such that, for any  > 0, there exists an open neighborhood U ⊆ Ω of X such that Z f n g dσ − cV ol(Q) < V ol(Q) ∂Q

for all cube Q of Rm+1 with x ∈ Q ⊂ U and such that, for some a priori fixed positive number C, V ol(Q) ≥ Cdiam(Q)m+1 . Here, n is the outward unit normal to the boundary of the cube Q, dσ is the surface measure of ∂Q and V ol stands for the usual Lebesgue measure in Rm+1 . Definition 4.8 The ordered pair (f, g) is said to be absolutely continuous on Ω if for any cube Q ⊂ Ω and any  > 0 there exists δ > 0 such that X Z i∈J

∂Qi

f η g dσ ≤ 

for any finite subdivision (Qi )i∈I of Q, and any subset J ⊂ I for which

X

V ol(Qi ) ≤

i∈J

δ. Furthermore, we may go further and discuss a Stokes analogue formula. Let Ω be a bounded Lipschitz domain in Rm+1 and let f, g be two Clifford algebra-valued continuous functions on Ω such that D(f |g) is also continuous on Ω. We have Z

Z Z

f η g dσ = ∂Ω

Ω

D(f |g) dV.

Clifford wavelets  61

Besides, if the pair (f, g) is differentiable, it is easy to check the analogue of Leibnitz rule D(f |g) = D(f |1)g + f D(l|g). Denote Dg = D(l|g) and f D = D(f |1). It is straightforward that Z

Z Z

f η gdσ =

(f D)g + f (Dg)dV.

∂Ω

Ω

We also set Df = D(f |1) and f D = D(1|f ). It should be pointed out that at any point of differentiability a ∈ Ω of the Clifford 0

algebra-valued function f =

X

fI eI , we may check that

I 0

(Df )(a) =

n XX ∂fI I

and

j=0

∂xj

(a)ej eI

0

(f D)(a) =

n XX ∂fI I

j=0

∂xj

(a)eI ej .

Notice that, by linearity considerations, it suffices to treat the case of a scalar-valued function f . We can also assume that the point of differentiability is the origin of the system. In this latter case, expanding f into its first-order Taylor series around the origin X X f (x) = f (0) + xj (∂j f )(0) + o(|a|), a = ej xj ∈ Rm+1 j

j

and using the easily checked fact that Z

xj η dσ = ej V ol(Q), ∀j,

∂Q

the derivative expressions above may be proved. Finally, we may also derive a factorization for the Laplace operator ∆ on Rm+1 as ∆ = DD = DD. Definition 4.9 A function f is said to be left monogenic (respectively, right monogenic, or two-sided monogenic) if Df = 0 (respectively, f D = 0, or Df = f D = 0). It is, in fact, possible to relate the concept of monogenicity to the differential operators known in real analysis such as the divergence and the curl. Indeed, let F = u0 −

m X

uj ej be a Rm+1 -valued function defined on an open set Ω of Rm+1 . To show that

j=1

F is left (right) monogenic, it is equivalent to prove that the (m+1)-tuple U = (uj )nj=0 satisfies div U = 0 and curl U = 0 on Ω.

62  Wavelet Analysis: Basic Concepts and Applications

Equivalently, we may show that the 1-form ω = u0 dx0 − ul dx1 − · · · − um dxm satisfies d∗ ω = 0 and d∗ ω = 0 on Ω, where d and d∗ are the exterior differentiation operator and its formal transpose, respectively. In addition, if the domain Ω is simply connected, we may show easily that F is monogenic if and only if there exists a unique (modulo an additive constant) realvalued harmonic function U on Ω such that (uj )nj=0 = gradU in Ω (i.e. F = DU ) . Monogenic functions are also introduced as generalizations of holomorphic functions, which are solutions of the so-called Cauchy-Riemann operator. For a function F : Ω −→ C, Ω, being a bounded domain in C ' R2 such that F (z) = F (x + iy) = U (x, y) + iV (x, y) and being holomorphic, is equivalent to ∂z f = 0 ⇐⇒ ∂x U = ∂y V

and ∂y U = −∂x V.

(4.1)

The Cauchy-Riemann operator ∂z will be replaced by the Dirac operator ∂x on the Clifford algebra Rm or its complexification Cm . We will discuss in the remaining part the extension of functional analysis to the Clifford framework in more details and in a simple way especially for nonmathematicians. The first important function that should be discussed is the famous exponential function in the framework of Clifford analysis. Formally, the exponential function is defined by means of the series exp(x) =

∞ X xk k=0

k!

, x ∈ Rm .

Recall that for x ∈ Rm , we have |xk | ≤ 2m |x|k . We thus get a convergent series on the whole Clifford algebra Rm . Moreover, we may show easily that | exp x| ≤ exp(2m−1 |x|). Our main result concerning this function is the following. Theorem 7 exp : Rm+1 −→ Rm+1 \ {0} is well-defined and onto. Furthermore, for all integer N ≥ 1 and all u ∈ Rm , the equation xN = u has a solution in Rm . In the latter, we will consider only functions with values on a Clifford algebra but where the variable is a vector in Rm taken as a part of the Clifford algebra. We put x = (x0 , x) ∈ Rm+1 with x =

n X

ei xi ∈ Rm . We will also write

i=1

x = e0 x0 +

n X i=1

ei xi =

n X

ei xi .

i=0

The vector space Rm can be seen as the hyperplane Rm ' {x = (x0 , x) ∈ Rm+1 : x0 = 0}.

Clifford wavelets  63

A function f defined on the vector space Rm and taking values in the Clifford algebra Rm or Cm will be expressed as f (x) =

X

eA fA (x),

(4.2)

A

where fA are complex-valued functions and A ⊂ {1, 2, · · · , n}. We also recall that its conjugate f is X f (x) = eA fA (x) A

whenever the functions fA take values in R. However, the conjugate will be f † (x) =

X

e†A fA (x)†

A

for a function with values in the complex Clifford algebra Cm . The continuity and derivability of f are to be taken component-wise. We thus denote for an open domain Ω ⊂ Rm C (r) (Ω, Rm ) := {f : Ω −→ Rm ; f =

X

fA eA with fA ∈ C (r) (Ω, R)}

A

C

(r)

(Ω, Cm ) := {f : Ω −→ Cm ; f =

X

fA eA with fA ∈ C (r) (Ω, C)}.

A

The Clifford-valued function f belongs to the Lebesgue space Lp (Ω, Rm , dV (x)) if all the components fA ∈ Lp (Ω, R, dx), 1 ≤ p < ∞, with the norm |f |p = 2n

X Z

1/p

|fA |p dV (x)

Ω

A

being equivalent to the norm Z

1/p

p

kf kp =

|f | dV (x)

,

Ω

where dV (x) stands for the Lebesgue measure on Rm . We say that f, g ∈ Lp (Rm , Rm , dV (x) are equal if the set {x ∈ Rm ; f (x) 6= g(x)} is Lebesgue measure zero. The inner product on L2 (Rm , Rm , dV (x)) is given by hf, giL2 (Rm ,Rm ,dV (x)) =

Z

f (x)g(x)dV (x).

(4.3)

Rm

Using the decomposition (4.2), we get hf, giL2 (Rm ,dV (x)) =

XZ A,B

Rm

fA (x)gB (x)eA eB .

The inner product (4.3) satisfies the Cauchy-Schwartz inequality < f, g >L2 (Rm ,Rm ,dV (x)) ≤ kf kL2 (Rm ,Rm ,dV (x)) kgkL2 (Rm ,Rm ,dV (x)) .

(4.4)

64  Wavelet Analysis: Basic Concepts and Applications

We now introduce the Dirac operator which generalizes the Cauchy-Riemann one for the complex case. It is defined by ∂x =

n X

(4.5)

e k ∂k ,

k=1

where for k = 1, 2, ..., m, ∂k = operator as

∂ . It may be seen also as a general form of the Weyl ∂xk Dx = ∂0 + ∂x ,

(4.6)

∂ , known also as the Fueter-Delanghe operator (see [185, ∂x0 210]). Such an operator acts from the left on Clifford-valued functions as where analogously ∂0 =

∂x f (x) =

n X

ei ∂i f (x) =

i=1

X

ei eA ∂i fA (x)

i,A

and on the right as f ∂x (x) =

n X

∂i f (x)ei =

i=1

X

eA ei ∂i fA (x).

i,A

In the same way, we get Dx f = ∂0 f + ∂x f

and f Dx = f ∂0 + f ∂x .

In the sequel, we will need for many times the conjugate of the Dirac operator ∂x =

n X

e k ∂k .

(4.7)

k=1

As for the real and complex classical cases of functional analysis, we will here also discuss the expressions of Dirac operator by means of the spherical coordinates. This will be used widely in the construction of Clifford wavelets. For this aim, we need to recall the spin groups. It consists of the subgroup of R+ n composed of the products of even numbers of elements, where the spinor norm equals ±1. In other words, Spin(p, q) = {s ∈ Rp,q ; s =

2l Y

ω j with ω 2j = ±1, 1 ≤ j ≤ 2l}.

(4.8)

j=1

Such group acts on the Clifford-valued functions as follows: s ∈ Spin(n) → Ls : f (x) → sf (¯ sxs)s. Definition 4.10 A partial differential operator with constant coefficients is called spin-invariant if it commutes with Ls .

Clifford wavelets  65

Immediately, we notice that the Dirac operator is one of the differential operators that are spin-invariant. Proposition 4.11 The Dirac operator satisfies ∂x Ls = Ls ∂x and thus it is spininvariant. Proof 4.1 As the spinor s is independent of x, we may write ∂x Ls f (x) = ∂x (sf (¯ sxs)s) = s∂x (f (¯ sxs)) s. Now, again, because of the independence of the spinor s of x, we get ∂x (f (¯ sxs)) = (∂x f )(sxs). Consequently, ∂x Ls f (x) = Lx ∂x f (x). Definition 4.12 A function f ∈ C 1 (Ω, Rm ) is called left-monogenic (respectively, right-monogenic) on Ω iff Dx f = (∂x0 + ∂x )f (x0 + x) = 0. Respectively, f Dx = 0. Example 8 The function E : Rm \ {0} −→ Rm defined by E(x) =

where |x|2 = Rm \ {0}.

n X

x , |x|n

|xk |2 is the Euclidean norm, is both left- and right-monogenic on

k=1

Since the Clifford product is not commutative, the two notions are not equivalent. The definition 4.12 is equivalent to a linear system of 2n first order PDE. Example 9 For the simple case n = 1 on the Clifford algebra R1 ' C, the left monogenicity may be written as Dx f (x0 , x1 ) = [∂0 + i∂1 ] f (x0 , x1 ) = 0. Writing f in the form f (x0 , x1 ) = u(x0 , x1 ) + iv(x0 , x1 ), we get ∂0 U = −∂1 V = 0 and ∂0 V = ∂1 U. which is the Cauchy-Riemann conditions for a complex-valued function to be holomorphic. Furthermore, the two-dimensional Laplacian may be obtained as the product of D and its complex conjugate D: ∆2 = DD.

66  Wavelet Analysis: Basic Concepts and Applications

We now return to the object of the section and continue to exploit the spin group action on the Dirac operator. We write x = rη, η =

n X

ei ηi , ηi =

i=0

n X xi , i = 0, 1, · · · , n, r2 = |xi |2 . r i=0

Using the spherical coordinates, the Dirac operator and its conjugate may be written as 1 1 ∂x = η(∂r + ∂η ) and ∂x = η(∂r + ∂η ), r r where ∂η is the so-called spherical Dirac operator acting on the sphere S m and expressed by ∂η = x ∧ ∂x . Consider next Γη = η∂η

and Γ?η = η∂η

(4.9)

called also the spherical Dirac operators, and their adjoints f=∂ η Γ η η

We obtain

f? = ∂ η. and Γ η η

1 ∂x = η ∂r + Γη r 



(4.10)

1 f? = (∂r + Γ )η r η

and its conjugate 1f ∂x = ∂r + Γ η η. r As a result, we obtain the decomposition of the Laplacian ∆m as n ∆m = ∂r2 + ∂r + ∆η , r where ∆η is the well-known Laplace-Beltrami operator. Considering next the identity operator I, we obtain a simplified form of the Laplace-Beltrami operator 







∆η = (n − 1) I − Γ?η Γ?η . Denote now E =

n X

xi ∂i = r∂r , the Euler operator. We my check easily that

i=0

xDx = E − x ∧ ∂x + (x0 ∂x − x∂0 ) . On the other hand, observe that xDx = E + Γη and that x∂x + ∂x x = −2E − n. We get the simple expression Γη = x ∧ ∂x = −

X

eij (xi ∂j − xj ∂i ).

i 0 and η are the spherical coordinates of x and A, B are two scalar-valued functions satisfying the so-called Vekua system ∂x0 A − ∂r B =

n−1 B r

and ∂x0 B + ∂r A = 0.

See, for instance, [171]. In the next part, we propose to review some useful integral transforms such as Cauchy integral, Stokes theorem and Cauchy-Clifford integral formula. Let Ω ⊂ Rm be an open set. Let also U ⊂ Ω be a compact and orientable piecewise differentiable bounded domain in Rm with Lipschitzian boundary ∂U . The surface element on ∂U will be denoted by dσx =

n X

(−1)j ej dx[j] ,

j=1

where dx[j] = dx1 ∧ dx2 ∧ · · · dxj−1 ∧ dxj+1 ∧ · · · ∧ dxn . Let η(x) denote the unit outward pointing normal-vector to the boundary ∂U at x ∈ ∂U . We know that dσx = η(x)dS(x), with dS(x) being the surface element on ∂U . The following result is a variant of the Stokes theorem in the Clifford framework.

Clifford wavelets  69

Theorem 12 (Clifford-Stokes Theorem) Let Ω ⊂ Rm be an open set. Let also U ⊂ Ω be a compact and orientable piecewise differentiable bounded domain in Rm with Lipschitzian boundary ∂U . Let f, g ∈ C 1 (Ω). Then Z

Z

f (x)dσx g(x) = ∂U

[(∂x f )g + f (∂x g)]dV (x). U

In particular, for f ≡ 1 we have Z

Z

dσx g(x) =

∂x g(x)dV (x).

∂U

U

Moreover, whenever f is left-monogenic and g is right-monogenic on Ω then Z

g(x)dσx f (x) = 0. ∂U

Proof 4.3 Left as an exercise to the reader. Theorem 13 Let f be left-monogenic on an open U ⊂ Ω as in Theorem 12 and let y ∈ U . Then Z 1 f (y) = E(x − y)η(x)f (x)dσx . an ∂U Similarly, whenever f is right-monogenic, we have 1 f (y) = an where E(x) = operator

x 1 ωm |x|m

Z

f (x)η(x)E(x − y)dσx ,

∂U

is the fundamental solution (Green function) of the Dirac ∂x E(x) = δ(x).

ωm =

m 2π 2 m Γ( 2 )

is the area of the unit sphere in Rm .

Proof 4.4 We only prove the result for the left-monogenic function. We consider the sphere S n−1 (y, r) for r > 0 chosen small enough such that the disc whose boundary is S n−1 (y, r) is included in U . Applying Theorem 12 on E and f , we get Z

E(x − y)η(x)f (x)dσx =

∂U

Z S n−1 (y,r)

E(x − y)η(x)f (x)dσx .

Observing that S n−1 (y, r), we immediately have η(x) =

y−x y−x x = = . |x| r x − y

Consequently, on S n−1 (y, r), E(x − y)η(x) =

x−yy−x r2 1 = = n−1 . n n+1 r r r r

70  Wavelet Analysis: Basic Concepts and Applications

As a result, Z S n−1 (y,r)

1

Z

E(x − y)η(x)f (x)dσx =

S n−1 (y,r)

Z

= S n−1 (y,r)

Z

= S n−1 (y,r)

rn−1

f (x)dσx

f (x) − f (y) + f (y) dσx rn−1 f (x) − f (y) dσx + rn−1 f (x) − f (y)

Z

=

S n−1 (y,r)

f (x) − f (y)

Z S n−1 (y,r)

n−1 x − y

f (y)dσx

Z

n−1 dσx + f (y) x − y

S n−1 (y,r)

1 = an

Z

S n−1 (y,r)

f (y) dσx + an

dσx

Z S n−1

dσx

f (x) − f (y)

Z

= S n−1 (y,r)

n−1 dσx + an f (y). x − y

By continuity, we have f (x) − f (y)

Z

lim

r→0 S n−1 (y,r)

n−1 dσx = 0, x − y

Hence, we finally obtain 1 f (y) = ωm

4.7

Z

E(x − y)η(x)f (x)dσx .

∂U

CLIFFORD-FOURIER TRANSFORM

In this section, we propose to review some basic concepts of the Clifford-Fourier transform. Recall that the classical Fourier transform can be seen as the operator exponential   X   ∞ π π k k 1 −i H , (4.13) F = exp −i H = 2 k! 2 k=0 where H is the scalar-valued operator  −1  ∆n + x2 + n (4.14) 2 called Hermite operator. A first characterization of these operators is the following dealing with the invariance under the Fourier transform.

H=

Proposition 4.14 The operators H and exp(−i π2 H) are Fourier invariant in the sense that [) = H(fb) H(f and

π π \ exp(−i H)(f ) = exp(−i H)(fb). 2 2

Clifford wavelets  71

Proof 4.5 We have 1 −1 (∆n f (x) + x2 f (x) + nf (x))e−ix·ξ dV (x) n m 2 2 (2π) R Z Z −1 1 1 −ix·ξ 2 = [ dV (x) + ∂ f (x)e x2 f (x)e−ix·ξ dV (x) n n 2 (2π) 2 Rm x (2π) 2 Rm Z 1 +n f (x)e−ix·ξ dV (x)] n (2π) 2 Rm i −1 h 2 b ξ f (ξ) + ∂ξ2 fb(ξ) + nfb(ξ) = 2 = H(fb)(ξ). Z

[)(ξ) = H(f

We now investigate the second point. We have 1 π \ exp(−i H)(f )(ξ) = n 2 (2π) 2

Z

1 = n (2π) 2

Z

= =

∞ X k=0 ∞ X k=0

Rm

(

Rm


k −i π2

k! −i π2 k!

