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An Introduction to Wavelets Through Linear Algebra By Michael W. Frazier

An Introduction to Wavelets Through Linear Algebra by Michael W. Frazier

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Summary

Wavelet theory is on the boundary between mathematics and engineering, making it ideal for demonstrating to students that mathematics research is thriving in the modern day. Intended to be as elementary an introduction to wavelet theory as possible, the text does not claim to be a thorough or authoritative reference on wavelet theory.

An Introduction to Wavelets Through Linear Algebra Summary

An Introduction to Wavelets Through Linear Algebra by Michael W. Frazier

Wavelet theory is on the boundary between mathematics and engineering, making it ideal for demonstrating to students that mathematics research is thriving in the modern day. Students can see non-trivial mathematics ideas leading to natural and important applications, such as video compression and the numerical solution of differential equations. The only prerequisites assumed are a basic linear algebra background and a bit of analysis background. Intended to be as elementary an introduction to wavelet theory as possible, the text does not claim to be a thorough or authoritative reference on wavelet theory.

Table of Contents

Preface Acknowledgments Prologue: Compression of the FBI Fingerprint Files 1 Background: Complex Numbers and Linear Algebra 1.1 Real Numbers and Complex Numbers 1.2 Complex Series, Euler's Formula, and the Roots of Unity 1.3 Vector Spaces and Bases 1.4 Linear Transformations, Matrices, and Change of Basis 1.5 Diagonalization of Linear Transformations and Matrices 1.6 Inner Products, Orthonormal Bases, and Unitary Matrices 2 The Discrete Fourier Transform 2.1 Basic Properties of the Discrete Fourier Transform 2.2 Translation-Invariant Linear Transformations 2.3 The Fast Fourier Transform 3 Wavelets on $bZ_N$ 3.1 Construction of Wavelets on $bZ_N$: The First Stage 3.2 Construction of Wavelets on $bZ_N$: The Iteration Step 3.3 Examples and Applications 4 Wavelets on $bZ$ 4.1 $\ell ^2(bZ)$ 4.2 Complete Orthonormal Sets in Hilbert Spaces 4.3 $L^2([-\pi ,\pi ))$ and Fourier Series 4.4 The Fourier Transform and Convolution on $\ell ^2(bZ)$ 4.5 First-Stage Wavelets on $bZ$ 4.6 The Iteration Step for Wavelets on $bZ$ 4.7 Implementation and Examples 5 Wavelets on $bR$ 5.1 $L^2(bR)$ and Approximate Identities 5.2 The Fourier Transform on $bR$ 5.3 Multiresolution Analysis and Wavelets 5.4 Construction of Multiresolution Analyses 5.5 Wavelets with Compact Support and Their Computation 6 Wavelets and Differential Equations 6.1 The Condition Number of a Matrix 6.2 Finite Difference Methods for Differential Equations 6.3 Wavelet-Galerkin Methods for Differential Equations Bibliography Index

Additional information

NLS9781475772999
9781475772999
1475772998
An Introduction to Wavelets Through Linear Algebra by Michael W. Frazier
New
Paperback
Springer-Verlag New York Inc.
2013-03-23
503
N/A
Book picture is for illustrative purposes only, actual binding, cover or edition may vary.
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