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ANHA Series Preface | p. v |
Preface | p. ix |
List of Contributors | p. xvii |
Introduction: Mathematical Aspects of Time-Frequency Analysis | p. 1 |
Aims of Time-frequency Analysis | p. 1 |
Signal and Model | p. 2 |
Transforms | p. 3 |
Signal Manipulations-Filters | p. 8 |
Why Discretizing? Techniques, Challenges, Pitfalls | p. 8 |
Basic Methods of Time-Frequency Analysis: Orthonormal Bases and Generalized Fourier Series | p. 10 |
Schauder Basis in Banach Spaces | p. 10 |
Generalized Fourier Series | p. 14 |
The Fourier Integral Transform | p. 15 |
Definition and Properties | p. 15 |
The Plancherel Transform | p. 26 |
The Theorem of Paley-Wiener | p. 29 |
Discretization: The Poisson Summation Formula and the Sampling Theorem | p. 30 |
Windowed Fourier Transform | p. 31 |
The Short-Time Fourier Transform (STFT) | p. 31 |
The Gabor Transform | p. 32 |
The Heisenberg Uncertainty Principle | p. 36 |
Discretization: Gabor Frames | p. 38 |
Shortcomings of the Windowed Fourier Transform | p. 38 |
The Wavelet Transform | p. 39 |
Definition and Properties | p. 39 |
Scale Discretization-The Dyadic Wavelet Transform | p. 42 |
Multiresolution Analyses | p. 43 |
Other Multiscale Transforms | p. 45 |
Tensor Product Wavelets in 2D | p. 45 |
Some Wavelet-Type Transforms | p. 46 |
Moving to Other Manifolds-Wavelets on the Sphere | p. 47 |
Exercises | p. 49 |
B-Spline Generated Frames | p. 51 |
Introduction | p. 51 |
Bessel Sequences in Hilbert Spaces | p. 52 |
General Bases and Orthonormal Bases | p. 54 |
Riesz Bases | p. 56 |
Frames and Their Properties | p. 59 |
Frames and Riesz Bases | p. 62 |
B-Splines | p. 64 |
Frames of Translates | p. 65 |
Basic Gabor Frame Theory | p. 67 |
Tight Gabor Frames | p. 70 |
The Duals of a Gabor Frame | p. 72 |
Explicit Construction of Dual Gabor Frame Pairs | p. 73 |
Wavelets and the Unitary Extension Principle | p. 77 |
Exercises | p. 84 |
Continuous and Discrete Reproducing Systems that Arise from Translations. Theory and Applications of Composite Wavelets | p. 87 |
Introduction | p. 87 |
Unified Theory of Reproducing Systems | p. 91 |
Unified Theorem for Reproducing Systems | p. 92 |
Continuous Wavelet Transform | p. 98 |
Admissible Groups | p. 101 |
Wave Packet Systems | p. 102 |
Affine Systems with Composite Dilations | p. 104 |
Affine System with Composite Dilations | p. 108 |
Other Examples | p. 110 |
Continuous Shearlet Transform | p. 116 |
Edge Analysis Using the Shearlet Transform | p. 120 |
A Shearlet Approach to Edge Analysis and Detection | p. 121 |
Discrete Shearlet System | p. 123 |
Optimal Representations Using Shearlets | p. 126 |
Exercises | p. 129 |
Wavelets on the Sphere | p. 131 |
Introduction | p. 131 |
Scale-Space Premises | p. 132 |
Directional Correlations | p. 132 |
Harmonic Analysis | p. 133 |
Affine Transformations | p. 136 |
Continuous Formalism | p. 141 |
Generic Wavelets | p. 141 |
Stereographic Wavelets | p. 145 |
Kernel Wavelets | p. 149 |
Discretization of Variables | p. 152 |
Analysis Algorithms | p. 153 |
Pixelization | p. 153 |
Fast Algorithms | p. 155 |
Discrete Formalism | p. 159 |
Discrete Wavelets | p. 159 |
Other Constructions | p. 165 |
Reconstruction Algorithm | p. 167 |
Multiresolution | p. 167 |
Fast Algorithm | p. 168 |
Applications | p. 169 |
Cosmic Microwave Background Analysis | p. 169 |
Human Cortex Image Denoising | p. 170 |
Conclusion | p. 173 |
Exercises | p. 174 |
Wiener's Lemma: Theme and Variations. An Introduction to Spectral Invariance and Its Applications | p. 175 |
Introduction | p. 175 |
Wiener's Lemma-Classical | p. 177 |
Definitions from Banach Algebras | p. 178 |
Absolutely Convergent Fourier Series | p. 178 |
Wiener's Lemma | p. 179 |
Proof of Wiener's Lemma | p. 180 |
Abstract Concepts-Inverse-Closedness | p. 182 |
Convolution Operators | p. 188 |
Exercises for Section 5.2 | p. 193 |
Variations | p. 195 |
Weighted Versions of Wiener's Lemma | p. 195 |
Matrix Algebras | p. 200 |
Absolutely Convergent Series of Time-Frequency Shifts | p. 208 |
Convolution Operators on Groups | p. 216 |
Pseudodifferential Operators | p. 220 |
Time-Varying Systems and Wireless Communications | p. 227 |
Exercise for Section 5.3 | p. 234 |
references | p. 235 |
Index | p. 245 |
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