π exp(−i H)(f )(x)e−ix·ξ dV (x) 2 ∞ X −i π2

k!

k=0

(


k

1 n (2π) 2

Z

)

H (f )(x) e−ix·ξ dV (x) k

) k

−ix·ξ

H (f )(x)e

dV (x)

Rm


k

\ k (f )(ξ). H

Now, by iterating on the parameter k and using the first point above, we get \ \ k (f ) = H {H k−1 (f )} · · · = Hk H 0 (f ) = Hk fb. \ H As a result, ∞ X −i π2 π \ exp(−i H)(f )(ξ) = 2 k! k=0


k

π Hk fb(ξ) = exp(−i Hfb). 2

To extend the Fourier transform to Clifford algebras, we start by exploiting the Hermite operators. The first step consists in splitting the Laplace operator into product of Dirac operators to obtain Clifford-valued operators. See, for example, [81, 85, 86, 190]. Denote



1 1 ∂x − x ∂x + x and Φ2 = ∂x + x ∂x − x . Φ1 = 2 2 The operators Φ1 and Φ2 may be expressed otherwise as in the following proposition. Proposition 4.15 Φ1 and Φ2 satisfy the following assertions: i. Φ1 = H + Γx . ii. Φ2 = H − Γx + n.

72  Wavelet Analysis: Basic Concepts and Applications n 2

iii. Φ1 + Φ2 = 2 H + iv. Φ1 − Φ2 = 2 Γx −




.


n 2

.

Proof 4.6 We recall firstly that Γx =

−1 2 (x∂x

− ∂x x − n). Consequently,



1 ∂x − x ∂x + x 2  1 2 ∂x + ∂x x − x∂x − x2 = 2  1
1 2 = ∂x − x2 − n − x∂x − ∂x x − n 2 2 = H + Γx .

Φ1 =

Similarly, we have

1 ∂x + x ∂x − x 2  1 2 ∂x − x2 − n + x∂x − ∂x x + n = 2
1 =H+ x∂x − ∂x x + 2n − n 2 = H − Γx + n.

Φ2 =

The assertions iii and iv follow from i and ii. Proposition 4.16 Φ1 and Φ2 are Fourier-invariant,in the sense that df = Φ fb and Φ 1 1

df = Φ fb. Φ 2 2

Proof 4.7 We prove firstly that b \ Γ x (f ) = Γξ (f ).

Indeed, we have  −1  [ [ \ Γ x∂x f − ∂x xf − nfb (ξ) x (f )(ξ) = 2  −1  d c − nfb (ξ) = ∂ξ ∂x f − ξ xf 2  −1  = ∂ξ ξ − ξ∂ξ − n fb(ξ) 2 = Γξ (fb)(ξ).

Consequently, df = Hf d + Γd b Φ 1 x f = Φ1 f .

Similarly for Φ2 . Now, we are able to define the Clifford-Fourier transform.

Clifford wavelets  73

Definition 4.17 The Clifford-Fourier transform is the couple F = (F+ , F− ) of exponential operators π π F+ = exp(−i H+ ) and F− = exp(−i H− ). 2 2 To join the classical form of the Fourier transform and to obtain a one operator F from the pair (F+ , F− ), researchers considered the rule F 2 = F+ F− . By letting 1 H = (H+ + H− ), 2 we come back to the classical form F = exp(−i π2 H). To join the operators Φ1 and Φ2 , a simple choice may be H+ = Φ1 −

n 2

and H− = Φ2 −

n . 2

Another alternative proposed in [75, 190] consists of taking H+ = Φ1

and H− = Φ2 − n.

As a consequence, we may define an integral representation for the new CliffordFourier transform as 1 π n F+ [f ](ξ) = Γξ − e−ix·ξ f (x)dV (x) exp −i n m 2 2 2 (2π) R    Z π 1 n F− [f ](ξ) = exp i Γξ − e−ix·ξ f (x)dV (x). n 2 2 (2π) 2 Rm 



Z



(4.15)

In terms of the pair of operators, one may set π F+ = exp −i H+ 4 1 2





π and F− = exp −i H− 4 1 2





and thus put 1

1

F = F+2 F−2 . Hereafter, we discuss some properties of the Clifford-Fourier transform. Proposition 4.18 For two Clifford algebra-valued functions f and g and a, b ∈ Cm , we have F± [f a + gb] = F± [f ] a + F+ [g] b. For λ > 0, we have F± [f (λ•)] (ξ) =

ξ 1 F [f ] ( ). ± λn λ

74  Wavelet Analysis: Basic Concepts and Applications

Proof 4.8 The first assertion is obvious. For the second assertion, we will show firstly that ξ 1 F [f (λ•)] (ξ) = n F [f ] ( ). λ λ We have Z 1 F [f (λ•)] (ξ) = f (λx)e−ix·ξ dV (x). n m 2 (2π) R By setting y = λx, we get F [f (λ•)] (ξ) =

1 1 n n λ (2π) 2

Z

ξ

f (y)e−ix· λ dV (y) =

Rm

1 b ξ f ( ). λn λ

It follows that 

and

n π Γξ − 2 2 

F+ [f (λ•)] (ξ) = exp −i

π n F− [f (λ•)] (ξ) = exp i Γξ − 2 2 







1 b ξ f( ) λn λ

1 b ξ f ( ). λn λ

Observe next that Γx = Γ x . As a result, λ



π n Γξ − 2 2 λ 

F+ [f (λ•)] (ξ) = exp −i



ξ 1 b ξ 1 f ( ) = n F+ [f ] ( ). n λ λ λ λ

Similarly for F− [f (λ•)] (ξ). Proposition 4.19 F± [•f (•)] (ξ) = ∓(∓i)n ∂ξ F+ [f ] (ξ). Proof 4.9 We have π n Γξ − F [•f (•)] (ξ) 2 2    π n = i exp −i Γξ − ∂ξ F [f ] (ξ) 2 2 (∞  ) X (−i π )k  n k 2 =i Γξ − ∂ξ F [f ] (ξ). k! 2 k=0 





F+ [•f (•)] (ξ) = exp −i

Observe next that ∂ξ Γξ + Γξ ∂ξ = (n − 1)∂ξ h

i

and that Γξ ∂ξ = ∂ξ n − 1 − Γξ . We deduce that 

n Γξ − ∂ξ = ∂ξ 2 



n−2 − Γξ . 2 

(4.16)

Clifford wavelets  75

Consequently, 

n Γξ − 2

k



∂ξ = ∂ξ

n−2 − Γξ 2

k

.

As a result, we obtain (

F+ [•f (•)] (ξ) = i

 ∞ X (−i π2 )k

k!

k=0 (∞ X

(−i π2 )k =i k! k=0 (

=i

∞ X (−i π2 )k

k!

k=0

n Γξ − 2

k )

n Γξ − 2

k





∂ξ

∂ξ F [f ] (ξ) )

∂ξ F [f ] (ξ)

n−2 − Γξ 2

k )

F [f ] (ξ)

π n F [f ] (ξ) Γξ − + 1 2 2      π n π ∂ξ exp i Γξ − F [f ] (ξ) = i exp i 2 2 2 = −∂ξ F− [f ] (ξ). 





= i∂ξ exp i

The same techniques hold for F− . In fact, we may have more general rules. Proposition 4.20 For k ∈ N, we have h

i

F+ x2k f (ξ) = (−1)k ∂ξ2k F+ [f ] (ξ) h

i

F+ x2k+1 f (ξ) = −(−1)k ∂ξ2k+1 F− [f ] (ξ). In fact, the explicit form of the kernel of (4.15) is already a difficult problem. We know just few cases where the expression is simplified such as the case where n = 2 in which Z 1 F+ [f ](ξ) = exp(ξ ∧ x)f (x)dV (x) 2π R2 and Z 1 F− [f ](ξ) = exp(x ∧ ξ)f (x)dV (x). 2π R2 For even dimensions, in general, a first step has been developed in [86] in terms of the Bessel function. For example, for n = 4 we have the kernels (ξ ∧ x) π |x ∧ ξ|−1/2 (1 + x · ξ)J1/2 (|x ∧ ξ|) + J3/2 (|x ∧ ξ|)(x · ξ) 2 |x ∧ ξ|

r

(ξ ∧ x) π |x ∧ ξ|−1/2 (1 − x · ξ)J1/2 (|x ∧ ξ|) + J3/2 (|x ∧ ξ|)(x · ξ) . 2 |x ∧ ξ|

K+ (x, ξ) = K− (x, ξ) =

!

r

Recently, a general form has been developed in [183] for all even dimensions.

!

76  Wavelet Analysis: Basic Concepts and Applications

4.8

CLIFFORD WAVELET ANALYSIS

Clifford wavelets or wavelets on Clifford algebras are the last variants of wavelet functions developed by researchers in order to overcome many problems that are not well investigated by classical transforms. The challenge in such concepts is not the wavelet functions themselves but the structure of Clifford algebras and their flexibility to include different forms of vector analysis in the same time. In the present section, we propose to develop two main methods to construct Clifford wavelets. The first one is based on spin groups and thus includes the factor of rotations in the wavelet analysis provided with the translation and dilatation factors. This method generalizes in some sense the first attempt in developing multidimensional wavelets such as Cauchy ones. See J. P. Antoine and his team works [11, 12, 13]. The second part is concerned with the development of wavelet analysis from monogenic functions, mainly polynomials. These constitute an extension of orthogonal polynomials to the case of Clifford algebras. Recall that orthogonal polynomials are widely applied in wavelet theory on Euclidean spaces. Extensions to the case of Clifford framework are few, especially the works of Ghent University group on Clifford analysis, Brackx et al. and recently more general extensions due to Arfaoui, Ben Mabrouk, Cattani in [24, 25]. 4.8.1

Spin-group based Clifford wavelets

As in the classical cases of wavelets on Euclidean spaces, we seek some properties that must be satisfied for a Clifford algebra-valued function to be a mother wavelet. Definition 4.21 (Clifford Wavelet) Let ψ ∈ L1 ∩ L2 (Rm , Rm , dV (x)) be such that ψψ ∈ L1 ∩ L2 (Rm , Rm , dV (x)). The function ψ is said to be a Clifford mother wavelet iff the following assertions hold simultaneously. h

i†

b b i. ψ(ξ) ψ(ξ)

is scalar.

ii. The admissibility condition h

Aψ = (2π)n

Z

i†

b b ψ(ξ) ψ(ξ)

|ξ|n

Rm

dV (ξ) < ∞.

The function ψ is said to be admissible and Aψ is its admissibility constant. We notice here also that being admissible as a mother wavelet the function ψ should satisfy some oscillation property such as b ψ(0) = 0 ⇐⇒

Z

ψ(x)dV (x). Rm

Example 14 A generalized Clifford Mexican hat wavelet has been provided in [78] by considering 1 x2 ψ(x) = exp( x2 )Hn (x) = (−1)n ∂x exp( ), 2 2

Clifford wavelets  77

where Hn is the Clifford version of the Hermite polynomial of degree n. We get n

b = (2π) 2 (−i)n ξ n exp( ψ(ξ)

ξ2 ). 2

Consequently, its admissibility constant will be b 2 ψ(ξ) n dV (ξ) Aψ = (2π)n R m ξ Z 2 2 ξ = (2π)2n exp( ) dV (ξ) 2 Rm Z ∞ Z

2

rm−1 e−r dr < ∞.

= (2π)n ωm

0

Now, starting with an admissible Clifford mother wavelet, we generate a whole set of wavelets by the action of general group of translations, dilations and spin-rotations, which generalizes the affine group applied in the case of real wavelets. For (a, b, s) ∈ R+ × Rm × Spin(n), denote ψ a,b,s (x) =

1 s(x − b)s )s. n sψ( a a2

It is straightforward that whenever ψ is admissible, the copies ψ a,b,s are also admissible and satisfy precisely Aψa,b,s =

an/2 Aψ < ∞. (2π)n

We will now see that these copies permit further to approximate any function in L2 (Rm , Rm , dV (x)). Proposition 4.22 The set Λψ = ψ a,b,s : a > 0, b ∈ Rm , s ∈ Spin(n) is dense in L2 (Rm , Rm , dV (x)). 



Proof 4.10 Let f ∈ L2 (Rm , Rm , dV (x)) be an analyzed function such that < ψ a,b,s , f >L2 (Rm ,Rm ,dV (x)) = 0, ∀a > 0, b ∈ Rm and s ∈ Spin(n). We shall prove that f = 0. Indeed, we already know from Parseval identity of the Clifford-Fourier transform that a,b,s , fb > 2 m < ψ a,b,s , f >L2 (Rm ,Rm ,dV (x)) =< ψ[ L (R ,Rm ,dV (x)) = 0.

On the other hand, observe that n

a,b,s , fb > 2 m < ψ[ L (R ,Rm ,dV (x)) = a 2

Z

h

Rm

This yields necessarily that † b s ψ(asξs) sfb(ξ) = 0.

h

i†

b eib·ξ s ψ(asξs) sfb(ξ)dV (ξ) = 0.

i

78  Wavelet Analysis: Basic Concepts and Applications

Now, as for fixed ξ 6= 0 in Rm , we already have n

o

asξs, a > 0 and s ∈ Spin(n) = Rm .

It results that fb = 0 and so f = 0. We now introduce the Clifford wavelet continuous transform of functions. Definition 4.23 The Clifford wavelet continuous transform of a function f in L2 (Rm , Rm , dV (x)) with respect to an admissible mother wavelet ψ is Tψ [f ] (a, b, s) =< ψ

a,b,s

Z

, f >L2 (Rm ,Rm ,dV (x)) =

i†

h

ψ a,b,s (x) f (x)dV (x).

Rm

Such transform possesses, as for other integral transforms of functions, many useful properties as stated in the following proposition. Proposition 4.24 The Clifford continuous wavelet transform is i. Translation-invariant, in the sense that Tψ [f (• − c] (a, b, s) = Tψ [f ] (a, b − c, s). ii. Dilation-invariant, in the sense that 

Tψ

1 • a b ) (a, b, s) = Tψ [f ] ( , , s). n f( λ λ λ2 λ 

iii. Spin-rotation-invariant, in the sense that Tψ [Lt f ] (a, b, s) = tTψ [f ] (a, tbt, ts)t. Proof 4.11 i. Observe that s(x − b)s s ψ a Rm

1 Tψ [f (• − c] (a, b, s) = n a2

Z

 

1 = n a2

Z

"

†

sf (x − c)dV (x).

s(y − (b − c))s a

s ψ Rm

!#†

sf (y)dV (y)

= Tψ [f ] (a, b − c, s). ii. We have 

Tψ

1 • 1 ) (a, b, s) = n n f( λ2 λ a2 

 

Z

s ψ Rm

s(x − b)s a

"

λ n = ( )2 a

Z

1 = a n (λ) 2

Z

†

s

s ψ

s(λy − b)s a

"

s(y − λb )s

Rm

s ψ Rm

a b = Tψ [f ] ( , , s). λ λ

a λ

1 x )dV (x) n f( λ2 λ

!#†

sf (y)dV (y)

!#†

sf (y)dV (y)

Clifford wavelets  79

iii. Observe that 1 n a2

Z

1 = n a2

Z

1 = n a2

Z

Tψ [Lt f ] (a, b, s) =

 

s ψ Rm

"

s ψ Rm

"

s ψ Rm

1 =t n a2

s(x − b)s a

stf (txt)tdV (x)

s(tyt − b)s a

!#†

s(tyt − b)s a

!#†

"

Z

†

stf (tyt)tdV (y) stf (tyt)tdV (y)

st(y − tbt)ts a

{ts} ψ

Rm

!#†

{st} f (tyt)tdV (y)t

= tTψ [f ] (a, tbt, ts)t We now propose to develop Parseval-Plancherel-type rules, which are the most important formula in wavelet theory as they permit reconstruction of functions from their wavelet transforms. We firstly introduce an inner product relative to the continuous Clifford wavelet transform. Let n o Hψ = Tψ [f ] , f ∈ L2 (Rm , Rm , dV (x)) be the image of L2 (Rm , Rm , dV (x)) relative to the operator Tψ . We define the inner product by 1 [Tψ [f ] , Tψ [g]] = Aψ

Z

Z Z

(Tψ [f ] (a, b, s))† Tψ [g] (a, b, s)

Spin(n) Rm R+

da dV (b)ds, an+1

where ds stands for the Haar measure on Spin(n). Proposition 4.25 Tψ : L2 (Rm , Rm , dV (x)) −→ Hψ is an isometry. Proof 4.12 We shall prove that [Tψ [f ] , Tψ [g]] =< f, g >L2 (Rm ,Rm ,dV (x)) . Denote Φψ (a, s, ξ) [f ] (−b) = and Φψ (a, s, ξ) [g] (−b) =

h

 i† b b ψ(asξs) sf (ξ) (−b)

h

 i† b ψ(asξs) sgb(ξ) (−b).

We immediately obtain n n Tψ [f ] (a, b, s) = a 2 s(2π) 2 Φψ\ (a, ξ, s) [f ] (−b)

and

n n Tψ [g] (a, b, s) = a 2 s(2π) 2 Φψ\ (a, ξ, s) [g] (−b).

(4.17)

80  Wavelet Analysis: Basic Concepts and Applications

Applying Parseval formula, we get D

E

\ \ Φψ (a, •, s) [f ], Φψ (a, •, s) [g] = hΦψ (a, •, s) [f ] , Φψ (a, •, s) [g]i .

