A First Course on Wavelets

by Hernandez; Eugenio

• ISBN13: 9780849382741
• ISBN10: 0849382742
• Edition: 1st
• Format: Hardcover
• Copyright: 1996-09-12
• Publisher: CRC Press

Summary

Wavelet theory had its origin in quantum field theory, signal analysis, and function space theory. In these areas wavelet-like algorithms replace the classical Fourier-type expansion of a function. This unique new book is an excellent introduction to the basic properties of wavelets, from background math to powerful applications. The authors provide elementary methods for constructing wavelets, and illustrate several new classes of wavelets.The text begins with a description of local sine and cosine bases that have been shown to be very effective in applications. Very little mathematical background is needed to follow this material. A complete treatment of band-limited wavelets follows. These are characterized by some elementary equations, allowing the authors to introduce many new wavelets. Next, the idea of multiresolution analysis (MRA) is developed, and the authors include simplified presentations of previous studies, particularly for compactly supported wavelets.Some of the topics treated include:o Several bases generated by a single function via translations and dilationso Multiresolution analysis, compactly supported wavelets, and spline waveletso Band-limited waveletso Unconditionality of wavelet baseso Characterizations of many of the principal objects in the theory of wavelets, such as low-pass filters and scaling functionsThe authors also present the basic philosophy that all orthonormal wavelets are completely characterized by two simple equations, and that most properties and constructions of wavelets can be developed using these two equations. Material related to applications is provided, and constructions of splines wavelets are presented.Mathematicians, engineers, physicists, and anyone with a mathematical background will find this to be an important text for furthering their studies on wavelets.

Table of Contents

Bases for L2(R)

Preliminaries

Orthonormal Bases Generated by a Single Function: The Balian-Low Theorem

Smooth Projections on L2(R)

Local Sine and Cosine Bases and the Construction of Some Wavelets

The Unitary Folding Operators and the Smooth Projections

Notes and References

Multiresolution Analysis and the Construction of Wavelets

Multiresolution Analysis

Construction of Wavelets from a Multiresolution Analysis

The Construction of Compactly Supported Wavelets

Better Estimates for the Smoothness of Compactly Supported Wavelets

Notes and References

Band-Limited Wavelets

Orthonormality

Completeness

The LemariT-Meyer Wavelets Revisited

Characterization of Some Band-Limited Wavelets

Notes and References

Other Constructions of Wavelets

Franklin Wavelets on the Real Line

Spline Wavelets on the Real Line

Orthonormal Bases of Piecewise Linear Continuous Functions for L2(T)

Orthonormal Bases of Periodic Splines

Periodization of Wavelets Defined on the Real Line

Notes and References

Representation of Functions by Wavelets

Bases for Banach Spaces

Unconditional Bases for Banach Spaces

Convergence of Wavelet Expansions in LP(R)

Pointwise Convergence of Wavelets Expansions

H1 and BMO on R

Wavelets as Unconditional Bases for H1(R) and LP(R) with 1