Wavelets, Multiscale Systems And Hypercomplex

From a mathematical point of view it is fascinating to realize that most, if not all, of the notions arising from the theory of analytic functions in the open unit disk have counterparts when one replaces the integers by the nodes of a homogeneous tree. It is also fascinating to realize that a whole function theory, different from the classical theory of several complex variables, can be developped when one considers hypercomplex (Clifford) variables, Fueter polynomials and the Cauchy-Kovalevskaya product, in place of the classical polynomials in three independent variables.This volume contains a selection of papers on the topics of Clifford analysis and wavelets and multiscale analysis, the latter being understood in a very wide sense. The theory of wavelets is mathematically rich and has many practical applications.Contributors: R. Abreu-Blaya, J. Bory-Reyes, F. Brackx, Sh. Chandrasekaran, N. de Schepper, P. Dewilde, D.E. Dutkay, K. Gustafson, H. Heyer, P.E.T. Jorgensen, T. Moreno-García, L. Peng, F. Sommen, M.W. Wong, J. Zhao, H. Zhu

Editorial Introduction

ix

R. ABREU-BLAYA, J. BORY-REYES AND T. MORENO-GARCIA

Teodorescu Transform Decomposition of Multivector Fields on Fractal Hypersurfaces

1. Introduction

1

(2)

2. Preliminaries

3

(3)

2.1. Clifford algebras and multivectors

3

(1)

2.2. Clifford analysis and harmonic multivector fields

4

(1)

2.3. Fractal dimensions and Whitney extension theorem

5

(1)

3. Jump problem and monogenic extensions

6

(3)

4. K-Multivectorial case. Dynkin problem and harmonic extension

9

(1)

5. Example

10

(5)

5.1. The curve of B. Kats

11

(1)

5.2. The surface Γ*

11

(2)

5.3. The function u*

13

(1)

5.4. Proof of properties a)··· e)

13

(2)

References

15

(2)

F. BRACKX, N. DE SCHEPPER AND F. SOMMEN

Metric Dependent Clifford Analysis with Applications to Wavelet Analysis

1. Introduction

17

(4)

2. The metric dependent Clifford toolbox

21

(28)

2.1. Tensors

21

(3)

2.2. From Grassmann to Clifford

24

(6)

2.3. Embeddings of Rm

30

(2)

2.4. Fischer duality and Fischer decomposition

32

(4)

2.5. The Euler and angular Dirac operators

36

(7)

2.6. Solid g-spherical harmonics

43

(2)

2.7. The g-Fourier transform

45

(4)

3. Metric invariant integration theory

49

(10)

3.1. The basic language of Clifford differential forms

49

(6)

3.2. Orthogonal spherical monogenics

55

(4)

3.2.1. The Cauchy-Pompeiu formula

55

(2)

3.2.2. Spherical monogenics

57

(2)

4. The radial g-Clifford-Hermite polynomials and associated CCWT

59

(7)

4.1. The radial g-Clifford-Hermite polynomials

59

(3)

4.2. The g-Clifford-Hermite wavelets

62

(1)

4.3. The g-Clifford-Hermite Continuous Wavelet Transform

63

(3)

References

66

(3)

P. DEWILDE AND SH. CHANDRASEKARAN

A Hierarchical Semi-Separable Moore-Penrose Equation Solver

1. Introduction

69

(2)

2. HSS representations

71

(2)

3. Preliminaries

73

(3)

4. HSS row absorption procedure

76

(3)

Complexity calculation

78

(1)

5. An HSS Moore-Penrose reduction method

79

(4)

6. Discussion and conclusions

83

(1)

Acknowledgements

84

(1)

References

84

(3)

D.E. DUTKAY AND P.E.T. JORGENSEN

Methods from Multiscale Theory and Wavelets Applied to Nonlinear Dynamics

1. Introduction

87

(1)

2. Connection to signal processing and wavelets

88

(2)

3. Motivating examples, nonlinearity

90

(2)

MRAs in geometry and operator theory

92

(1)

3.1. Spectrum and geometry: wavelets, tight frames and Hilbert spaces on Julia sets

92

(4)

3.1.1. Background

92

(3)

3.1.2. Wavelet filters in nonlinear models

95

(1)

3.2. Multiresolution analysis (MRA)

96

(3)

3.2.1. Pyramid algorithms and geometry

98

(1)

3.3. Julia sets from complex dynamics

99

(1)

4. Main results

100

(11)

4.1. Spectral decomposition of covariant representations: projective limits

107

(4)

5. Remarks on other applications

111

(11)

Acknowledgements

122

(1)

References

122

(5)

K. GUSTAFSON

Noncommutative Trigonometry

1. Introduction

127

(1)

2. The first (active) period 1966-1972

128

(3)

3. The second (intermittent) period 1973-1993

131

(2)

4. The third (most active) period 1994-2005

133

(7)

5. Related work: Discussion

140

(2)

6. Noncommutative trigonometry: Outlook

142

(8)

6.1. Extensions to matrix and operator algebras

143

(3)

6.2. Multiscale system theory, wavelets, iterative methods

146

(2)

6.3. Quantum mechanics

148

(2)

References

150

(7)

H. HEYER

Stationary Random Fields over Graphs and Related Structures

1. Introduction

157

(1)

2. Second-order random fields

158

(5)

2.1. Basic notions

158

(1)

2.2. Spatial random fields with orthogonal increments

159

(2)

2.3. The Karhunen representation

161

(2)

3. Stationarity of random fields

163

(8)

3.1. Graphs, buildings and their associated polynomial structures

163

(2)

3.1.1. Distance-transitive graphs and Cartier polynomials

163

(1)

3.1.2. Triangle buildings and Cartwright polynomials

164

(1)

3.2. Stationary random fields over hypergroups

165

(3)

3.3. Arnaud-Letac stationarity

168

(3)

References

171

(2)

M.W. WONG AND H. ZHU

Matrix Representations and Numerical Computations of Wavelet Multipliers

1. Wavelet multipliers

173

(2)

2. The Landau-Pollak-Slepian operator

175

(1)

3. Frames in Hilbert spaces

176

(3)

4. Matrix representations of wavelet multipliers

179

(1)

5. Numerical computations of wavelet multipliers

180

(2)

References

182

(1)

J. ZHAO AND L. PENG

Clifford Algebra-valued Admissible Wavelets Associated to More than 2-dimensional Euclidean Group with Dilations

1. Introduction

183

(1)

2. Clifford algebra-valued admissible wavelet transform

184

(4)

3. Examples of Clifford algebra-valued admissible wavelets

188

(1)

Acknowledgement

189

(1)

References

189

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Wavelets, Multiscale Systems And Hypercomplex Analysis

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