Consequently, [Tψ [f ] , Tψ [g]] Z Z

Z

1 = (2π)n Aψ

 da (Φψ (a, ξ, s) [f ] (ξ)) Φψ (a, ξ, s) [g] (ξ)dV (b) ds a Rm

Spin(n) R+

Z (Z

Z

1 = (2π)n Aψ

†

) h † h i† i† da b b b ( ψ(asξs) sf (ξ) ds ψ(asξs) sb g (ξ)dV (ξ) a Rm

Spin(n) R+

 h i† da b b sψ(asξs) ψ(asξs) sb g (ξ)dV (ξ) ds a Spin(n) Rm + R   Z h Z i † Z h i† da  1 b b = fb(ξ) sψ(asξs) ψ(asξs) s ds gb(ξ)dV (ξ).  Spin(n) (2π)n Aψ Rm a 

1 = (2π)n Aψ

Z Z

Z

h

fb(ξ)

i†

R+

Observing now that Z

Z

Spin(n)

h i† da Aψ b b sψ(asξs) s ds = , ψ(asξs) a (2π)n

(4.18)

R+

we get immediately 1 [Tψ [f ] , Tψ [g]] = (2π)n Aψ

Z

h

 i† Z



Z

fb(ξ)

Rm



S n−1

R+

da  Γ(t, ν)dS(ν) gb(ξ)dV (ξ), t 

† b b where we denoted Γ(t, ν) = ψ(tν) . Otherwise, by taking u = tν, we obtain ψ(tν)

h

[Tψ [f ] , Tψ [g]] =

1 (2π)n Aψ

Z

Z

i†

=

h

Rm

i

h

 Z i† 

fb(ξ)

Rm

 

Rm

h

i†

b b ψ(u) ψ(u)

|u|n

  

dV (u) gb(ξ)dV (ξ)  

fb(ξ) gb(ξ)dV (ξ)

=< fb, gb > =< f, g > .

(4.19)

Corollary 4.26 The operator Tψ : L2 (Rm , Rm , dV (x)) −→ L2 (R+ × Rm × Spin(n),

1 dadV (b)ds ) Aψ an+1

is an isometry. More precisely, we have the Parseval-Plancherel equality Z

Z Z

Spin(n) Rm R+

(Tψ [f ] (a, b, s))2

da dV (b)ds = Aψ kf k22 . an+1

Clifford wavelets  81

As a result of the last corollary and as in the real case, we have here a Clifford wavelet reconstruction formula. Proposition 4.27 For all f ∈ L2 (Rm , Rm , dV (x)), we have 1 Aψ

f (x) =

Z

Z Z

ψ a,b,s (x)Tψ [f ] (a, b, s)

Spin(n) Rm R+

da dV (b)ds an+1

in L2 (Rm , Rm , dV (x)) . Proof 4.13 Let f and g be two square integrable Clifford-valued functions with Clifford wavelet transforms Tψ [f ] and Tψ [g], respectively. From 4.25 and using 4.23, we have < f, g >L2 (Rm ,Rm ,dV (x)) Z Z Z 1 da = (Tψ [f ] (a, b, s))† Tψ [g] (a, b, s) n+1 dV (b)ds Aψ a Spin(n) Rm R+

1 = Aψ = =

Z

Z Z

[Tψ [f ] (a, b, s)] † Tψ [g] (a, b, s)

Spin(n) Rm R+

1 Aψ

Z

1 Aψ

Z

=< f,

Z Z

Spin(n)

1 Aψ

< ψ a,b,s , f >†L2 (Rm ,Rm ,dV (x)) Tψ [g](a, b, s)

da dV (b)ds an+1

< f, ψ a,b,s >L2 (Rm ,Rm ,dV (x)) Tψ [g](a, b, s)

da dV (b)ds an+1

Rm R+

Z Z

Spin(n)

da dV (b)ds an+1

Rm R+

Z Z

Z Spin(n)

Then g(x) =

ψ a,b,s (x)Tψ [g](a, b, s)

Rm R+

1 Aψ

Z

Z Z

Spin(n)

da dV (b)ds >L2 (Rm ,Rm ,dV (x)) . an+1

ψ a,b,s (x)Tψ [g](a, b, s)

Rm R+

da dV (b)ds. an+1

The last result in this section deals with reproducing kernels relative to the continuous Clifford wavelet transform. 

−(n+1) Theorem 15 A function F (a, b, s) ∈ L2 R+ × Rm × Spin(n), A−1 dadV (b)ds ψ a is the Clifford wavelet transform of a square integrable function f iff

1 F (a, b, s) = Cψ

Z Spin(n)

Z Rm

Z

+∞



†

Kψ (a, b, s; a ˜, ˜b, s˜)

0

d˜ a F (˜ a, ˜b, s˜) n+1 dV (˜b)d˜ s, a ˜

  ˜ ˜, ˜b, s˜) = Tψ ψ a,b,s (˜ a, ˜b, s˜) =< ψ a˜,b,˜s , ψ a,b,s > is the reproducing where Kψ (a, b, s; a kernel.

Proof 4.14 Left to the readers as an exercise.



82  Wavelet Analysis: Basic Concepts and Applications

4.8.2

Monogenic polynomial-based Clifford wavelets

In this section, we propose to review a second method to construct wavelets on Clifford algebras. The idea is based on the so-called monogenic polynomials, which constitute an extension of orthogonal polynomials on Clifford algebras. There are, in fact, many types of such polynomials and different associated wavelets have been obtained. We will focus here on just one way due to the well-known Gegenbauer polynomials, known also as ultraspheroidal polynomials. Other classes may be developed by the readers by similar techniques. Furthermore, we may refer to [24, 25, 21, 20, 75, 79, 80, 82, 83, 86, 87, 76, 77, 78, 81, 85] for other existing classes of polynomials and associated wavelets in both the classical context and the Clifford one. Gegenbauer polynomials are defined on the orthogonality interval I =] − 1, 1[ relatively to the weight function 1

ωp (x) = (1 − x2 )p− 2 via Rodrigues rule as Gpm (x) =

(−1)m Γ(p + 12 )Γ(m + 2p) 1 dm ωp+m (x). 2m m!Γ(2p)Γ(p + m + 21 ) ωp (x) dxm

Denote next p Rm =

(−1)m Γ(p + m + 21 )Γ(p + 21 )Γ(m + 2p) 2m m!Γ(2p)

and akm,p =

k (−1)k Cm Γ(p + m + 12 − k)Γ(p +

By splitting

1

1 2

+ k)

.

1

ωp (x) = (1 − x)p− 2 (1 + x)p− 2 and applying Leibnitz derivation rule, we get the explicit form p Gpm (x) = Rm

m X

akm,p (1 − x)m−k (1 + x)k .

k=0

For m = 0 and m = 1, this gives, respectively, Gp0 (x) = 1 For m = 2, we get Gp2 (x)

Gp1 (x) = 2px.

and

1 = 2p(p + 1) x − 2p + 2 

2



and for m = 3, 4 3 = p(p + 1)(p + 2) x3 − x . 3 2p + 4 Gegenbauer polynomials Gpm may also be introduced via the induction rule stated for p ≥ −1 2 by Gp3 (x)





mGpm (x) = 2x(m + p − 1)Gpm−1 (x) − (m + 2p − 2)Gpm−2 (x), with initial polynomials Gp0 and Gp1 as above.

(4.20)

Clifford wavelets  83

Gegenbauer wavelets are examples of wavelets on the interval and depend on four parameters j, n, m, p. The parameter j ∈ N is related to the level of resolution, n ∈ {1, 2, 3, ..., 2j−1 } is related to the translation, m = 0, 1, 2, ..., M − 1, M > 0, 1 is the degree of the Gegenbauer polynomial, and finally a reel parameter p > − is 2 related to the order of Gegenbauer polynomials. The Gegenbauer mother wavelet is defined on [0, 1) by ψ m,p (x) = Gpm (x). Next, the translation-dilation copies of ψ m,p are defined by   √1 2 2j Gp (2j x − 2n + 1) m,p m ψj,n (x) = Lpm 

0

Remark 4.28

2n 2n − 2 ≤t< j, j 2 2 , elsewhere,

,

1 • For p = , we get Legendre wavelets. 2

• For p = 0 and p = 1, we obtain the Chebyshev wavelet of first and second kind, respectively. m,p To obtain the mutual orthogonality of Gegenbauer wavelets ψj,n , the weight function associated with the Gegenbauer polynomials has to be dilated and translated as for the Gegenbauer wavelets. Thus, we obtain a translation-dilation copy of the weight ω as 1 ωj,n (x) = ω(2j x − 2n + 1) = (1 − (2j x − 2n + 1)2 )p− 2 .

At fixed level of resolution, we get

ωj,n (x) =

   ωj,1 (x)          ωj,2 (x)       

, ,

ωj,3 (x)

                 

,

0≤x
= Cj,n

Z 0

1

m,p ωj,n (x)ψj,n (x) f (x) dx.

For more details, we may refer to [315], [316], [325], [333].

(4.21)

84  Wavelet Analysis: Basic Concepts and Applications

In this section, we propose to revisit the Gegenbauer polynomials associated with the real weight function ω(x) = (1 + x2 )α , α ∈ R, extended to the context of Clifford algebra-valued polynomials. Consider the Clifford algebra-valued weight function ωα (x) = (1 + |x|2 )α , α ∈ R. The general Clifford-Gegenbauer polynomials, denoted by G`,m,α (x), are generated by the CK-extension F ∗ (t, x) defined by F ∗ (t, x) =

∞ ` X t `=0

`!

G`,m,α (x) ωα−` (x); t ∈ R, x ∈ Rm .

As for the real case of orthogonal polynomials, we impose a left monogenic property on F ∗ in Rm+1 to obtain a recursive relation on the general Clifford-Gegenbauer polynomials G`,m,α . Hence, since F ∗ is monogenic, we have (∂t + ∂x )F ∗ (t, x) = 0.

(4.22)

The first part related to the time derivative is evaluated as ∂t F ∗ (t, x) =

∞ ` X t `=0

`!

G`+1,m,α (x) ωα−`−1 (x).

For the second part ∂x F ∗ (t, x), we shall use the following technical lemma. Lemma 4.29 For all n ∈ N, we have ∂x (xn ) = γn,m xn−1 , where

(

γn,m =

−n if n is even. −(m + n − 1) if n is odd.

So, now, observing that ∂x (x) = −m,

∂x (x2 ) = −2x and ∂x (|x|2 ) = 2x,

we get ∂x F ∗ (t, x) =

∞ ` X t `=0

`!




∂x G`,m,α (x)ωα−` (x) + G`,m,α (x)∂x ωα−` (x) .

Observing again that ∂x ωα−` (x) = 2(α − `)x ωα−`−1 (x), the monogenicity property (4.22) leads to the recurrence relation G`+1,m,α (x)ωα−`−1 (x) + ωα−` (x)∂x G`,m,α (x) +2(α − `)ωα−`−1 (x)xG`,m,α (x) = 0,

Clifford wavelets  85

or equivalently G`+1,m,α (x) = −2(α − `)xG`,m,α (x) − (1 + |x|2 )∂x G`,m,α (x).

(4.23)

Starting from G0,m,α (x) = 1, we obtain as examples G1,m,α (x) = −2αx, G2,m,α (x) = 2α[(2(α − 1) + m)x2 − m], G3,m,α (x) = [−4α((2α − 1) + m)(α − 1)]x3 + 4α(α − 1)(m + 2)x. The Clifford-Gegenbauer polynomials may also be introduced via the Rodrigues formula subject to the next proposition. Proposition 4.30 G`,m,α (x) = (−1)` ω`−α (x)∂x` ( ωα (x)). Proof. We proceed by recurrence on `. For ` = 1, we have ∂x ωα (x) = 2α x ωα−1 (x) = (−1)(−2αx) ωα−1 (x) = (−1) ωα−1 (x)Gα1,m (x), which means that Gα1,m = (−1) ω1−α (x)∂x ωα (x). For ` = 2, we get (2)

∂x ωα (x) = 2α[2(α − 1)x2 (1 − x2 )α−2 − m(1 − x2 )α−1 ] = (−1)2 ωα−2 [2α[2(α − 1) + m]x2 − m] = (−1)2 ωα−2 (x) G2,m,α (x). Hence, 2 (2) Gα,β 2,m = (−1) ω2−α (x)∂x ωα (x).

So, assume the recurrence hypothesis ` (`) Gα,β `,m (x) = (−1) ω`−α (x)∂x ωα (x).

Denote =(x) = −2(α − `)x(−1)` ω`−α (x)∂x(`) ωα (x), and 0

b∈Rm

Ca,b (f )Ca,b (g)

da dV (b) = Aµ,α `,m am+1

Z

f (x)g(x)dV (x). Rm

Proof. Using the Clifford-Fourier transform, we observe that m ^ \ b ψ(a.)(b), Ca,b (f )(b) = a 2 fb(.)

e where, h(u) = h(−u), ∀ h. Thus,     m m \ \ b b Ca,b (f )Ca,b (g) = fb(.)a 2 ψ(a.) (−b) gb(.)a 2 ψ(a.) (−b).

92  Wavelet Analysis: Basic Concepts and Applications

Consequently, Z

< Ca,b (f ), Ca,b (g) > = a>0

Z

Z Rm

Z

= a>0

= Aµ,α `,m

da dV (b) m m \ \ b b 2 ψ(a.) 2 ψ(a.) fb(.)a gb(.)a am+1

Rm

fb(b)gb(b)

2 b am |ψ(ab)| da dV (b) am+1

Z ω∈Rm

fd (b)gb(b)dV (b)

b b> = Aµ,α `,m < f , g

= < f, g > . Proof of Theorem 16. It follows immediately from Lemma 4.41.

4.9

SOME EXPERIMENTATIONS

In this section, we reproduce in brief an analogue application as in [24] based on the following two-dimensional Clifford weight function ω(x) = (1 + |x|2 )3/2 e−|x|

2 /2

,

where x = e1 x1 + e2 x2 , x1 , x2 ∈ R and (e1 , e2 ) is the canonical basis of R2 equipped with the product rule ei ej + ej ei = 2δij , where δij stands for the Kronecker symbol. By denoting r2 = x21 + x22 = |x|2 , we get as examples the Clifford mother wavelets (non-normalized) 2 ψ0,2 (x) = (1 + r2 )3/2 e−r /2 , ψ1,2 (x) = (r2 − 2)(1 + r2 )3/2 e−r

2 /2

x,

ψ2,2 (x) = (−4 + 6r4 − 2r6 )(1 + r2 )3/2 e−r

2 /2

,

and ψ3,2 (x) = (1 − 16r2 + 21r4 + 15r6 + r8 )(1 + r2 )3/2 e−r

2 /2

x.

These wavelets are represented in Figure 4.2. Some examples of these Clifford wavelets have been applied to some biosignals in [24, 25]. Interested readers may be referred to such references. We here conduct another different experimentation by conducting a wavelet decomposition of a 2D image by means of the Clifford wavelet ψ1,2 (x) above a single level (1-level) decomposition. The illustrations in Figure 4.3 have been obtained from the child Mohamed Amine Ben Mabrouk’s photo (son of the author Anouar Ben Mabrouk). The image has been processed for a 1-level wavelet decomposition. The upper right-hand sub-figure illustrates the 1-level approximation of the analyzed original image. The second line sub-figures and the left-hand sub-figure are the components of the 1-level detail projection of the original image.

Clifford wavelets  93

Figure 4.2

Some 2D Clifford wavelets.

The next experimentation concerns the application of 3D Clifford wavelets for 3D image processing tested on magnetic resonance images. We will apply here in the case where m = 3 the following 3D Clifford wavelets ψ0,3 (x) = (1 − x)ζ(ρ), h

i

ψ1,3 (x) = (ρ4 − ρ2 + 1) + (ρ2 − 2)x ζ(ρ), i

h

ψ2,3 (x) = (ρ6 + 4ρ4 + 12ρ2 + 5) + (−ρ6 − ρ4 − 2ρ2 + 2)x ζ(ρ) ψ3,3 (x) =

(ρ10 − 6ρ6 + 35ρ4 + 45ρ2 + 1) +(ρ8 − 2ρ6 − 2ρ4 + 2ρ2 + 5)x ζ(ρ),



ζ(ρ) = (1 + ρ2 )3/2 exp(−

ρ2 ) 2

94  Wavelet Analysis: Basic Concepts and Applications

Figure 4.3

ψ1,2 (x)-Clifford wavelet 1-level decomposition of Mohamed Amine’s photo.

where x = e1 x1 + e2 x2 + e3 x3 by means of the canonical basis (e1 , e2 , e3 ) of R3 equipped already with the extra operations ei ej + ej ei = −2δij and where ρ = |x| = q

x21 + x22 + x23 . As in [24], we examined the method using in the present case the 3D Clifford wavelet ψ3,3 (x) above. The experimentation has resulted in Figure 4.4.

Clifford wavelets  95

Figure 4.4

The 3D ψ3,3 (x) Clifford wavelet brain processing at the level J = 2.

96  Wavelet Analysis: Basic Concepts and Applications

4.10

EXERCISES

Exercise 1.

Show ∀x ∈ Rm and ∀Ak ∈ Rm , a k-grade, the following assertions. a. x · Ak = [xAk ]k−1 = 21 (xAk − (−1)k Ak x). b. x ∧ Ak = [xAk ]k+1 = 12 (xAk + (−1)k Ak x). c. Generalize the previous assertions to Ak · Bl = [Ak Bl ]|k−l| , with Ak · Bl = 0 if kl = 0 Ak ∧ Bl = [Ak Bl ]|k+l| for all k-grade element Ak and l-grade element Bl in Rm . Exercise 2.

a. Show that the main involution and the reversion commute: f∗ = a e∗ , ∀a ∈ Rn . a

b. Show that for all a =

n X

[a]k ∈ Rm =

m M

Rkm , we have

k=0

k=0

e= a

X

∗

a =

X

a=

X

(−1)k [a]k ,

k

(−1)k ([a]2k + [a]2k+1 ) ,

k

(−1)k ([a]2k − [a]2k+1 ) .

k

Exercise 3.

