Four Short Courses on Harmonic Analysis

This state-of-the-art textbook examines four research directions in harmonic analysis and features some of the latest applications in the field, including cosmic microwave background analysis, human cortex image denoising, and wireless communication. The work is the first one that combines spline theory (from a numerical or approximation-theoretical view), wavelets, frames, and time-frequency methods leading up to a construction of wavelets on manifolds other than Rn.Written by internationally renowned mathematicians, the interdisciplinary chapters are expository by design, enabling the reader to understand the theory behind modern image and signal processing methodologies. The main emphasis throughout the book is on the interdependence of the four modern research directions covered. Each chapter ends with exercises that allow for a more in-depth understanding of the material and are intended to stimulate the reader toward further research. A comprehensive index completes the work.Topics covered:* Frames and bases in mathematics and engineering* Wavelets with composite dilations and their applications* Wavelets on the sphere and their applications* Wiener's Lemma: theme and variationsFour Short Courses on Harmonic Analysis is intended as a graduate-level textbook for courses or seminars on harmonic analysis and its applications. The work is also an excellent reference or self-study guide for researchers and practitioners with diverse mathematical backgrounds working in different fields such as pure and applied mathematics, image and signal processing engineering, mathematical physics, and communication theory.

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
Table of Contents provided by Ingram. All Rights Reserved.

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Four Short Courses on Harmonic Analysis

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