An Introduction to Wavelets Through Linear Algebra (Undergraduate Texts in Mathematics)

Frazier, Michael W.

ISBN 10: 1475772998 / ISBN 13: 9781475772999

Editore: Springer, 2013

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Codice articolo ABLIING23Mar2716030093959

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Titolo: An Introduction to Wavelets Through Linear ...

Casa editrice: Springer

Data di pubblicazione: 2013

Legatura: Brossura

Condizione: New

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Riassunto:

This text was originally written for a "Capstone" course at Michigan State University. Basic wavelet theory seems to be a perfect topic for such a course. It is on the boundary between mathematics and engineering. 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. This text is intended to be as elementary an introduction to wavelet theory as possible. It is not intended as a thorough or authoritative reference on wavelet theory.

Contenuti:

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