Consider the Pauli matrices σ0 =

1 0 0 1

!

, σ1 =

0 1 1 0

!

, σ2 =

0 −i i 0

!

, σ3 =

1 0 0 −1

!

.

a. Compute the products σi σj for all i, j. b. Let A be the algebra spanned by σi , i = 1, 2, 3, 4. Prove that A is isomorphic to M2 (C). Exercise 4.

Consider the Dirac matrices γ0 =

σ0 0 0 −σ0

!

, γ1 =

0 σ1 −σ1 0

!

, γ2 =

0 σ2 −σ2 0

!

, γ3 =

0 σ3 −σ3 0

!

.

Clifford wavelets  97

a. Compute γi γj + γj γi , i, j = 0, 1, 2, 3. b. Characterize the Clifford algebra A spanned by σi , i = 1, 2, 3, 4. c. Prove that A is isomorphic to M4 (C). Exercise 5.

Show that for a Clifford number x ∈ Rm | exp x| = exp(Re(x)). Exercise 6.

Let f be a vector field on Rm . Show that 1 a. div f = ∂x · f = (∂x f + f ∂x ). 2 1 b. curl f = ∂x ∧ f = (∂x f − f ∂x ). 2 Exercise 7.

Show that 



∂x [f (x)]+ = ∂x f (x)

−





and ∂x [f (x)]− = ∂x f (x)

+

Exercise 8.

Consider the Clifford-Hermite weight function W (x) = exp(−

|x|2 ). 2

Denote F ∗ (t, x) its CK-extension and Hl (x) the Clifford-Hermite polynomials generated by F ∗ (t, x). 1) By applying the characteristics of F ∗ (t, x) show that Hl+1 (x) − xHl (x) + ∂x Hl (x) = 0, ∀l ∈ N. 2) Show that Hl (x) satisfies the Rodrigues formula Hl (x) = (−1)l exp(

|x|2 l |x|2 )∂x ) exp(− ) . 2 2 



3) Prove finally the following mutual orthogonality for the polynomials Hl (x): Z

exp(− Rm

|x|2 )Hl (x)Hk (x)dV (x) = δl,k γl , 2

where γl,m is a constant depending only on l and m (to be explicated).

98  Wavelet Analysis: Basic Concepts and Applications

Exercise 9.

We conserve the same assumptions and notations as in Exercise 8. Consider for l ∈ N the function |x|2 )Hl (x). ψl (x) = exp(− 2 1) Compute the Clifford-Fourier transform of ψl denote by F[ψl ](ξ), ∀ξ ∈ Rm . 2) Compute the admissibility constant 

(2π)m Al = ωm

Z Rm

†

F[ψl ](ξ) F[ψl ](ξ) |ξ|m

dV (ξ),

where ωm is the volume of the unit sphere Sm in Rm . 3) Show finally that Z Rm

xj ψl (x)dV (x) = 0, for j = 0, 1, . . . , l − 1.

Exercise 10.

Let m ∈ N decomposed into a sum m = p + q in such a way the vector variable x ∈ Rm = Rp+q will be decomposed as a sum x = y + z, with y ∈ Rp and z ∈ Rq . Consider next the Clifford-Hermite weight W (x) = exp(−α

|y|2 |w|2 ) exp(−β ), α, β > 0. 2 2

Let Fα,β ∗ (t, x) be its CK-extension and Hlα,β (x) the Clifford-Hermite polynomials generated by Fα,β ∗ (t, x). By following the same steps as Exercise 8 and Exercise 9, 1) Provide a recurrence rule for the polynomials Hlα,β (x). 2) Provide a Rodrigues formula for Hlα,β (x). 3) Prove finally a mutual orthogonality relation for Hlα,β (x). 4) Provide an analogue ψlα,β (x) for the Clifford-Hermite wavelet. 5) Compute the Clifford-Fourier transform of the new wavelet ψlα,β (x). 6) Compute its admissibility constant. 7) Show finally that ψlα,β (x) has l vanishing moments. 8) Try to express Hl1,1 (x) by means of Hl (y) and Hl (z).

CHAPTER

5

Quantum wavelets

5.1

INTRODUCTION

Quantum wavelets are special types of wavelet functions characterized by special properties that may not be satisfied by other functions. In the present context, our aim is to develop wavelet functions based on some special functions such as Bessel one. Bessel functions form an important class of special functions and are applied almost everywhere in mathematical physics. They are also known as cylindrical functions, or cylindrical harmonics, because of their strong link to the solutions of the Laplace equation in cylindrical coordinates. The purpose of this chapter is to present some wavelet analysis in the framework of quantum theory (q-theory). We aim precisely to review details on q-Bessel functions introduced in the context of q-theory, which makes a variant of Bessel functions on the real line. Famous relations associated with wavelet transforms such as Plancherel/Parseval ones as well as reconstruction formula will be investigated. A first step in wavelet/Bessel transform interconnection has been conducted in [216], where the theory of continuous wavelet transform has been extended to a class of generalized one associated with a class of singular differential operators. This class contains, in particular, the so-called Bessel function (see also [347]). Such an extension is natural as Bessel functions may yield good bases for functional spaces similar to known cases such as Fourier, Haar and orthogonal polynomials. Besides, Bessel functions are applied in various domains, especially partial differential equations, wave motion, diffusion, etc. Quantum or q-theory, in fact, provides a discrete refinement of continuous harmonic analysis on suitable sub-spaces such as Rq composed of the discrete grid ±q n , n ∈ Z, q ∈ (0, 1), which is a dense subset in R.

5.2 BESSEL FUNCTIONS There are three main ways to introduce Bessel functions. The first one is due to a special type of second-order singular and linear differential equations where the Bessel functions form a generator set of solutions. The second one resembles orthogonal polynomials and provides Bessel functions as solutions of three-level recurrent

99

100  Wavelet Analysis: Basic Concepts and Applications

singular and linear functional equation. The third one provides Bessel functions via the Rodriguez derivation formula. Definition 5.1 The Bessel equation is a linear differential equation of second order written in the form ! 1 0 v2 y” + y + 1 − 2 y = 0, x x where v is a positive constant. Theorem 17 Bessel’s differential equation has a solution of the form Jv (x) =

 v X x

2

k≥0

(−1)k k!Γ(v + k + 1)

 2k

x 2

.

Γ is the Gamma Euler’s function. The function Jv is called Bessel function of the first kind of order v. Proof. Put y = xp

X

ai x i =

i≥0

X

ai xi+p ,

i≥0

where p is a real parameter. By replacing y and its derivatives in Definition 5.1, we get a0 (p2 − v 2 ) = 0, a1 ((p + 1)2 − v 2 ) = 0 and ai ((i + v)2 − v 2 ) + ai−2 = 0, ∀ i ≥ 2. For p = v, we get a1 = 0 and ai = −

ai−2 , ∀ i ≥ 2. i(2v + i)

Hence, a2k+1 = 0 for all k ≥ 0, and a2k = (−1)k Taking a0 =

a0 , ∀k ≥ 0. 22k k!(v + k)(v + k − 1) . . . (v + 1)

1 and observing that 2v Γ(v + 1) Γ(v + k + 1) = (v + k)(v + k − 1) . . . (v + 1)Γ(v + 1),

we get a2k = (−1)k So as the theorem.

1 22k+v k!Γ(v

+ k + 1)

, k ≥ 0.

Quantum wavelets  101

Remark 5.2 1. For p = −v, the solution of Bessel’s equation in Definition 5.1 is called Bessel’s function of the first kind with order −v and is denoted J−v (x) with  −v X  2k x x (−1)k J−v (x) = . 2 k!Γ(k − v + 1) 2 k≥0 2. The same solution can be obtained by choosing p+1 = v in the proof of Theorem 17. 3. For v = n ∈ N, we have Jn = (−1)n J−n . 4. The Bessel function of the second kind of order α denoted usually Yα is given by  cos(πα)Jα (x) − J−α (x)   , f or α ∈ /Z  sin(πα) Yα (x) = cos(πv)Jv (x) − J−v (x)    lim , f or α ∈ Z. v→α sin(πv) Proposition 5.3 Let λ and µ be two different roots of the Bessel function Jv (x). The following orthogonality property holds, Z

1

x Jv (λx)Jv (µx) dx = 0. 0

Proof. Denote yv,λ (x) = Jv (λx)

and

yv,µ (x) = Jv (µx).

Then, yv,λ and yv,µ are solutions of the following Bessel-type differential equations 0 (xyv,λ )0 (x) + (λ2 x −

and

v2 )yv,λ (x) = 0 x

(5.1)

v2 )yv,µ (x) = 0. (5.2) x and equation (5.2) by yv,λ and integrating the

0 (xyv,µ )0 (x) + (µ2 x −

Multiplying equation (5.1) by yv,µ difference on (0, 1) we get Z 1h 0

i

0 0 (xyv,λ )0 (x)yv,µ (x) − (xyv,µ )0 (x)yv,λ (x) dx + (λ2 − µ2 )

Z

1

xyv,λ (x)yv,µ (x)dx = 0. 0

(5.3) Next, an integration by parts in the first integral yields Z 1h 0

i

0 0 0 0 (xyv,λ )0 (x)yv,µ (x) − (xyv,µ )0 (x)yv,λ (x) dx = yv,λ (1)yv,µ (1) − yv,µ (1)yv,λ (1).

Recall now that yv,λ (1) = Jv (λ) = yv,µ (1) = Jv (µ) = 0.

102  Wavelet Analysis: Basic Concepts and Applications

This implies that Z 1h 0

i

0 0 (xyv,λ )0 (x)yv,µ (x) − (xyv,µ )0 (x)yv,λ (x) dx = 0.

Hence, it remains in (5.3) (λ2 − µ2 )

Z

1

x Jv (λx)Jv (µx) dx = 0. 0

Since λ 6= µ, we get Z

1

x Jv (λx)Jv (µx) dx = 0. 0

Figure 5.1

Graphs of Bessel functions Jv of the first kind for v = 0, 1, 2, 3.

Quantum wavelets  103

Figure 5.2

Graphs of Bessel functions Yv of the second kind for v = 0, 1, 2, 3.

We now review the notions of Bessel and Fourier-Bessel transforms of functions. The readers may refer to [217] for more details. We denote the inner product in L2 (R+ , dx) by Z ∞

hf, gi =

f (x) g(x) dx 0

and the associated norm by k.k2 . Similarly, we denote the inner product in L2 (R+ , ξdξ) by Z ∞

hf, giξ =

f (ξ) g(ξ) ξdξ 0

and the associated norm by k.kξ,2 . Definition 5.4 Let f ∈ L2 (R+ , dx). The Bessel transform of f is defined by B(f )(ξ) =

Z

+∞

√ f (x) xJv (xξ) dx, ∀ ξ > 0,

0

where Jv is the Bessel function of first kind and index v. We immediately have the following characteristics. Proposition 5.5

1. For all f ∈ L2 (R+ , dx), B(f ) ∈ L2 (R+ , ξdξ).

104  Wavelet Analysis: Basic Concepts and Applications

2. The Bessel transform B is invertible and B −1 (g)(x) =

Z

√ g(ξ) x Jv (xξ) ξ dξ, ∀g ∈ L2 (R+ , ξdξ).

+∞

0

Proof. 1. Let f and g be in L2 (R+ , dx). We have < B(f ), B(g) >ξ =

+∞

Z

B(f )(ξ)B(g)(ξ)ξdξ

0

√ √ x yf (x)g(y)Jv (xξ)Jv (yξ)ξdx dy dξ

Z

= R3+

√ √ δ(x − y) x yf (x)g(y) dx dy x Z √ √ 1 = x xf (x)g(x) dx x R+ Z

=

R2+

= < f, g > . So, taking g = f , we get kB(f )kξ,2 = kf k2 , which means that B is an isometry. e ). We will prove that B(B(f e )) = f . Indeed, 2. Denote the right-hand quantity by B(f e ))(ξ) = B(B(f

Z

+∞

√ e )(x) xJv (xξ) dx B(f

+∞

Z

0

Z

+∞

= 0

Z

√ √ f (η) xJv (xη) η xJv (xξ) dη dx

0 +∞

f (η)η

= 0

δ(η − ξ) dη = f (ξ). η

Definition 5.6 The Fourier-Bessel transform of order v is defined by FB(f )(ξ) =

Z

∞

f (x) Jv (xξ) xdx; ∀ f.

(5.4)

0

Lemma 5.7 Fourier-Bessel transform and Bessel transform are related via the equality √ FB(f )(ξ) = B( .f )(ξ). Indeed, For t > 0, we have Z √ B( tf )(ξ) =

+∞

√

√ tf (t) tJv (tξ) dt

Z0 +∞

=

f (t) t Jv (tξ) dt Z0 ∞

=

f (t) Jv (tξ) tdt 0

= FB(f )(ξ).

Quantum wavelets  105

5.3

BESSEL WAVELETS

There are in literature several approaches to introduce Bessel wavelets. We refer, for instance, to [315, 317]. For 1 ≤ p < ∞ and µ > 0, denote 

Lpσ (R+ )

kf kpp,σ

= f : R+ → R;

Z

=

∞

p



|f (x)| dσ(x) < ∞ ,

0

where dσ(x) is the measure defined by dσ(x) =

x2µ 1

2µ− 2 Γ(µ + 12 )

χ[0,∞[ (x) dx.

Denote also

1 1 1 jµ (x) = 2µ− 2 Γ(µ + ) x 2 −µ Jµ− 1 (x), 2 2 1 where Jµ− 1 (x) is the Bessel function of order v = µ − 2 . 2 Denote next Z ∞ jµ (xt) jµ (yt) jµ (zt) dσ(t). D(x, y, z) =

0

Definition 5.8 For a 1-variable function f , we define a translation operator ∞

Z

D(x, y, z) f (x) dσ(z), ∀ 0 < x, y < ∞,

τx f (y) = fe(x, y) = 0

and for a 2-variables function f , we define a dilation operator 1 x y Da f (x, y) = 2µ+1 f ( , ). a a a Proposition 5.9 Z

∞

jµ (zt) D(x, y, z) dσ(z) = jµ (xt) jµ (yt), ∀ 0 < x, y < ∞, 0 ≤ t < ∞

0

and

Z

∞

D(x, y, z) dσ(z) = 1. 0

Proof. We will develop the proof of the first point. The second is left to the reader as a simple exercise. So, simple observations permit us to write Z

∞

jµ (zt)D(x, y, z)dσ(z) 0

= where

(xyt)1/2−µ Cµ2

∞

Z 0

Z

∞

s3/2−µ J1/2−µ (xs)J1/2−µ (ys)zJ1/2−µ (xzt)J1/2−µ (zs)dsdz,

0

1 1 = 2µ− 2 Γ(µ + 12 ). Using Fubini’s rule, we obtain Cµ Z

∞

jµ (zt)D(x, y, z)dσ(z) 0

δ(t − s) (xyt)1/2−µ ∞ 3/2−µ s J1/2−µ (xs)J1/2−µ (ys) ds = Cµ2 s 0 (xyt)1/2−µ 3/2−µ 1 = t J1/2−µ (xt)J1/2−µ (yt) 2 Cµ t = jµ (xt)jµ (yt). Z

106  Wavelet Analysis: Basic Concepts and Applications

Definition 5.10 The Bessel wavelet copies Ψa,b are defined from the Bessel mother wavelet Ψ ∈ L2σ (R+ ) by 1

Ψa,b (x) = Da τb Ψ(x) =

∞

Z

a2µ+1

0

b x D( , , z) Ψ(z) dσ(x), ∀a, b ≥ 0. a a

The continuous Bessel wavelet transform (CBWT) of a function f ∈ L2σ (R+ ), at the scale a and the position b, is defined by (BΨ f )(a, b) =

1

∞

Z

a2µ+1

∞

Z 0

0

b t f (t) Ψ(z) D( , , z) dσ(z) dσ(t). a a

It is well known in Bessel wavelet theory that such a transform is a continuous function according to the variable (a, b). The following result is a variant of Parseval/Plancherel rules for the case of Bessel wavelet transforms. Theorem 18 [315] Let Ψ ∈ L2σ (R+ ) and f, g ∈ L2σ (R+ ).Then Z

∞

Z

∞

(BΨ f )(a, b) (BΨ g)(a, b) 0

0

whenever AΨ =

∞

Z

dσ(a) dσ(b) = AΨ hf, gi, a2µ+1

2 b t−2µ−1 |Ψ(t)| dσ(t) < ∞.

0

Proof. We have Z

+∞

f (t) Ψa,b (t) dσ(t)

(BΨ f )(a, b) = 0

1

=

Z

a2σ+1

∞

0

Z 0

∞

b t f (t) Ψ(z) D( , , z) dσ(z) dσ(t). a a

Now observe that b t D( , , z) = a a

+∞

Z 0

b t jµ ( u) jµ ( u) jµ (zu) dσ(u). a a

Hence, (BΨ f )(a, b) =

=

=

1

Z

a2µ+1 1

R3+

b t f (t)Ψ(z)jµ ( u)jµ ( u)jµ (zu)dσ(u)dσ(z)dσ(t) a a

u b fb( )Ψ(z)jµ ( u)jµ (zu)dσ(u)dσ(z) a a R2+

Z

a2µ+1 1

u b b fb( )Ψ(u)j µ ( u)dσ(u) a a R+

Z

a2µ+1 Z

=

b fb(η)Ψ(aη)j µ (bη)dσ(η) R+

=





b fb(η)Ψ(aη) (b).

Quantum wavelets  107

As a result, Z

(BΨ f )(a, b)(BΨ g)(a, b)

R2+

dσ(a) dσ(b) a2η+1

Z

=

b b fb(η)Ψ(aη) gb(η)Ψ(aη)dσ(η)

R2+

2 dσ(a) b |Ψ(aη)|

Z

Z

=

fb(η)gb(η)

R2+

dσ(a) a2σ+1 

a2σ+1

R+

dσ(η)

Z

= CΨ R+

fb(η)gb(η)dσ(η)

= CΨ hfb, gbi = CΨ hf, gi. Theorem 19 Let Ψ ∈ L2σ (R+ ) and f ∈ L2σ (R+ ). Then f (x) = in the

L2σ -sense

1 AΨ

Z Z

(BΨ f )(a, b) Ψ(

whenever AΨ =

∞

Z

x − b dσ(a)dσ(b) ) a a2µ+1

2 b t−2µ−1 |Ψ(t)| dσ(t) < ∞.

