Introduction to Wavelets Through Linear Algebra

ISBN-10: 0387986391
ISBN-13: 9780387986395
Edition: 1999
List price: $79.95 Buy it from $26.26
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Description: The mathematical theory of wavelets is less than 15 years old, yet already wavelets have become a fundamental tool in many areas of applied mathematics and engineering. This introduction to wavelets assumes a basic background in linear algebra  More...

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Book details

List price: $79.95
Copyright year: 1999
Publisher: Springer
Publication date: 8/17/2001
Binding: Hardcover
Pages: 503
Size: 6.25" wide x 9.25" long x 1.00" tall
Weight: 1.936
Language: English

The mathematical theory of wavelets is less than 15 years old, yet already wavelets have become a fundamental tool in many areas of applied mathematics and engineering. This introduction to wavelets assumes a basic background in linear algebra (reviewed in Chapter 1) and real analysis at the undergraduate level. Fourier and wavelet analyses are first presented in the finite-dimensional context, using only linear algebra. Then Fourier series are introduced in order to develop wavelets in the infinite-dimensional, but discrete context. Finally, the text discusses Fourier transform and wavelet theory on the real line. The computation of the wavelet transform via filter banks is emphasized, and applications to signal compression and numerical differential equations are given. This text is ideal for a topics course for mathematics majors, because it exhibits and emerging mathematical theory with many applications. It also allows engineering students without graduate mathematics prerequisites to gain a practical knowledge of wavelets.

Preface
Acknowledgments
Prologue: Compression of the FBI Fingerprint Files
Background: Complex Numbers and Linear Algebra
Real Numbers and Complex Numbers
Complex Series, Euler's Formula, and the Roots of Unity
Vector Spaces and Bases
Linear Transformations, Matrices, and Change of Basis
Diagonalization of Linear Transformations and Matrices
Inner Products, Orthonormal Bases, and Unitary Matrices
The Discrete Fourier Transform
Basic Properties of the Discrete Fourier Transform
Translation-Invariant Linear Transformations
The Fast Fourier Transform
Wavelets on $bZ_N$
Construction of Wavelets on $bZ_N$: The First Stage
Construction of Wavelets on $bZ_N$: The Iteration Step
Examples and Applications
Wavelets on $bZ$
$ ell ^2(bZ)$
Complete Orthonormal Sets in Hilbert Spaces
$L^2([- pi , pi ))$ and Fourier Series
The Fourier Transform and Convolution on $ ell ^2(bZ)$
First-Stage Wavelets on $bZ$
The Iteration Step for Wavelets on $bZ$
Implementation and Examples
Wavelets on $bR$
$L^2(bR)$ and Approximate Identities
The Fourier Transform on $bR$
Multiresolution Analysis and Wavelets
Construction of Multiresolution Analyses
Wavelets with Compact Support and Their Computation
Wavelets and Differential Equations
The Condition Number of a Matrix
Finite Difference Methods for Differential Equations
Wavelet-Galerkin Methods for Differential Equations
Bibliography
Index

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