The Wavelet Transform

Cover......Page 1
Title......Page 4
Preface......Page 8
Contents......Page 12
1.2 Introduction to distribution theory......Page 16
1.3 The Fourier transform......Page 20
1.4 The Fourier transformation in F\'(R^n)......Page 21
1.5 The Hilbert transform......Page 23
1.6 Wavelets......Page 24
1.7 The continuous wavelet transform......Page 27
1.8 Asymptotic expansion......Page 31
2.1 Introduction......Page 36
2.2 Approximation properties......Page 41
2.3 Wavelet transform, Hilbert transform and Fractional integrals......Page 44
2.4 Wavelet transform on weighted L^P spaces......Page 52
2.5 Boundedness of the wavelet transform on L^P with different weights......Page 54
2.6 Wavelet transform and operators .........Page 56
2.7 The Wavelet transform on F\'_p......Page 59
2.8 The wavelet transform on L^P(R^n)......Page 62
3.1 Introduction......Page 64
3.2 The wavelet transform of tempered distributions......Page 65
3.3 Composition of wavelet transforms......Page 67
3.4 Weighted Sobolev Spaces......Page 71
3.5 Wavelet transforms on weighted Sobolev spaces......Page 72
3.6 Abelian theorems for wavelet transform of functions......Page 74
3.7 Abelian theorems for wavelet transform of distributions......Page 76
3.8 An Application......Page 80
4.1 Introduction......Page 82
4.2 Wavelet Transform on Spaces of Type S......Page 84
4.3 The Wavelet Transform of Generalized Functions......Page 89
4.4 The Wavelet Transform of Tempered Ultradistributions......Page 90
4.5 Wavelet Transform of Gevrey Functions of Compact Support......Page 93
4.6 Band Limited Wavelets......Page 95
5.2 The spaces .........Page 98
5.3 The wavelet transformation on W -spaces......Page 101
6.1 Introduction......Page 108
6.2 The generalized Sobolev space .........Page 109
6.3 The wavelet transform on generalized Sobolev space .........Page 111
6.4 Asymptotic behaviour for small dilation parameters......Page 115
6.5 Local convergence......Page 118
6.6 Example......Page 121
7.2 The generalized translation and convolution......Page 124
7.3 Special cases......Page 127
7.4 Convolution for the wavelet transform......Page 133
7.5 Convolution for the discrete wavelet transform......Page 135
7.6 Existence theorems......Page 137
7. 7 An Approximation Theorem......Page 142
8.1 Introduction......Page 144
8.2 The Wavelet Convolution Product......Page 145
8.3 Existence Theorems......Page 147
8.5 Approximations in L^P and B_(p,k) Spaces......Page 150
9.1 Introduction......Page 152
9.2 Asymptotic Expansions of the Generalized Stieltjes Transform......Page 157
9.3 Asymptotic expansions for T_2 and T_3......Page 158
9.4 Asymptotic Expansions of..........Page 160
9.5 Asymptotic Expansion of Mexican Hat Wavelet Transform......Page 165
10.1 Introduction......Page 170
10.2 Asymptotic expansion for large a......Page 171
10.3 Asymptotic expansion of Morlet wavelet transform......Page 173
10.4 Asymptotic Expansion of Mexican Hat Wavelet Transform......Page 176
10.5 Asymptotic expansion of Haar wavelet transform......Page 177
10.6 Asymptotic expansion for small a......Page 179
10.7 Asymptotic expansion for small a continued......Page 183
Bibliography......Page 186
Subject Index......Page 190