0

The proof is an immediate consequence of the previous theorem.

5.4

FRACTIONAL BESSEL WAVELETS

Definition 5.11 The Hankel transform, also called the Bessel-Fourier transform, of a function ϕ ∈ L1 (R+ ) is defined by Z b (hµ ϕ)(y) = ϕ(y) =

∞

0

1 1 (xy) 2 Jµ (xy)ϕ(x)dx, x ∈ R+ , µ ≥ − . 2

Its inverse formula is b (h−1 = ϕ(x) = µ ϕ(y))(x)

∞

Z 0

1

b (xy) 2 Jµ (xy)ϕ(y)dy, y ∈ R,

where Jµ is the Bessel function of first kind and with order µ. Definition 5.12 The fractional Hankel transform, also called fractional BesselFourier transform, of parameter θ on L1 (R+ ) is defined by (hθµ ϕ)(y)

=

ϕbθµ (y)

Z

= 0

∞

Kµθ (x, y)ϕ(x)dx,

where Kµθ is the kernel Kµθ (x, y)

=

 1 θ i (x2 +y 2 ) cot θ  (xy cos θ) 2 Jµ (xy cos θ), cµ e 2  1 2

(xy) Jµ (xy),

  

δ(x − y),

θ 6=qnπ, π θ= , 2 θ = nπ, ∀n ∈ Z

108  Wavelet Analysis: Basic Concepts and Applications

and δs = 1 for s = 0 and 0 elsewhere is the Dirac mass or the Kronecker symbol, and finally   exp i(1 + µ)( π2 − θ) θ , θ 6= nπ. cµ = sin θ Definition 5.13 Let f, g ∈ L1 (R+ ). We define the fractional Hankel convolution by +∞

Z

(f ]θ g)(x) = 0

f (y) (τxθ g)(y) dy,

where the fractional Hankel translation τxθ g is given by (τxθ g)(y)

θ

= g (x, y) = e

−i (x2 +y 2 ) cot θ 2

∞

Z

g(z) Dµθ (x, y, z) dz,

0

and

i

Dµθ (x, y, z)

2

2

2µ−1 5 (x, y, z)2µ−1 cθµ e 2 (x +y +z = 1√ (xyz)µ− 2 π Γ(µ + 21 )

2 ) cos θ

,

where 5(x, y, z) is the area of the triangle of vertices x, y, z. Proposition 5.14 1. Dµθ (x, y, z) is symmetric in x, y, z. 2. For ξ = 0, we obtain ∞

Z 0

1

1 θ Dµ (x, y, z) z µ+ 2 dz ≤

(xy)µ+ 2

1

2µ Γ(µ + 1) |sin θ|µ+ 2

.

Lemma 5.15 For ϕ ∈ L2 (R), we have 1

1

kx−µ− 2 (τxθ ϕ)(y)kL2 ≤

µ+ 12

| sin θ|

2µ Γ(µ + 1)

kϕkL2 .

Proof. We have (τxθ ϕ)(y)

θ

= ϕ (x, y) = e

−i (x2 +y 2 ) cot θ 2

Z

∞

ϕ(z) Dµθ (x, y, z) dz,

0

Using Proposition 5.14. above, we obtain θ (τ ϕ)(y) x Z ∞ ≤ ϕ(z) Dµθ (x, y, z) dz 0

≤

∞

Z 0

Z

≤

1 1 −1 ϕ(z) z 2 (µ+ 12 ) Dθ (x, y, z) 2 z 12 (µ+ 21 ) Dθ (x, y, z) 2 dz µ µ

∞

z 0

−(µ+ 21 )

 12 Z θ |ϕ(z)| Dµ (x, y, z) dz 2

z

(µ+ 21 )

0

! 1 Z 2

1

≤

∞

(xy)µ+ 2

z

1

2µ Γ(µ + 1) |sin θ|µ+ 2

∞

0

−(µ+ 21 )

 12 θ Dµ (x, y, z) dz

 12 θ |ϕ(z)| Dµ (x, y, z) dz . 2

Quantum wavelets  109

Hence, ∞

Z 0

2 θ (τx ϕ)(y) dy 1

≤

xµ+ 2

Z 0

xµ+ 2

Z

z

−(µ+ 21 )

∞

Z  1 2

µ+ 2

2µ Γ(µ + 1)| sin θ|

1 θ Dµ (x, y, z) y µ+ 2 dy

(xz)µ+ 2

2

|ϕ(z)|

1

2µ Γ(µ + 1)| sin θ|µ+ 2

0

x2(µ+ 2 ) 

∞

1

∞

1

2µ Γ(µ + 1)| sin θ|µ+ 2

Z 0

1

=

1

z −(µ+ 2 ) |ϕ(z)|2 dz

1

2µ Γ(µ + 1)| sin θ|µ+ 2 1

≤

∞

dz

|ϕ(z)|2 dz.

0

Consequently, 1

1

kx−µ− 2 (τxθ ϕ)(y)kL2 ≤

µ+ 12

| sin θ|

kϕkL2 .

2µ Γ(µ + 1)

Lemma 5.16 If ϕ ∈ L2 (R), then ∞

Z 0

2 θ (τy ϕ)(x) dx ≤ 

1

y 2(µ+ 2 ) 2µ

Z  1 2

µ+ 2

Γ(µ + 1)| sin θ|

∞

|ϕ(z)|2 dz.

0

Definition 5.17 The function space Hµ (R+ ) is composed of infinitely differentiable function ϕ on R+ with complex values, which satisfies



1

µ Γm,k (ϕ) = sup xm (x−1 D)k [x−(µ+ 2 ) ϕ(x)] < ∞, ∀µ ∈ R, m, k ∈ N∗ . x∈R+

Definition 5.18 The function space Hµ,θ (R+ ) is composed of class function C ∞ , ϕ on R+ with complex values, which satisfies for k, m ∈ N∗



θ Υm,k (ϕ) = sup xm ∆kµ,x ϕ(x) < ∞,

∀θ 6= nπ, n ∈ Z,

x∈R+

where

#

"

∆µ,k

d2 d 1 − 4µ2 = + 2ix cos θ +( ) + i cos θ − x2 cos2 θ , 2 dx dx 4x2

and ∆kµ,x

=x

−2k

2k X

l=2k X

r=0

l=0

al x

2l

!

x−1

d dx

r

,

where the constant al depends only on µ and θ. Proposition 5.19 Let Kµθ (x, y) be the fractional Hankel transform kernel. Then, we have

110  Wavelet Analysis: Basic Concepts and Applications

1. ∆rµ,x Kµθ (x, y) = (−y 2 cos2 θ)r Kµθ (x, y), r ∈ N∗ . 



2. hθµ (∆∗µ,x )r ϕ(x) (y) = (−y 2 cos2 θ)r hθµ ϕ(y) and ϕ ∈ Hµ (R+ ), where

"

∆∗µ,x

d 1 − 4µ2 d2 = − 2ix cos θ + − i cos θ − x2 cos2 θ dx2 dx 4x2

#

is the fractional Bessel operator of parameter θ. Proposition 5.20 Let ϕ ∈ L1 (R+ ). Then, ϕbθµ satisfies the following properties 1. ϕbθµ ∈ L∞ (R+ ) with kϕbθµ kL∞ ≤q Aµ,θ kϕkL1 , where Aµ,θ is a positive constant that depends on µ and θ. 2.

lim

y−→±∞

ϕbθµ (y) = 0.

3. ϕbθµ is continuous on R+ . Proof. 1) It follows from Definition 5.12 that ϕbθµ (y) =

∞

Z 0

Kµθ (x, y)ϕ(x)dx.

In addition, we have kϕbθµ kL∞

= sup essy∈R |ϕbθµ (y)|, ∞

Z

= sup essy∈R ≤ sup essy∈R

0

Z 0

≤ Aµ,θ

Z

∞



Kµθ (x, y)ϕ(x)dx , |Kµθ (x, y)||ϕ(x)|dx,

∞

|ϕ(x)|dx,

0

= Aµ,θ kϕ|kL1 < ∞. 2) It follows from Proposition 5.19 and Assertion 2 for r = 1 that 



hθµ ∆∗µ,x ϕ(x) (y) = (−y 2 cos2 θ) hθµ ϕ(y). Hence, hθµ ϕ(y) = ϕbµθ (y) =

−y 2

  1 hθµ ∆∗µ,x ϕ(x) (y). 2 cos θ

Consequently, |ϕbµθ (y)| =

  1 ∗ θ |h ∆ ϕ(x) (y)| −→ 0 as y −→ ±∞. µ µ,x | − y 2 cos2 θ|

Quantum wavelets  111

3) Let h > 0. We shall show that kϕbµθ (y + h) − ϕbµθ (y)k∞ → 0 as h −→ ±0. Indeed, sup |ϕbµθ (y + h) − ϕbµθ (y)| y∈R

=

Z sup

∞

Z sup

∞

0

y∈R

=

y∈R

≤

0

Kµθ (x, y

[Kµθ (x, y ∞

Z

|cθµ | sup y∈R 0

+ h)ϕ(x)dx −

∞

Z 0

+ h) −



Kµθ (x, y)ϕ(x)dx

Kµθ (x, y)] ϕ(x)dx

h

1

eih( 2 cot θ+y cot θ) (x(y + h) cos θ) 2

1

Jµθ (x(y + h) cos θ) − (xy cos θ) 2 Jµθ (xy cos θ) |ϕ(x)| dx ≤ Aµ,θ

∞

Z

|ϕ(x)|dx.

0

In addition, when h → 0, we have h i 1 h 1 1 (x cos θ) 2 eih( 2 cot θ+y cot θ) (y + h) 2 Jµθ (x(y + h) cos θ) − y 2 Jµθ (xy cos θ) → 0.

Hence, sup |ϕbµθ (y + h) − ϕbµθ (y)| → 0 as h → 0. y∈R

Proposition 5.21 (Parseval formula) Let Φ and Ψ be the fractional Hankel transforms of ϕ and ψ, respectively, that is to say Φ(y) = (hθµ ϕ)(y) and Ψ(y) = (hθµ ψ)(y). Then, we have Z

∞

∞

Z

ϕ(x) ψ(x)dx = sin θ

Φ(y) Ψ(y)dy.

0

0

In particular,

∞

Z

2

|ϕ(x)| dx = sin θ

0

Z

∞

|Φ(y)|2 dy.

0

Proof. Observe firstly that ψ(x) = (cθµ ) sin θ Denote next

Z

∞

i

2 +y 2 ) cot θ

e− 2 (x

0

i

2 +y 2 ) cot θ

Eθ (x) = e− 2 (x

1

(xy cos θ) 2 Jµθ (xy cos θ)(hθµ ψ)(y)dy. 1

(xy cos θ) 2 .

112  Wavelet Analysis: Basic Concepts and Applications

We get hϕ, ψi = cθµ sin θ Z

∞

Z 0 ∞

(hθµ ψ)(y)

Z

∞

0



Eθ (x)Jµθ (xy cos θ)ϕ(x)dx dy

(hθµ ϕ)(y)(hθµ ψ)(y)dy

= sin θ 0

∞

Z

= sin θ

Φ(y) Ψ(y)dy. 0

Hence, the essential part is proved. Now, for ϕ = ψ, we get the particular case. Definition 5.22 For θ 6=qnπ, n ∈ Z, we define fractional mother wavelet by 1 x−b ψa,b,θ (x) = √ ψ a a 



e

−i (x2 −b2 ) cot θ 2

.

The continuous fractional Bessel wavelet transform (CFrBWT) is the generalization of the continuous Bessel wavelet transform (CBWT) of parameter θ. Definition 5.23 A fractional Bessel wavelet is a function ψ ∈ L2 (R+ ) satisfying the admissibility condition Z

Cµ,ψ,θ = 0

∞

1 x−2µ−2 |(hθµ ψ)(x)|2 dx < ∞, ∀µ ≥ − . 2

Definition 5.24 By dilation and translation operations of ψ ∈ L2 (R+ ), we obtain a θ fractional Bessel wavelet family ψa,b (x) given by i b2 x2 1 1 θ ψa,b (x) = √ Da τbθ ψ(x) = √ e− 2 ( a2 + a2 ) cot θ a a

Z 0

∞

b x ψ(z) Dµθ ( , , z) dz. a a

Lemma 5.25 If ψ ∈ L2 (R+ ), we have 1

θ kψb,a kL2 ≤

1

b(µ+ 2 ) a−(µ+ 2 ) 1

| sin θ|µ+ 2 2µ Γ(µ + 1)

kψkL2 .

Proof. Using Definitions 5.14 and 5.24 and the inequality of Cauchy-Schwarz, we

Quantum wavelets  113

obtain θ ψa,b (x)|

=

∞

Z 1 − i ( b2 + x2 ) cot θ √ e 2 a2 a2 a

0

≤ ≤

≤

b x |ψ(z)| |Dµθ ( , , z)| dz a a

∞

1 1 1 1 1 1 b x b x |ψ(z)| |z − 2 (µ+ 2 ) | |Dµθ ( , , z)| 2 |z 2 (µ+ 2 ) | |Dµθ ( , , z)| 2 dz a a a a

Z

1 √ a

Z

1 √ a

Z

0

0

∞

2

|ψ(z)| |z

−(µ+ 21 )

0 ∞

Z 0

≤

x , z) dz a a

∞

1 √ a

1 √ a



b ψ(z) Dµθ ( ,

x , z)| dz a a

b | |Dµθ ( ,

1

2

b |Dµθ ( ,

1 x , z)| |z (µ+ 2 ) | dz a a

1

2

!1

1

(bx)µ+ 2

2

1

1

a2(µ+ 2 ) 2µ Γ(µ + 1)| sin θ|µ+ 2 ∞

Z

2

|ψ(z)| |z

−(µ+ 12 )

0

Denote next

b | |Dµθ ( ,

x , z)| dz a a

1

2

.

1

hθ,µ a,b =

bµ+ 2 1

a2µ+2 | sin θ|µ+ 2 2µ Γ(µ + 1)

.

We get Z

∞

0

≤ hθ,µ a,b ≤

θ |ψa,b (x)|2 dx

Z

∞

1

z −(µ+ 2 ) |ψ(z)|2 dz

0

θ,µ
2

ha,b

Z

∞

0

Z

∞

1 b x |Dµθ ( , , z)| xµ+ 2 dx a a

|ψ(z)|2 dz.

0

Hence, the lemma is proved. Definition 5.26 We define the transform of the continuous fractional Bessel wavelet Bψθ of f ∈ L2 (R+ ) by D

E

θ (Bψθ f )(a, b) = f, ψb,a .

Proposition 5.27 Let f, ψ ∈ L2 (R+ ). The continuous fractional Bessel wavelet transform Bψθ of f satisfies (Bψθ f )(a, b)

Z

= 0

∞

θ,µ Ea,b (x) Jµ (bx cos θ) (hθµ e

−i (.)2 2

cot θ

f )(x) (hθµ ψ)(ax) dx,

114  Wavelet Analysis: Basic Concepts and Applications

where θ,µ Ea,b (x) = a−µ sin θ cθµ e

−i 1 ( 2 a2

−1)(ax)2 cot θ

1

1

x−µ− 2 (bx cos θ) 2 .

Proof. We have θ (Bψθ f )(a, b) = hf, ψb,a i

Z

∞ θ dt f (t)ψb,a

= 0

Z

∞

= 0

Denote next

i b2 t2 1 f (t) √ e− 2 ( a2 + a2 ) cot θ a



∞

Z 0

b t ψ(z) Dµθ ( , , z) dz a a



1 i 2 ξ (t cos θ) 2 e− 2 t cot θ , a i 1 2 1 1 b b,µ σa,θ (ξ) = e− 2 ( a2 −1)ξ cot θ ξ (−µ− 2 ) ( ξ cos θ) 2 a i

2 2+ ξ a2

i

1

σa,θ (ξ, t) = e 2 (t

) cot θ

and b,µ ea,θ σ (ξ) = e− 2 ( a2 −1)ξ

2

cot θ (−µ− 12 )

ξ

1 b ( ξ cos θ) 2 . a

We obtain (Bψθ f )(a, b) 1 √ θ cµ a

=

Z

∞

0



cθµ

∞

Z 0

ξ σa,θ (ξ, t) Jµθ (t

a



cos θ)f (t)dt

b,µ (ξ) Jµθ ( ab ξ cos θ)(hθµ ψ(z))(ξ)dξ ×σa,θ

1 √ θ cµ a

=

Z 0

∞

i b ξ 2 b,µ ea,θ σ (ξ)Jµθ ( ξ cos θ) × (hθµ e− 2 (.) cot θ f )( )(hθµ ψ(z))(ξ)dξ. a a

For x = aξ , we obtain Proposition 5.27. In addition, we have 

i 2

hθµ e− 2 b

cot θ





1

i

= a−µ sin θ x−µ− 2 e 2 a

(Bψθ f )(a, b)

2 x2

i

cot θ 2

(hθµ e− 2 (.)

cot θ



f )(x))hθµ ψ(ax) .

Theorem 20 If ψ1 , ψ2 are two wavelets and (Bψ1 f )(a, b), (Bψ2 g)(a, b) their respective continuous fractional Bessel wavelet transforms, then Z 0

∞

Z 0

∞

(Bψ1 f )(a, b) (Bψ2 g)(a, b)

where 1 Cµ,ψ hf, gi 1 ,ψ2 ,θ

Z

= 0

∞

db da 1 = sin2 θ Cµ,ψ hf, gi, 1 ,ψ2 ,θ a2

a−2µ−2 (hθµ ψ1 )(a) (hθµ ψ2 )(a) da < ∞.

