Presenting the subject from the point of view of signal analysis, A First Course in Wavelets with Fourier Analysis, 2nd Edition provides a self-contained mathematical treatment of the subject that is accessible to a broad audience. Expanded applications to signal processing, exercises, and complete proofs of the presented theory, as well as solutions to selected exercises in the back of the book, make this an accessible reference for mathematicians, signal processing engineers, scientists, and advanced undergraduates.

Albert Boggess, PhD. is Professor of Mathematics at Texas AM University. Dr. Boggess has over twenty-five years of academic experience and has authored numerous publications in his areas of research interest, which include overdetermined systems of partial differential equations, several complex variables, and harmonic analysis. Francis J. Narcowich, PhD. is Professor of Mathematics and Director of the Center for Approximation Theory at Texas AM University. Dr. Narcowich serves as an Associate Editor of both the SIAM Journal on Numerical Analysis and Mathematics of Computation, and he has written more than eighty papers on a variety of topics in pure and applied mathematics. He currently focuses his research on applied harmonic analysis and approximation theory.

Preface and Overview	p. ix
Inner Product Spaces	p. 1
Motivation	p. 1
Definition of Inner Product	p. 2
The Spaces L² and l²	p. 4
Definitions	p. 4
Convergence in L² Versus Uniform Convergence	p. 8
Schwarz and Triangle Inequalities	p. 11
Orthogonality	p. 13
Definitions and Examples	p. 13
Orthogonal Projections	p. 15
Gram-Schmidt Orthogonalization	p. 20
Linear Operators and Their Adjoints	p. 21
Linear Operators	p. 21
Adjoints	p. 23
Least Squares and Linear Predictive Coding	p. 25
Best-Fit Line for Data	p. 25
General Least Squares Algorithm	p. 29
Linear Predictive Coding	p. 31
Exercises	p. 34
Fourier Series	p. 38
Introduction	p. 38
Historical Perspective	p. 38
Signal Analysis	p. 39
Partial Differential Equations	p. 40
Computation of Fourier Series	p. 42
On the Interval -¿ ≤ x ≤ ¿	p. 42
Other Intervals	p. 44
Cosine and Sine Expansions	p. 47
Examples	p. 50
The Complex Form of Fourier Series	p. 58
Convergence Theorems for Fourier Series	p. 62
The Riemann-Lebesgue Lemma	p. 62
Convergence at a Point of Continuity	p. 64
Convergence at a Point of Discontinuity	p. 69
Uniform Convergence	p. 72
Convergence in the Mean	p. 76
Exercises	p. 83
The Fourier Transform	p. 92
Informal Development of the Fourier Transform	p. 92
The Fourier Inversion Theorem	p. 92
Examples	p. 95
Properties of the Fourier Transform	p. 101
Basic Properties	p. 101
Fourier Transform of a Convolution	p. 107
Adjoint of the Fourier Transform	p. 109
Plancherel Theorem	p. 109
Linear Filters	p. 110
Time-Invariant Filters	p. 110
Causality and the Design of Filters	p. 115
The Sampling Theorem	p. 120
The Uncertainty Principle	p. 123
Exercises	p. 127
Discrete Fourier Analysis	p. 132
The Discrete Fourier Transform	p. 132
Definition of Discrete Fourier Transform	p. 134
Properties of the Discrete Fourier Transform	p. 135
The Fast Fourier Transform	p. 138
The FFT Approximation to the Fourier Transform	p. 143
Application: Parameter Identification	p. 144
Application: Discretizations of Ordinary Differential Equations	p. 146
Discrete Signals	p. 147
Time-Invariant, Discrete Linear Filters	p. 147
Z-Transform and Transfer Functions	p. 149
Discrete Signals & Matlab	p. 153
Exercises	p. 156
Haar Wavelet Analysis	p. 160
Why Wavelets?	p. 160
Haar Wavelets	p. 161
The Haar Scaling Function	p. 161
Basic Properties of the Haar Scaling Function	p. 167
The Haar Wavelet	p. 168
Haar Decomposition and Reconstruction Algorithms	p. 172
Decomposition	p. 172
	p. 176
Filters and Diagrams	p. 182
Summary	p. 185
Exercises	p. 186
Multiresolution Analysis	p. 190
The Multiresolution Framework	p. 190
Definition	p. 190
The Scaling Relation	p. 194
The Associated Wavelet and Wavelet Spaces	p. 197
Decomposition and Reconstruction Formulas: A Tale of Two Bases	p. 201
Summary	p. 203
Implementing Decomposition and Reconstruction	p. 204
The Decomposition Algorithm	p. 204
The Reconstruction Algorithm	p. 209
Processing a Signal	p. 213
Fourier Transform Criteria	p. 214
The Scaling Function	p. 215
Orthogonality via the Fourier Transform	p. 217
The Scaling Equation via the Fourier Transform	p. 221
Iterative Procedure for Constructing the Scaling Function	p. 225
Exercises	p. 228
The Daubechies Wavelets	p. 234
Daubechies' Construction	p. 234
Classification, Moments, and Smoothness	p. 238
Computational Issues	p. 242
The Scaling Function at Dyadic Points	p. 244
Exercises	p. 248
Other Wavelet Topics	p. 250
Computational Complexity	p. 250
Wavelet Algorithm	p. 250
Wavelet Packets	p. 251
Wavelets in Higher Dimensions	p. 253
Exercises on 2D Wavelets	p. 258
Relating Decomposition and Reconstruction	p. 259
Transfer Function Interpretation	p. 263
Wavelet Transform	p. 266
Definition of the Wavelet Transform	p. 266
Inversion Formula for the Wavelet Transform	p. 268
Technical Matters	p. 273
Proof of the Fourier Inversion Formula	p. 273
Technical Proofs from Chapter 5	p. 277
Rigorous Proof of Theorem 5.17	p. 277
Proof of Theorem 5.10	p. 281
Proof of the Convergence Part of Theorem 5.23	p. 283
Solutions to Selected Exercises	p. 287
MATLAB“ Routines	p. 305
General Compression Routine	p. 305
Use of MATLAB's FFT Routine for Filtering and Compression 306
Sample Routines Using MATLAB's Wavelet Toolbox	p. 307
MATLAB Code for the Algorithms in Section 5.2	p. 308
Bibliography	p. 311
Index	p. 313
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