Over the last two years, wavelet methods have shown themselves to be of considerable use to harmonic analysts and, in particular, advances have been made concerning their applications. The strength of wavelet methods lies in their ability to describe local phenomena more accurately than a traditional expansion in sines and cosines can. Thus, wavelets are ideal in many fields where an approach to transient behaviour is needed, for example, in considering acoustic or seismic signals, or in image processing. Yves Meyer stands the theory of wavelets firmly upon solid ground by basing his book on the fundamental work of Calderón, Zygmund and their collaborators. For anyone who would like an introduction to wavelets, this book will prove to be a necessary purchase.

Preface to the English Edition

ix

Introduction

xi

1 Fourier series and integrals, filtering and sampling

1

(17)

Introduction

1

(1)

Fourier series

1

(6)

Fourier integrals

7

(2)

Filtering and sampling

9

(5)

``Wavelets'' in the work of Lusin and Calderon

14

(4)

2 Multiresolution approximations of L2 (Rn)

18

(48)

Introduction

18

(4)

Multiresolution approximation: definition and examples

22

(3)

Riesz bases and orthonormal bases

25

(4)

Regularity of the function φ

29

(1)

Bernstein's inequalities

30

(2)

A remarkable identity satisfied by the operator Ej

32

(9)

Effectiveness of a multiresolution approximation

41

(4)

The operators Dj = Ej+1 - Ej, j &epsis; Z

45

(5)

Besov spaces

50

(4)

The operators Ej and pseudo-differential operators

54

(3)

Multiresolution approximations and finite elements

57

(3)

Example: the Littlewood-paley multiresolution approximation

60

(2)

Notes and comments

62

(4)

3 Orthonormal wavelet bases

66

(64)

Introduction

66

(5)

The construction of wavelets in dimension 1

71

(10)

Construction of wavelets in dimension 2 by the tensor product method

81

(2)

The algorithm for constructing multi-dimensional wavelets

83

(4)

Computing two-dimensional wavelets

87

(6)

The general existence theorem for wavelet bases

93

(2)

Cancellation of wavelets

95

(1)

Wavelets with compact support

96

(11)

Wavelets with compact support in higher dimensions

107

(2)

Wavelets and spaces of functions and distributions

109

(3)

Wavelet series and Fourier series

112

(12)

Notes and comments

124

(6)

4 Non-orthogonal wavelets

130

(8)

Introduction

130

(1)

Frames (or ``skew structures'')

131

(1)

Ingrid Daubechies' criterion

132

(1)

Riesz Bases and Lp convergence

133

(5)

5 Wavelets, the Hardy space H1 and its dual BMO

138

(25)

Introduction

138

(3)

Equivalent definitions of the space H1 (Rn)

141

(3)

Atomic decomposition at the coefficient level

144

(4)

Back to earth

148

(2)

Atoms and molecules

150

(1)

The space BMO of John and Nirenberg

151

(5)

Maurey's theorem

156

(1)

Notes and complementary remarks

157

(6)

6 Wavelets and spaces of functions and distributions

163

(45)

Introduction

163

(1)

Criteria for belonging to Lp (Rn) and to Lp,s (Rn)

164

(12)

Hardy spaces Hp (Rn) with 0 < p ≤ 1

176

(2)

Holder spaces

178

(8)

The Beurling algebra

186

(3)

The hump algebra

189

(3)

The space generated by special atoms

192

(5)

The Bloch space B0∞∞

197

(1)

Characterization of continuous linear operators T : B0,1 1 → B0,1 1

198

(1)

Wavelets and Besov spaces

199

(3)

Holomorphic wavelets and Bochkariev's theorem

202

(5)

Conclusion

207

(1)

Bibliography

208

(12)

New references on wavelets and their applications

220

(1)

Index

221

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