Quantum wavelets  115

Proof. Denote µ,θ Ea,b (x) = a−µ sin θ e

−i 1 ( 2 a2

−1)(ax)2 cot θ −µ− 12

θ Ea,µ (x) = a−2µ−2 sin3 θ e i

Eb,θ (x) = e 2 (b and

1

(bx cos θ) 2 ,

x

−i (1−a2 )x2 2

2 +x2 ) cot θ

i

2 )y 2

Ea,θ (y) = e 2 (2−a

1

x−µ− 2 ,

cot θ

1

(bx cos θ) 2

cot θ

1

y −µ− 2 .

Applying Proposition 5.27, we get ∞

∞

db da a2  Z0 ∞ Z0 ∞  Z ∞ −i 2 µ,θ Ea,b (x) Jµ (bx cos θ) (hθµ e 2 (.) cot θ f )(x) (hθµ ψ1 )(ax) dx = cθµ Z

Z

(Bψ1 f )(a, b) (Bψ2 g)(a, b)

0

0



×

cθµ

Z

Z

0

∞

∞ Z0 ∞

= ×

θ Ea,µ (x) (hθµ e

−i (.)2 2

0Z ∞ 0 θ cµ Eb,θ (x)Jµ (bx cos θ) 0

µ × Ea,θ (y) (hθµ e

Z

∞

Z

∞

Z

0 ∞

= 0

× cθµ

0

= sin3 θ

−i (.)2 2

cot θ

cot θ



g)(y) (hθµ ψ2 )(ay) dy db

da a2

f )(x) hθµ ψ1 (ax) Z

× cθµ

−i (.)2 2

∞

Eb,θ (y)Jµ (by cos θ) 0

cot θ

f )(x) hθµ ψ1 (ax)

µ (y) Eb,θ (x)Jµ (bx cos θ)(hθµ )−1 (Ea,θ

Z

∞Z

∞

0

, (hθµ e θ Ea,µ (x) (hθµ e

µ × hθµ (hθµ )−1 (Ea,θ (y) (hθµ e

Z

∞

Z 0

0

∞

−i (.)2 2

−i (.)2 2

cot θ

−i (.)2 2

cot θ

Z

= sin θ 0

∞

a−2µ−2 x−2µ−1 (hθµ e

(hθµ e

−i (.)2 2

cot θ

∞

× = sin3 θ Cµ,ψ1 ,ψ2 ,θ

Z 0

∞

cot θ

−i (.)2 2

−i (.)2 2

(−2µ−2)

(ax) 0

f )(x) hθµ ψ1 (ax)

cot θ cot θ

g)(x)(hθµ ψ2 )(ax)dx da

g)(x) 

(hθµ ψ1 )(ax) (hθµ ψ2 )(ax)x da

(hθµ f )(x)hθµ g(x) dx

= sin3 θ Cµ,ψ1 ,ψ2 ,θ hhθµ f, hθµ gi = sin2 θ Cµ,ψ1 ,ψ2 ,θ hf, gi.

g)(y)(hθµ ψ2 )(ay))(b)db dx da

f )(x) hθµ ψ1 (ax)

−i (.)2 2

f )(x) (hθµ e

Z

cot θ

g)(y)(hθµ ψ2 )(ay))(x)dx da

×(hθµ e 3

cot θ

g)(y)(hθµ ψ2 )(ay)dy db dx da

θ Ea,µ (x) (hθµ e

0

= sin3 θ

−i (.)2 2

µ,θ Ea,b (y) Jµ (by cos θ) (hθµ e

dx

116  Wavelet Analysis: Basic Concepts and Applications

Proposition 5.28 Z

1.

∞

For ψ = ψ1 = ψ2 , we get

∞

Z

(BΨ f )(a, b) (Bψ g)(a, b) 0

0

db da 1 = sin2 θ Cµ,ψ,θ hf, gi. a2

2. For f = g and ψ = ψ1 = ψ2 , there holds that Z

∞

∞

Z

|(BΨ f )(a, b)|2

0

0

db da 1 = sin2 θ Cµ,ψ,θ kf k2 . a2

Theorem 21 Let f ∈ L2 (R+ ), we have 1 f (t) = 2 1 sin θ Cµ,ψ,θ

∞

Z

∞

Z 0

0

θ (Bψθ f )(a, b) ψb,a (t)

db da , a > 0. a2

Proof. For g ∈ L2 (R+ ), sin

2

Z

1 θ Cµ,ψ,θ hf, gi

∞

Z

∞

Z

∞

= 0

0

Z

∞

= 0

0 ∞

Z 0

Z

∞

= 0

So, 1 hf, gi = 2 1 sin θ Cµ,ψ,θ

Z

(Bψθ f )(a, b)

∞

Z

Z

∞

∞

Z

db da a2

∞ θ dt g(t) ψb,a ,

0



db da a2

db da g(t) dt a2 0 0  Z ∞ db da θ θ (Bψ f )(a, b) ψb,a (t) 2 , g(t) . a 0

Z

=

(Bψθ f )(a, b) (Bψθ g)(a, b)

∞

0

0

θ (Bψθ f )(a, b) ψb,a (t)

θ (BΨ f )(a, b) ψb,a (t)



db da , g(t) . a2 

Theorem 22 If ψ ∈ L2 (R+ ), then Z

∞

0

h

i

(Bψθ f )(a, b) (Bψθ g)(a, b) db = a−2µ sin2 θ hF, Gi,

where i

2 )x2 ) cot θ

i

2 )x2 ) cot θ

F (x) = e− 2 ((2−a and

G(x) = e− 2 ((2−a

1

i

2 )x2 )2

cot θ

1

i

2 )x2 )2

cot θ

x−µ− 2 (hθµ e− 2 ((2−a

x−µ− 2 (hθµ e− 2 ((2−a

Proof. Denote firstly i b2 ω2 1 Γθa,b (ω) = √ e− 2 ( a2 + a2 ) cot θ a

and θ σa,µ (x) = e

−i ((2−a2 )x2 ) cot θ 2

1

x−µ− 2 .

f )(x)(hθµ )ψ(ax)

g)(x)(hθµ )ψ(ax).

Quantum wavelets  117

Using Propositions 5.27 and (20), we obtain ∞

Z

h

0 ∞

Z

= 0 ∞

Z

i

(Bψθ f )(a, b) (Bψθ g)(a, b) db

θ θ i db hf, ψb,a i hg, ψb,a ∞

Z

= 0

0 ∞

Z

"Z

θ (ω) dω f (ω) ψb,a

∞

=



f (ω) 0

0

Z

∞

× 0

Γθa,b (ω)



g(σ) Γθa,b (σ)

= sin3 θ a−2µ

Z

∞

Z

0

Z 0

0 ∞

∞

θ g(σ) ψb,a (σ) dσ



db

ω , z) dz dω a a

b ψ(z) Dµθ ( ,



#

b σ ψ(z) Dµθ ( , , z) dz dσ db a a 



−i 2 θ (x) (hθµ e 2 (.) cos θ f )(x) (hθµ ψ)(ax))(b) hθµ (σa,µ

0 θ (y) (hθ e × hθµ (σa,µ µ

∞

Z

 Z

−i (.)2 2

cot θ

g)(y) (hθµ ψ)(ay))(b) db

∞

(hθµ F )(b) (hθµ G)(b) db 0 sin3 θ a−2µ hhθµ F, hθµ Gi

= sin3 θ a−2µ =

= sin2 θ a−2µ hF, Gi. Theorem 23 Let ψ ∈ L2 (R+ ) is a Bessel wavelet and f is an integrable function at the edges, then the convolution (ψ ∗θ f ) is a fractional Bessel wavelet given by (ψ ∗θ f )(x) =

Z

∞

0

1

(τxθ ψ)(y) y (−µ+ 2 ) f (y) dy.

Proof. To show that the convolution (ψ ∗θ f ) is a fractional Bessel wavelet, it suffices to show that it satisfies the admissibility condition of Definition 5.23. Indeed, Z

∞

|(ψ ∗θ f )(x)|2 dx

0

2 Z ∞ θ (−µ+ 12 ) dx f (y) dy (τ ψ)(y) y x 0 0 Z ∞  Z ∞ 2 1 θ (−µ+ 21 ) 12 f 2 (y) dy dx f (y) || dy (τ ψ)(y) y x 0 0  1 #2 Z ∞ "Z ∞ 2  12 Z ∞ 2 θ (−µ+ 21 ) |f (y)| (τx ψ)(y) y |f (y)| dy dx dy Z

= ≤ ≤

∞

0

0

Z

∞

0

 Z

∞ Z

|f (y)| dy

= 0

Z

≤

0

0

2 Z

∞

|f (y)| dy 0

0

∞

∞

 2  θ (−µ+ 21 ) (τ ψ)(y) y dx |f (y)| dy x

2  θ (−µ+ 21 ) (τ ψ)(x) y y dx .

118  Wavelet Analysis: Basic Concepts and Applications

Using Lemma 5.25, we obtain 1

k(ψ ∗θ f )(x)kL2 ≤

µ+ 12

| sin θ|

kψL2 kf kL1 < ∞.

2µ Γ(µ + 1)

Consequently, (ψ ∗θ f ) ∈ L2 (R+ ), which implies that hθµ (ψ ∗θ f ) exists. Moreover, ∞

Z 0

2



x−2µ−2 hθµ (ψ ∗θ f ) dx

≤ | cos θ|

∞

Z

2 −3µ− 52 θ (hµ f )(x cos θ) (hθµ ψ)(x cos θ) dx x   1

0

≤ | cos θ| sup x−µ− 2 |(hθµ f )(x cos θ)|2 x

∞

Z

×

0

x−2µ−2 |(hθµ ψ)(x cos θ)|2 ) dx 

0



1

= Cµ,ψ,θ sup x−µ− 2 |(hθµ f )(x cos θ)|2 < ∞. x

So (ψ ∗θ f ) is a fractional Bessel wavelet. Theorem 24 If f, ψ ∈ L2 (R+ ) and (Bψθ f )(a, b) is the continuous fractional Bessel wavelet transform, then 1. The function (a, b) 7−→ (Bψθ f )(a, b) is continuous on R+ × R+ 2. The following holds 1

k(Bψθ f )(a, b)kL∞

1

b(µ+ 2 ) a−(µ+ 2 )

≤

1

| sin θ|µ+ 2 2µ Γ(µ + 1)

kf kL2 kψkL2 .

Proof. 1) Let us show that (Bψθ f )(a, b) is continuous on R+ × R+ . Let (b0 , a0 ) be an arbitrary but fixed point. Using the H¨older’s inequality, we obtain

=

θ (Bψ f )(a, b) − (Bψθ f )(b0 , a0 )   Z ∞ Z ∞ 1 θ b t θ b0 t √ f (t) ψ(z) Dµ ( , , z) − Dµ ( , , z) dt dz a a a a a 0

≤

≤

∞

1 √ a

Z

1 √ a

Z

Z

∞

×

∞

Z

0

0 ∞

0

−µ− 12

2

|f (t)| dt

∞

Z

0 −µ− 12

0

  f (t)ψ(z) Dθ ( b , t , z) − Dθ ( b0 , t , z) dt dz µ µ a a a a 0

t

z 0

0

z

µ+ 21

0

2

|ψ(z)| dz

Z

∞

µ+ 21

t 0

b Dµθ ( ,

t b0 t , z) − Dµθ ( , , z) dz a a a0 a0



b Dµθ ( ,



0



t b0 t , z) − Dµθ ( , , z) dt a a a0 a0 

1

1

2

.

2

Quantum wavelets  119 1

Denote next Kµθ = | sin θ|µ+ 2 2µ Γ(µ + 1). Using Proposition 5.14, we obtain Z

∞

z

µ+ 12

0

1



µ+ 1

1



b0 2  b0 t tµ+ 2  bµ+ 2 b t − Dµθ ( , , z) − Dµθ ( , , z) dz ≤ a a a0 a0 Kµθ a2(µ+ 12 ) a2(µ+ 12 ) 0





and similarly Z

∞

µ+ 12

t 0



1

1 t b0 t z µ+ 2 µ+ 1 , z) − Dµθ ( , , z) dz ≤ abµ+ 2 − a0 b0 2 , θ a a a0 a0 Kµ

b Dµθ ( ,







Moreover, from the continuity of Dµθ ( ab , at , z) and the monotonic convergence theorem, we get lim lim (Bψθ f )(a, b) − (Bψθ f )(b0 , a0 ) = 0. b−→b0 a−→a0

Hence, the continuity of (Bψθ f )(a, b).

5.5

QUANTUM THEORY TOOLKIT

This section is devoted to the introduction of the quantum theory tools. Useful definitions, notations and properties of q-derivatives and q-integrals will be introduced. For 0 < q < 1, denote n + e+ Rq = {±q n , n ∈ Z}, R+ q = {q , n ∈ Z} and Rq = Rq

[

{0}.

Definition 5.29 [224] The q-shifted factorial are defined by (a, q)0 = 1, (a, q)n =

n−1 Y

(1 − aq k ) = (1 − a)(1 − aq)......(1 − aq n−1 ),

k=0

(a, q)∞ =

lim (a, q)n =

n−→+∞

+∞ Y

(1 − aq k ),

k=0

and the symbol of Pochhammer (a)n is given by (a)0 = 1, (a)n = a(a + 1)....(a + n − 1),

n ∈ N∗ .

For all complex numbers a1 , a2 , ..., ar , we define (a1 , a2 , ...., ak ; q)n = (a1 , q)n (a2 , q)n ....(ak , q)n . Remark 5.30 The q-shifted factorial is considered as the q-analogue of the Pochhammer symbol as it satisfies lim−

q−→1

(q a , q)n = (a)n . (1 − q)n

120  Wavelet Analysis: Basic Concepts and Applications

e + , the q-Jackson integrals from 0 to a and from 0 to +∞ are Definition 5.31 On R q defined, respectively, by (see [73], [213]) a

Z

X

f (x)dq x = (1 − q) a

0

and

f (aq n ) q n

n≥0

Z

∞

f (x)dq x = (1 − q)

0

X

f (q n ) q n

n∈N

provided that the sums converge absolutely. On [a, b], the integral is given by b

Z

b

Z

f (x)dq x −

f (x)dq x = a

0

Z

a

f (x)dq x. 0

This allows us to introduce next the functional space e + ) = {f : kf kq,p,v < ∞}, Lq,p,v (R q

where Z

kf kq,p,v =

∞

|f (x)|p x2v+1 dq x

1

p

,

0

−1 e + ) be the space of functions defined on R e + , continuous where v > fixed. Let Cq0 (R q q 2 in 0 and vanishing at +∞, equipped with the induced topology of uniform convergence such that kf kq,∞ = sup |f (x)| < ∞. e+ x∈R q e + ) designates the space of functions that are continuous at 0 and Finally, Cqb (R q e +. bounded on R q e + ) is defined by Definition 5.32 The q-derivative of a function f ∈ Lq,p,α (R q

Dq f (x) =

   f (x) − f (qx) ,

(1 − q)x  f 0 (0) ,

x 6= 0 else,

provided that f is differentiable at 0. Lemma 5.33 [19] If f is differentiable, note that lim Dq f (x) =

q−→1

df (x) . dx

The operator Dq f is then the q-analogue of the classical derivative.

Quantum wavelets  121

Proof. Let f be a differentiable function. Then lim

h−→0

f (qx + h) − f (qx) = f 0 (qx). h

For h = x − qx, when q −→ 1 we have h −→ 0. Hence, f (qx + h) − f (qx) h−→0 h f (x) − f (qx) = lim q−→1 (1 − q)x

lim Dq f (x) =

q−→1

=

lim

df (x) . dx

Proposition 5.34 [19] The q-derivative of a function is a linear operator. That is, for any functions f and g and any constants a and b, we have Dq (af + bg)(x) = aDq f (x) + bDq g(x). Proof. For all x 6= 0 ,we have Dq (af + bg)(x) = =

(af + bg)(x) − (af + bg)(qx) (1 − q) x a f (x) + b g(x) − a f (qx) − b g(qx) (1 − q) x

= a

f (x) − f (qx) g(x) − g(qx) + b (1 − q) x (1 − q) x

= a Dq f (x) + b Dq g(x). Proposition 5.35 [19] For x 6= 0, the q-derivative of product is given by Dq (f (x)g(x)) = f (qx)Dq g(x) + Dq f (x)g(x), and for g(x) 6= 0, we have 

Dq

f (x) g(x)



=

g(qx)Dq f (x) − f (qx)Dq g(x) . g(qx)g(x)

122  Wavelet Analysis: Basic Concepts and Applications

Proof. For x 6= 0, we have Dq (f g)(x) =

(f g)(x) − (f g)(qx) (1 − q) x

=

f (x) g(x) − f (qx) g(x) + f (qx) g(x) − f (qx) g(qx) (1 − q) x

=

f (qx) g(x) − f (qx) g(qx) f (x) g(x) − f (qx) g(x) + (1 − q) x (1 − q) x

=

g(x) − g(qx) f (x) − f (qx) g(x) + f (qx) (1 − q) x (1 − q) x

= Dq f (x) g(x) + f (qx) Dq g(x). Proposition 5.36 [19] 1. For any function f , we have Z

x

f (t)dq t = f (x).

Dq 0

2. The q-analogue of the rule of integration by parts is b

Z

f (x) Dq g(x)dq x = [f (b)g(b) − f (a)g(a)] −

b

Z

g(qx) Dq f (x)dq x, a

a

where the integration is understood in q-Jackson sense. Proof. Assertion 1. is somehow reproduced from [19] and is easy. We have ∞ X

"

x

Z

f (t)dq t = Dq (1 − q)x

Dq 0

# n

f (xq )q

n

n=0

2

(1 − q) x

∞ h X

i

f (xq n ) − f (xq n+1 ) q n

n=0

= = (1 − q)

∞ h X

(1 − q)x i

f (xq n ) − f (xq n+1 ) q n

n=0

= f (x). 2. Using the definition of the q-integral, we have Z

b

Z

f (x) Dq g(x)dq x = a

b

f (x) Dq g(x)dq x −

0

Z

a

f (x) Dq g(x)dq x. 0

Applying again the definition of the q-integral, we get Z

b

f (x) Dq g(x)dq x = 0

∞ X n=0

f (bq n )(g(bq n ) − g(bq n+1 )).

