An Introduction to wavelets through linear algebra

✍ Scribed by Frazier M.

• Publisher: Springer
• Year: 1999
• Tongue: English
• Leaves: 520
• Series: Undergraduate Texts in Mathematics
• Category: Library

✦ Synopsis

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

Cover
......Page 1
An Introduction to Wavelets through Linear Algebra
......Page 4
ISBN 0387986391
......Page 5
Preface
......Page 6
Aknowledgments
......Page 12
Contents
......Page 16
Prologue: Compression of the FBI Fingerprint Files
......Page 18
1.1 Real Numbers and Complex Numbers......Page 24
Exercises......Page 32
1.2 Complex Series, Euler’s Formula, and the Roots of Unity......Page 33
Exercises......Page 43
1.3 Vector Spaces and Bases......Page 46
Exercises......Page 54
1.4 Linear Transformations, Matrices, and Change of Basis......Page 57
Exercises......Page 69
1.5 Diagonalization of Linear Transformations and Matrices......Page 73
Exercises......Page 93
1.6 Inner Products, Orthonormal Bases, and Unitary Matrices......Page 96
Exercises......Page 111
2.1 De.nition and Basic Properties of the Discrete Fourier Transform......Page 118
Exercises......Page 139
2.2 Translation-Invariant Linear Transformations......Page 145
Exercises......Page 161
2.3 The Fast Fourier Transform......Page 168
Exercises......Page 179
3.1 Construction of Wavelets on ZN: The First Stage......Page 182
Exercises......Page 204
3.2 Construction of Wavelets on ZN: The Iteration Step......Page 213
Exercises......Page 235
3.3 Examples and Applications......Page 242
Exercises......Page 278
4.1 2(Z)......Page 282
Exercises......Page 285
4.2 Complete Orthonormal Sets in Hilbert Spaces......Page 288
Exercises......Page 294
4.3 L2([-p, p)) and Fourier Series......Page 296
Exercises......Page 309
4.4 The Fourier Transform and Convolution on 2(Z)......Page 315
Exercises......Page 324
4.5 First-Stage Wavelets on Z......Page 326
Exercises......Page 334
4.6 The Iteration Step for Wavelets on Z......Page 338
Exercises......Page 345
4.7 Implementation and Examples......Page 347
Exercises......Page 363
5.1 L2(R) and Approximate Identities......Page 366
Exercises......Page 376
5.2 The Fourier Transform on R......Page 379
Exercises......Page 393
5.3 Multiresolution Analysis and Wavelets......Page 397
Exercises......Page 411
5.4 Construction of Multiresolution Analyses......Page 415
Exercises......Page 439
5.5 Wavelets with Compact Support and Their Computation......Page 446
Exercises......Page 464
6.1 The Condition Number of a Matrix......Page 468
Exercises......Page 473
6.2 Finite Difference Methods for Differential Equations......Page 476
Exercises......Page 482
6.3 Wavelet-Galerkin Methods for Differential Equations......Page 487
Exercises......Page 498
Bibliography......Page 501
Index
......Page 508