Wavelets is a carefully organized and edited collection of extended survey papers addressing key topics in the mathematical foundations and applications of wavelet theory. The first part of the book is devoted to the fundamentals of wavelet analysis. The construction of wavelet bases and the fast computation of the wavelet transform in both continuous and discrete settings is covered. The theory of frames, dilation equations, and local Fourier bases are also presented.
The second part of the book discusses applications in signal analysis, while the third part covers operator analysis and partial differential equations. Each chapter in these sections provides an up-to-date introduction to such topics as sampling theory, probability and statistics, compression, numerical analysis, turbulence, operator theory, and harmonic analysis.
The book is ideal for a general scientific and engineering audience, yet it is mathematically precise. It will be an especially useful reference for harmonic analysts, partial differential equation researchers, signal processing engineers, numerical analysts, fluids researchers, and applied mathematicians.
Table of Contents
Construction of Orthonormal Wavelets, R.S. Strichartz
An Introduction to the Orthonormal Wavelet Transform on Discrete Sets, M. Frazier and A. Kumar
Gabor Frames for L2 and Related Spaces, J.J. Benedetto and D.F. Walnut
Dilation Equations and the Smoothness of Compactly Supported Wavelets, C. Heil and D. Colella
Remarks on the Local Fourier Bases, P. Auscher
Wavelets and Signal Processing
The Sampling Theorem, Phi-Transform, and Shannon Wavelets for R, Z, T, and ZN, M. Frazier and R. Torres
Frame Decompositions, Sampling, and Uncertainty Principle Inequalities, J.J. Benedetto
Theory and Practice of Irregular Sampling, H.G. Feichtinger and K. Gröchenig
Wavelets, Probability, and Statistics: Some Bridges, C. Houdré
Wavelets and Adapted Waveform Analysis, R.R. Coifman and V. Wickerhauser
Near Optimal Compression of Orthonormal Wavelet Expansions, B. Jawerth, C.-C. Hsiao, B. Lucier, and X. Yu
Wavelets and Partial Differential Operators
On Wavelet-Based Algorithms for Solving Differential Equations, G. Beylkin
Wavelets and Nonlinear Analysis, S. Jaffard
Scale Decomposition in Burgers' Equation, F. Heurtaux, F. Planchon, and V. Wickerhauser
The Cauchy Singular Integral Operator and Clifford Wavelets, L. Andersson, B. Jawerth, and M. Mitrea
The Use of Decomposition Theorems in the Study of Operators, R. Rochberg
"Wavelets is a presentation of the highest quality, of the state of the art in wavelet theory and some of its applications to signal processing and numerical analysis. It is generally easy to read and some of the chapters can be used as introductions to certain aspects of the subject."
-Physics Today, November 1994