Quantum wavelets  123

Similarly a

Z

f (x) Dq g(x)dq x = 0

∞ X

f (aq n )(g(aq n ) − g(aq n+1 )).

n=0

On the other hand, we get by using similar techniques b

Z

g(qx) Dq f (x)dq x = 0

and Z

∞ X

g(bq n+1 )(f (bq n ) − f (bq n+1 ))

n=0

a

g(qx) Dq f (x)dq x = 0

∞ X

g(aq n+1 )(f (aq n ) − f (aq n+1 )).

n=0

By regrouping all these equalities, we obtain Z

b

b

Z

g(qx) Dq f (x)dq x =

f (x) Dq g(x)dq x + a

a

∞ X

(In (b) − In (a) − Kn (b) + Kn (a),

n=0

where In (x) = f (xq n )(g(xq n )−g(xq n+1 )) and Kn (x) = g(xq n+1 )(f (xq n )−f (xq n+1 )). Next, observing that ∞ X

(In (b) − In (a) − Kn (b) + Kn (a) = f (b)g(b) − f (a)g(a),

n=0

we obtain Z

b

b

Z

g(qx) Dq f (x)dq x = [f (b)g(b) − f (a)g(a)] .

f (x) Dq g(x)dq x + a

a

So as Assertion 2. e + ) by Definition 5.37 The q-Bessel operator is defined for all f ∈ Lq,2,v (R q

∆q, v f (x) =

f (q −1 x) − (1 + q 2v )f (x) + q 2v f (qx) , x2

∀x 6= 0.

The following relation is easy to show and constitutes an analogue of Stokes rule. e + ) such that ∆q,v f, ∆q,v g ∈ Lq,2,v (R e + ), we have Lemma 5.38 For all f, g ∈ Lq,2,v (R q q Z

∞

∆q,v f (x) g(x) x 0

5.6

2v+1

Z

dq x =

∞

f (x) ∆q,v g(x) x2v+1 dq x.

0

SOME QUANTUM SPECIAL FUNCTIONS

We propose in this section to recall two basic functions that are applied almost everywhere in q-theory and its applications.

124  Wavelet Analysis: Basic Concepts and Applications

Definition 5.39 The q-analogue of the classical exponential function, called qexponential function, is defined by eq (x) = and Eq (x) =

∞ X

q

1 (1 − (1 − q)x)∞ q

k(k−1) 2

k=0

xk = (1 + (1 − q)x)∞ q , [k]q !

where

(q, q)k . (1 − q)k

[k]q ! =

Properties 5.40 The q-exponential functions satisfy the following properties: 1. Dq eq (x) = eq x. 2. Dq Eq (x) = Eq (qx). 3. eq (x)Eq (−x) = Eq (x)eq (−x) = 1. Proof. At the beginning of the twentieth century, Jackson introduced a q-analogue of the Gamma function. Definition 5.41 The q-Gamma Euler’s function is defined by Γq (x) =

(q, q)∞ (1 − q)1−x , x 6= 0, −1, −2, .... (q x , q)∞

Properties 5.42 The q-Gamma function satisfies the following properties: 1. The function Γq tends to the classical Euler function Γ when q tends to 1. That is, for all x, lim Γq (x = Γ(x). q→1

2. For all x, we have Γq (x + 1) = [x]q Γ(x), Γq (1) = 1. 3. Whenever q satisfies

log(1 − q) ∈ Z, log(q) we have the q-integral representation Z

∞

Γq (x) =

tx−1 E(−qt, q)dq t.

0

4. The Legendre duplication q-formula 1 1 Γq (2x)Γq2 ( ) = (1 + q)2x−1 Γq2 (x)Γq2 (x + ). 2 2

Quantum wavelets  125

Proof. Left to the reader. We now introduce the q-Bessel analogue of the Bessel function. Definition 5.43 [296, 297] The generalized q-Bessel-type function is defined by 3k ∞ (x/2)v X q 2 (k+v) (x2 /4)k . (q; q)v k=0 (q v+1 ; a)k (q; q)k

jv (x) =

The function jv is a q-analogue of the ordinary modified Bessel function as it satisfies the limit relation lim jv (x) = jv (x), q→1

where jv is the ordinary modified Bessel function. If v is an integer, it satisfies the equality jv (x) = j−v (x). The function jv (x) is an even (odd) function if the parameter v is an even (odd) integer, since j−v (x) = (−1)v jv (x), whenever v ∈ N. Definition 5.44 [195] The q-Bessel function is defined by jv (x, q 2 ) =

X

(−1)n

n≥0

q n(n+1) x2n . (q 2 , q 2 )n (q 2 , q 2 )n

Proposition 5.45 The q-Bessel function satisfies the following relations n

n

Jn (x, q) = (−1)n q 2 J−n (q 2 ; q), β

α

Jα (q 2 , q) = Jβ (q 2 , q), (1 − q v + x2 )Jv (x, q) = x{Jv−1 (x, q) + Jv+1 (x, q)}, (1 − q)Dq Jv (x, q) = q (

1−v ) 2

1

1

Jv−1 (xq 2 ,q ) − Jv+1 (xq 2 ,q ).

Proof. Left to the reader. Definition 5.46 [195] The normalized q-Bessel function is given by jα (x, q 2 ) =

X n≥0

(−1)n

q n(n+1) x2n . (q 2 , q 2 )n

(q 2α+2 , q 2 )n

1 The q-Bessel operator is defined as follows (α > − ), 2 ∆q, α f (x) =

f (q −1 x) − (1 + q 2α )f (x) + q 2α f (qx) , x2

∀x 6= 0.

126  Wavelet Analysis: Basic Concepts and Applications

q n(n+1) 1 and b2n = 2n −(1+q 2α )+q 2α q 2n , (q 2α+2 , q 2 )n (q 2 , q 2 )n q n ∈ N. Then, for all n ∈ N, we have b2n+2 an+1 + an = 0.

Lemma 5.47 Let an = (−1)n

Proposition 5.48 [213] For λ ∈ C, the problem ∆q, α u(x) = −λ2 u(x) and u(0) = 1, u0 (0) = 0 has as unique solution: the function x 7→ jα (λx, q 2 ). Proof. We have X

jα (λx, q 2 ) =

an λ2n x2n .

n≥0

Consequently, X

∆q, α jα (λx, q 2 ) =

n≥0

=

X

an

X X λ2n 2n 2α 2n 2n 2α x − (1 + q ) a λ x + q an λ2n q 2n x2n n q 2n n≥0 n≥0

an λ2n x2n−2



n≥0

=

X

x2  1 2α 2α 2n − (1 + q ) + q q q 2n

an λ2n b2n x2n−2

n≥0

= a0 b0 x−2 +

X

an λ2n b2n x2n−2

n≥1

=

X

an λ2n b2n x2n−2 .

n≥1

Hence, ∆q, α jα (λx, q 2 ) + λ2 jα (λx, q 2 ) = =

X n≥1 X

an λ2n b2n x2n−2 + λ2

X

X

X

an λ2n+2 x2n

n≥0

an+1 λ2n+2 x2n +

n≥0

=

an λ2n x2n

n≥0

ak+1 λ2k+2 x2k +

k≥0

=

X

X

an λ2n+2 x2n

n≥0

λ2n+2 x2n [b2n+2 an+1 + an ] .

n≥0

Using Lemma 5.47, we obtain ∆q, α jα (λx, q 2 ) + λ2 jα (λx, q 2 ) = 0,

Quantum wavelets  127

and jα (0, q 2 ) = 1, jα0 (0, q 2 ) = 0. The following relations are easy to show. The first is an analogue of Stokes rule. The second is an orthogonality relation for the normalized q-Bessel function. e + that Proposition 5.49 It holds for all x, y ∈ R q Z

∞

jα (xt, q 2 ) jα (yt, q 2 ) t2α+1 dq t =

0

where cq,α =

1 c2q,α

δq,α (x, y),

1 (q 2α+2 , q 2 )∞ 1 and δq,α (x, y) = δx,y . 2 2 1 − q (q , q )∞ (1 − q) x2(α+1)

Finally, the following preliminary result will be useful for the next. Lemma 5.50 Define the (q, v)-delta operator by δq,v (x, y) =

1 δx,y . It (1 − q) x2(|v|+1)

e + ) and all t ∈ R e + that holds for all f ∈ Lq,2,v (R q q Z

∞

f (t) =

f (x) δq,v (x, t) x2(|v|+1) dq x.

0

Proof. From the definition of the q-Jackson integral, we have Z

∞

f (x)δq,v (x, t)x2|v|+1 dq x = (1 − q)

0

∞ X

f (q n )δq,v (q n , s)q n(2|v|+2)

n=0

= (1 − q)f (q k )δq,v (q k , t)q k(2|v|+2) = f (q k ), where k is the unique integer such that t = q k .

5.7 QUANTUM WAVELETS In the present section, we propose to develop wavelet analysis based on the quantum Bessel functions. This class joins and extends the Bessel and q-Bessel wavelets to the q-theory framework. Definition 5.51 [213] The q-Bessel Fourier transform Fq,α is defined by Fq,α f (x) = cq,α

Z

∞

f (t) jα (xt, q 2 ) t2α+1 dq t,

0

where cq,α =

1 (q 2α+2 , q 2 )∞ . 1 − q (q 2 , q 2 )∞

Definition 5.52 The q-Bessel translation operator is defined by α Tq,x f (y)

Z

= cq,α 0

∞

Fq,α f (t) jα (xt, q 2 ) jα (yt, q 2 ) t2α+1 dq t.

128  Wavelet Analysis: Basic Concepts and Applications

e + ) a Fourier Lemma 5.53 The translation operator satisfies for all f ∈ Lq,2,α (R q invariance property ([1]) α e +. Fq,α (Tq,x f )(λ) = jα (λx, q 2 ) Fq,α f (λ), ∀λ, x ∈ R q

e + ), we have Proof. For f ∈ Lq,2,α (R q α Fq,α (Tq,x f )(λ) = cq,α

= c2q,α

Z

c2q,α

Z

= c2q,α

Z

=

∞

∞Z

h

∞

Z 0

α (Tq,x f )(t) jα (λt, q 2 ) t2α+1 dq t

i

Fq,α f (µ) jα (xµ, q 2 ) jα (tµ, q 2 ) µ2α+1 dq µ jα (λt, q 2 ) t2α+1 dq t

0

0 ∞

∞

Z

Fq,α f (λ)

2

2

jα (xµ, q ) jα (tµ, q ) µ

0

2α+1



dq µ jα (λt, q 2 ) t2α+1 dq t

0 ∞

Fq,α f (λ)

0

1 δq,α (x, t)jα (λt, q 2 ) t2α+1 dq t c2q,α

= jα (λx, q 2 ) Fq,α f (λ). Theorem 25 [1] The q-Bessel Fourier transform satisfies the following assertions: 2 + 1. For all f ∈ Lq,p,α (R+ q ), Fq,α f (x) = f (x), ∀x ∈ Rq . 2α+1 2. For all f ∈ Lq,2,α (R+ kf kq,2,α . q ), kFq,α f kq,2,α = q + 3. For all f ∈ Lq,p,α (R+ q ), p ≥ 1, we have Fq,α f ∈ Lq,p,v (Rq ). Furthermore, if 1 ≤ p ≤ 2, then 2

−1

p kFq,α f kq,p,v ≤ Bq,α kf kq,p,v ,

where Bq,α =

1 (−q 2 , q 2 )∞ (−q 2α+2 , q 2 )∞ . 1−q (q 2 , q 2 )∞

Proof. 1. If f ∈ Lq,p,α (R+ q ), then Fq,α f exists, and we have 2 Fq,α f

+∞

Z

= cq,α Z

Fq,α f (t) jα (xt, q 2 ) t2α+1 dq t

0 +∞

=



f (y) 0

Z

=

+∞

c2q,α

Z

∞

2

2

2α+1

jα (xt, q ) jα (yt, q ) t



dq t y 2α+1 dq y

0

f (y) δq (x, y) y 2α+1 dq y = f (x).

0

The computations are justified by the Fubini’s theorem: If p > 1, then we use the H¨older’s inequality Z 0

∞

Z

|f (y)|

∞

2

2

2α+1

|jα (xt, q ) jα (yt, q )|t 0



dq t y 2α+1 dq y

Quantum wavelets  129

Z

∞

p 2α+1

|f (y)| y

≤

 1 Z

∞

p

dq (y)

p 2α+1

|σ(y)| y

0

1

dq (y)

p

0

and

∞

Z

σ(y) =

|jα (xt, q 2 )jα (yt, q 2 )|t2α+1 dq t.

0

Then ∞

Z

p 2α+1

σ(y) y

Z

1

dq (y) =

p 2α+1

σ(y) y

dq (y) +

0

0

∞

Z

σ(y)p y 2α+1 dq (y).

1

Notice that Z

1

p 2α+1

σ(y) y

dq (y) ≤ kjα (., q

2

0

)kpq,∞

Z 0

1

∞

Z

2

2α+1

|jα (xt, q )|t

p

dq t

y 2α+1 dq y

0

≤ kjα (., q 2 )kpq,∞ kjα (., q 2 )kpq,1,α x−2(α+1)p y 2α+1 dq y < ∞ 



and Z

∞

p 2α+1

σ(y) y

)kpq,1,α

Z

≤ kjα (., q 2 )kpq,∞ kjα (., q 2 )kpq,1,v

Z

dq (y) ≤ kjα (., q

2

1

)kpq,∞

kjα (., q

2

∞

1

y 2α+1 dq y y 2(α+1)p

∞

1

dq y y 2(α+1)(p−1)+1

1

< ∞.

For p = 1, we get Z 0

∞

Z

|f (y)|

∞



|jα (xt, q 2 )jα (yt, q 2 )|t2α+1 dq t y 2α+1 dq y

0

≤ kf kq,1,α kjα (., q 2 )kq,∞ kjα (., q 2 )kq,1,α

1 x2(α+1)

.

2. We introduce the function Ψx as follows Ψx (t) = cq,α jα (tx, q 2 ). The inner product in the Hilbert space Lq,2,α (R) is defined for f, g ∈ Lq,2,α (R) by Z ∞

hf, gi =

f (t) g(t) t2α+1 dq t.

0

Using the orthogonality relation of jα (., q 2 ), we obtain x 6= y =⇒ hΨx , Ψy i = 0 and 1 2 kΨx k2q,2,α = x−2(α+1) . Next, Fq,α f (x) = hf, Ψx i and f ∈ Lq,2,α =⇒ Fq,α f = f. 1−q Then + hf, Ψx i = 0, ∀x ∈ R+ q =⇒ Fq,α f (x) = 0, ∀x ∈ Rq =⇒ f = 0.

130  Wavelet Analysis: Basic Concepts and Applications

Hence, {Ψx , x ∈ R+ q } form an orthogonal basis of the Hilbert space Lq,2,α and we + have {Ψx , x ∈ Rq } = Lq,2,α . Now, f ∈ Lq,2,α yields that f=

X x∈R+ q

1 hf, Ψx i Ψx . kΨx k2q,2,α

Consequently, X

kf k2q,2,α =

x∈R+ q

X 1 2 2 hf, Ψx i2 = (1 − q) x2(α+1) Fq,α f (x) = kFq,α f k2q,2,α . 2 kΨx kq,2,α x∈R+ q

3. This is an immediate consequence of the Assertion 1, the Riesz-Thorin theorem and the inversion formula. The following proposition summarizes some results about q-Bessel translation operator. Proposition 5.54 [213] The following assertions hold. e + ), we have 1. For any function f ∈ Lq,2,α (R q α α α Tq,x f (y) = Tq,y f (x) and Tq,x f (0) = f (x).

e q ), we have 2. For f, g ∈ Lq,p,α (R ∞

Z 0

and

α Tq,x (f )(y) g(y) y 2α+1 dq y =

α Tq,x jα (ty, q 2 )

Z

2

0

∞

α e+ f (y) Tq,x g(y) y 2α+1 dq y, ∀y ∈ R q

2

e +. = jα (tx, q ) jα (ty, q ), ∀t, x, y ∈ R q

e + ), we have 3. For Ψ ∈ Lq,2,α (R q α kTq,x Ψkq,2,α ≤

1 kΨkq,2,α . (q, q 2 )2∞

e + ), we have Proof. 1. For f ∈ Lq,2,α (R q α Tq,x f (y) = cq,α

∞

Z

Fq,α f (t) jα (xt, q 2 ) jα (yt, q 2 ) t2α+1 dq t

0 ∞

Z

= cq,α

Fq,α f (t) jα (yt, q 2 ) jα (xt, q 2 ) t2α+1 dq t

0 α = Tq,y f (x).

Similarly, we have α Tq,x f (0)

Z

∞

= cq,α

Fq,α f (t) jα (xt, q 2 ) jα (0, q 2 ) t2α+1 dq t

0

Z

= cq,α

∞

Fq,α f (t) jα (xt, q 2 ) t2α+1 dq t

0 2 = Fq,α f (x) = f (x).

Quantum wavelets  131

2. Denote for simplicity dq (st) = dq sdq t and dq (sty) = dq sdq tdq y. Denote also for e q,k = R eq × R eq × · · · × R e q . We have for f, g ∈ Lq,2,α (R e + ), k ≥ 1; R + + + + q Z

∞ α Tq,x f (y)g(y)dq y = c2q,α

Z e Rq,3 +

0

= c2q,α

Z e Rq,3 + Z

= cq,α Z = e Rq+

e Rq,2 +

f (s)g(y)jα (ts, q 2 )jα (tx, q 2 )jα (ty, q 2 )(tsy)2α+1 dq (sty)

g(y)jα (ty, q 2 )y 2α+1 dq yf (s)jα (ts, q 2 )jα (tx, q 2 )(ts)2α+1 dq (st)

Fq,α g(t)f (s)jα (ts, q 2 )jα (tx, q 2 )(ts)2α+1 dq tdq s

α Tq,x (g)(s)f (s)s2α+1 dq s.

The remaining part of Assertion 2 is an easy consequence of Lemma 5.53 and Theorem 25, Assertion 1. 3. Denote e q,v (x, y, t, s) = Kq,v (x, t, s)Kq,v (y, t, s) and Qq,v f (t, s) = Fq,v f (t)Fq,v f (s). K

We have v kTq,x f k2q,2,v = c2q,v

= c2q,v Z

∞

Z

0

∞

Z

∞

Z

∞

∞

e q,v (x, y, t, s)(tsy)2|v|+1 dq tdq sdq y Qq,v f (t, s)K

0

Qq,v f (t, s)

Z

∞

e q,v (x, y, t, s)y 2|v|+1 dq y(ts)2|v|+1 dq tdq s K

0

Qq,v f (t, s)δq,v (t, s)Kq,v (x, t, s)(ts)2|v|+1 dq tdq s ∞

Z

Fq,v f (t)ejq,v (xt, q 2 )t2|v|+1

Fq,v f (s)δq,v (t, s)ejq,v (xs, q 2 )s2|v|+1 dq sdq t

0

0

=

0

Z

0

0

Z

∞

Z0 ∞ Z0 ∞

= =

Z

∞

|Fq,v f (t)|2 |ejq,v (xt, q 2 )|2 t2|v|+1 dq t.

0

The second and the fourth equalities are simple applications of Fubini’s rule. The third and the fifth ones are applications of the second and the first assertions in Lemma 5.50, respectively. Next, observing that |ejq,v (xt, q 2 )| ≤

1 , (q, q 2 )2∞

we get Z

∞

|Fq,v f (t)|2 |ejq,v (xt, q 2 )|2 t2|v|+1 dq t ≤

0

=

∞ 1 |Fq,v f (t)|2 t2|v|+1 dq t 2 4 (q, q )∞ 0 1 kFq,v f k2q,2,v . (q, q 2 )4∞

Z

132  Wavelet Analysis: Basic Concepts and Applications

Definition 5.55 [213]A q-Bessel wavelet is an even function Ψ ∈ Lq,2,α (R+ q ) satisfying the following admissibility condition: Z

∞

Cα,Ψ =

|Fq,α Ψ(a)|2

0

dq a < ∞. a

e + ) is defined by The continuous q-Bessel wavelet transform of a function f ∈ Lq,2,α (R q α Cq,Ψ (f )(a, b) = cq,α

Z

∞

0

e+ f (x) Ψα(a,b) (x) x2α+1 dq x, ∀a ∈ R+ q , ∀b ∈ Rq ,

where Ψα(a,b) (x) =

√ α aTq,b (Ψa ); ∀a, b ∈ R+ q ,

and Ψa (x) =

x 1 Ψ( ). a2α+2 a

Proposition 5.56 [73] Let Ψ be a q-Bessel wavelet in Lq,2,α (R+ q ). Then, the function α + + e F : (a, b) 7→ Ψ(a,b) is continuous on Rq × Rq . + e+ Proof. It is clear that F is a mapping from R+ q × Rq into Lq,2,α (Rq ) and it is + e+ e+ continuous at all (a, b) ∈ R+ q × Rq . Now, fix a ∈ Rq . For b ∈ Rq , we have

kF (a, b) − F (a, 0)k2q,2,α α = kTq,b (Ψa ) − Ψa k2q,2,α α = q −4α−2 kFq,α (Tq,b (Ψa ) − Ψa )k2q,2,α

= q

−4α−2

Z∞ 2 2 1 − jα (xb, q 2 ) |Fq,α (Ψa )| (x)x2α+1 dq x. 0

e+ However, for all x ∈ R+ q and b ∈ Rq , we have 2 2 1 − jα (xb, q 2 ) |Fq,α (Ψa )| (x) ≤ (1 +

1 )2 |Fq,α (Ψa )|2 (x), 2 2 (q, q )∞

and Fq,α (Ψa ) ∈ Lq,2,α (R+ q ). So, the Lebesgue theorem leads to lim kF (a, b) − F (a, 0)kq,2,α = 0.

b−→0

Then, for all open neighborhood V of F (a, 0) in Lq,2,α (R+ q ), there exists an open + e neighborhood U of 0 in Rq such that ∀b ∈ U, F (a, b) ∈ V. e+ Thus, {a} × U is an open neighborhood of (a, 0) in R+ q × Rq and F ({a} × U ) ∈ V , which proves the continuity of F at (a, 0). The following result is a variant of Parseval-Plancherel rules for the case of q-Bessel wavelet transforms.

Quantum wavelets  133

Theorem 26 [213] Let Ψ be a q-Bessel wavelet in Lq,2,α (R+ q ). Then, 1. ; ∀g ∈ Lq,2,α (R+ q ), 1

∞

Z

Cα,Ψ

∞

Z

0

α |Cq,Ψ (g)(a, b)|2 b2α+1

0

dq a dq b = kgk2q,2,α , a2

2. ∀f1 , f2 ∈ Lq,2,α (R+ q ), Z

∞

0

f1 (x) f 2 (x) x2α+1 dq x =

1

∞

Z

Cα,Ψ

∞

Z 0

0

α Cq,Ψ (f1 )(a, b)

α (f )(a, b) b2α+1 Cq,Ψ 2

dq a dq b . a2

Proof. 1. By using Fubini’s theorem, and Assertion 3 in Theorem 25, we obtain q

4α+2

∞

Z

0

0

= q

4α+2

∞ Z

Z

=

|cαq,Ψ (g)(a, b)|2 b2α+1

∞

2

2

2



|Fq,α (g)(x)|

2 dq a

|Fq,α (Ψ)(ax)|

0 ∞

= Cα,Ψ

2α+1

0

∞

Z

dq a dq b a2

|Fq,α (g)(x)| |Fq,α (Ψa )| (x)x

0

Z

∞

Z

a





dq x

dq a a

x2α+1 dq x

|Fq,α (g)(x)|2 x2α+1 dq x

0

= Cα,Ψ q 4α+2 kgk2q,2,α . Assertion 2 is a direct consequence of Assertion 1. + Theorem 27 Let Ψ is a q-Bessel wavelet in Lq,2,α (R+ q ). Then, for all f ∈ Lq,2,α (Rq ), we have

cq,α f (x) = Cα,Ψ

Z 0

∞

Z 0

∞

α Cq,Ψ (f )(a, b) Ψα(a,b) (x) b2α+1

dq a dq b ; ∀x ∈ R+ q . a2

+ Proof. For x ∈ R+ q , we have h = δq,α belonging to Lq,2,α (Rq ). On the other hand, α according to Theorem 26, the definition of Cq,Ψ and the definition of the q-Jackson

134  Wavelet Analysis: Basic Concepts and Applications

integral, we have (1 − q)x2α+2 f (x) ∞

Z

=

f (t)h(t)t2α+1 dq t

0

=

=

1 Cα,Ψ cq,α Cα,Ψ

Z

∞

∞

0

0

Z

Z

∞Z

0

∞

0

= (1 − q)x2α+1

α α (h)(a, b)b2α+1 Cq,Ψ (f )(a, b)Cq,Ψ

α Cq,Ψ (f )(a, b)

cq,α Cα,Ψ

Thus, f (x) =

cq,α Cα,Ψ

∞

Z

0

∞

0

0

∞

Z

Z

Z

∞

0

Z 0

∞

dq adq b a2 

h(t)Ψα(a,b) (x) t2α+1 dq t

α Cq,Ψ (f )(a, b) Ψα(a,b) (x) b2α+1

α Cq,Ψ (f )(a, b) Ψα(a,b) (x) b2α+1

b2α+1

dq a dq b a2

dq a dq b . a2

dq a dq b , a2

which completes the proof.

5.8

EXERCISES

Exercise 1.

Let n ∈ N and f, g be two functions n-times q-differentiable. Show the q-Leibniz derivation rule Dqn (f g)(x) =

n X

n Ck,q Dqk (f )(xq n−k )Dq[ n − k](g)(x),

k=0

where n Ck,q =

(q; q)n (q; q)k (q; q)n−k

and where for l ∈ N, Dql = Dq ◦ Dq ◦ · · · ◦ Dq is the composition of the q-derivative operator Dq l-times. Exercise 2.

Show that for all real numbers x, y > 0, we have Z

∞

sJv (xs)Jv (ys)ds = 0

where δ(.) is the Kronecker symbol.

δ(x − y) x

Quantum wavelets  135

Exercise 3.

Denote dσ(x) =

x2µ 1

2µ− 2 Γ(µ + 12 )

χ[0,∞[ (x) dx

and

1 1 1 jµ (x) = 2µ− 2 Γ(µ + ) x 2 −µ Jµ− 1 (x), 2 2 where Jµ− 1 (x) is the Bessel function of order v = µ − 12 . Denote next 2

Z

∞

D(x, y, z) =

jµ (xt) jµ (yt) jµ (zt) dσ(t). 0

Show that

∞

Z

D(x, y, z) dσ(z) = 1. 0

Exercise 4.

Show that h

i

hθµ (∆∗µ,x )r δ(x − c) (y) = (−y 2 cos2 θ)r Kµθ (c, y), x, c ∈ R+ . Exercise 5.

Develop a proof for Proposition 5.14. Exercise 6.

Develop a proof for Lemma 5.16. Exercise 7.

Develop a proof for Lemma 5.47 Exercise 8.

Develop a proof for Proposition 5.49 Exercise 9.

Consider the q-Bessel function jα (x, q 2 ) =

X n≥0

(−1)n

q n(n+1) x2n (q 2α+2 , q 2 )n (q 2 , q 2 )n

and the q-Bessel operator defined for α > − ∆q, α f (x) =

1 by 2

f (q −1 x) − (1 + q 2α )f (x) + q 2α f (qx) , x2

∀x 6= 0.

136  Wavelet Analysis: Basic Concepts and Applications

Show that for all λ ∈ C, ∆q, α u(x) = −λ2 u(x) and u(0) = 1, u0 (0) = 0 has as unique solution: the function u(x) = jα (λx, q 2 ). Exercise 10.

With the same notations as in Exercise 9, denote the q-Bessel Fourier transform by Fq,α , defined as Z ∞

Fq,α f (x) = cq,α

f (t) jα (xt, q 2 ) t2α+1 dq t,

0

where cq,α is an appropriate constant. Define also a q-Bessel translation operator by ∞

Z

α Tq,x f (y)

= cq,α

Fq,α f (t) jα (xt, q 2 ) jα (yt, q 2 ) t2α+1 dq t.

0

e + ), 1) Show that for all f ∈ Lq,2,α (R q α e +. Fq,α (Tq,x f )(λ) = jα (λx, q 2 ) Fq,α f (λ), ∀λ, x ∈ R q

e + ) satisfying 2) A q-Bessel wavelet is an even function Ψ ∈ Lq,2,α (R q Z

∞

Cα,Ψ =

|Fq,α Ψ(a)|2

0

dq a < ∞. a

e + ) is defined by The continuous q-Bessel wavelet transform of a function f ∈ Lq,2,α (R q α Cq,Ψ (f )(a, b)

where Ψα(a,b) (x) =

√

∞

Z

= cq,α 0

f (x) Ψα(a,b) (x) x2α+1 dq x,

α aTq,b (Ψa ) and Ψa (x) =

1 a2α+2

x e+ Ψ( ), ∀a ∈ R+ q , ∀x, ∈ Rq . a

e+ Show that the function F : (a, b) 7→ Ψα(a,b) is continuous on R+ q × Rq . e + ), 3) Prove that ∀g ∈ Lq,2,α (R q

1 Cα,Ψ

∞

Z

∞

Z

0

0

α |Cq,Ψ (g)(a, b)|2 b2α+1

dq a dq b = kgk2q,2,α . a2

e + ), 4) Prove that ∀f1 , f2 ∈ Lq,2,α (R q Z 0

∞

f1 (x)f 2 (x)x2α+1 dq x =

1 Cα,Ψ

Z 0

∞

Z 0

∞

α Cq,Ψ (f1 , f2 )(a, b)b2α+1 dµq (a, b),

α α α (f )(a, b) and dµ (a, b) = where Cq,Ψ (f1 , f2 )(a, b) = Cq,Ψ (f1 )(a, b)Cq,Ψ 2 q

dq a dq b . a2

e + ), 5) Prove that for all f ∈ Lq,2,α (R q

cq,α f (x) = Cα,Ψ

Z 0

∞

Z 0

∞

α Cq,Ψ (f )(a, b) Ψα(a,b) (x) b2α+1

dq a dq b e +. ; ∀x ∈ R q a2

CHAPTER

6

Wavelets in statistics

6.1

INTRODUCTION

In Statistical modeling, especially of financial series, the familiar and most known models are gaussian and log-normal ones. The gaussian models are based on the price increment hypothesis, while the log-normal ones are based on the log-price increment hypothesis. One can then suggest that the statistics of price variations is gaussian. However, such a hypothesis is not sufficient as it does not estimate the importance of rare and intensive increments. Therefore, it is necessary to go back to the multi-scaling aspects of the problem by studying the stock exchange of the different financial products with the suitable concepts such as scale low and self-similar statistical aspects. Indeed, one can observe in practice that some return distributions have heavy tails than the gaussians. This fact has been noticed firstly by B. Mandelbrot when studying the cotton price. One can also remark that the tail indices of the distribution seem to increase with the time intervals. These facts and some others such as non-autocorrelation of some stock market models, clustering volatility and time scaling behaviors allow researches to think about some more general models to describe or to study financial series. There are many models that can be stated as classical ones such as brownian motion, fractional brownian motion, martingales and semi-martingales. For more details and backgrounds on these subjects and for a comprehensive literature on wavelet analysis, wavelet filter, time series, self-similarity, price models, volatility and related topics, we have listed an exhaustive updated list of references. Time series constitute a very delicate area of study due to the specific characteristics. Most of the time-varying series are non-linear, in particular, the financial and economic series, which present an intellectual challenge. Their behavior seems to change dramatically, and uncertainty is always present. The financial data available are always approximate. Increasingly, artificial intelligence techniques such as fuzzy systems, neural networks and genetic and hybrid algorithms have been used to successfully model complex fundamental relationships of non-linear time series. Such models, referred to as universal estimators, are theoretically capable of uniformly approximating any true continuous function over a compact set to any degree of accuracy. Consider the case of fuzzy systems, defined as techniques for reasoning uncertainty, and based on the theory of fuzzy sets developed by Zadeh (1973). Such 137

138  Wavelet Analysis: Basic Concepts and Applications

systems are characterized by linguistic interpretation. In addition, in the case of nonstationary financial series, the relationship between dependent and other independent variables is tainted with ambiguity. Introducing fuzzy logic into regression methods reduces the degree of uncertainty. It thus appears that the concept of vagueness covers indifferently the notions of knowledge badly specified, badly described, imperfect, partial, incomplete or approximate. Other than uncertainty and imprecision, most of the actual processes in the financial markets consist of complex combinations of sub-processes or components, which operate at different frequencies. On the other hand, the examination of a time series allows us, in general, to recognize three types of components: a trend, a seasonal component and a random variation. Other evolutionary characteristics, such as shocks, can also be observed but are less closely linked to the structure of the series. It is therefore useful to separate these components for two reasons. The first is to answer common sense questions such as that of the growth or the general decrease of the phenomenon observed. Extracting the trend and analyzing it will answer this question. It is also interesting to highlight the possible presence of a periodic variation, thanks to the analysis of the seasonal component. The second of these reasons is to rid the phenomenon of its tendency and its periodic variations to more easily observe the random phenomenon. Methods based on multi-scale wavelet transformation provide powerful analysis tools that decompose time series data into coefficients related to time and a specific frequency band. Wavelets are considered capable of isolating fundamental low-frequency dynamics from non-stationary time series, and are robust to the presence of noise. The interpretation of the complex financial time series devices is therefore made easy by first applying the wavelet transformation and later interpreting each sub-series individually.

6.2 6.2.1

WAVELET ANALYSIS OF TIME SERIES Wavelet time series decomposition

Wavelet analysis allows the representation of time series into species relative to the time and frequency information known as time-frequency decomposition. It consists in decomposing a series in different frequency components with a scale-adapted resolution and thus permits observation and analysis of data at different scales. A time series X(t) is projected onto Wj , yielding a component DXj (t) given by DXj (t) =

X

dj,k ψj,k (t),

(6.1)

k

where the dj,k are the detail coefficients of the series X(t). The following decomposition is proved for j ∈ Z: X(t) =

X

DXj (t) =

j

The component AXJ (t) =

X

X j≤J

DXj (t) +

X

DXj (t).

(6.2)

j>J

DXj (t) is called the approximation of X(t) at the level

j≤J

J and reflects the trend or the global shape of the series in the space VJ . Therefore,

Wavelets in statistics  139

it may be expressed as AXJ (t) =

X

aJ,k ϕJ,k (t),

(6.3)

k

where the aJ,k are the approximation coefficients of the series. As a result, we obtain the wavelet decomposition of X(t) at the truncation level J: X(t) = AXJ (t) +

X

DXj (t).

(6.4)

j>J

It is composed of one part reflecting the global behavior or the trend of the series and a second part reflecting the higher frequency oscillations or the fine scale deviations of the series near its trend. In practice, we cannot obviously compute the complete set of coefficients. We thus fix a maximal level of decomposition J and consider the decomposition XJ (t) = AXJ0 (t) +

X

DXj (t)

(6.5)

J0