Wavelet Transforms and Their Applications

This textbook is an introduction to wavelet transforms and accessible to a larger audience with diverse backgrounds and interests in mathematics, science, and engineering. Emphasis is placed on the logical development of fundamental ideas and systematic treatment of wavelet analysis and its applications to a wide variety of problems as encountered in various interdisciplinary areas. Topics and Features: \* This second edition heavily reworks the chapters on Extensions of Multiresolution Analysis and Newlands’s Harmonic Wavelets and introduces a new chapter containing new applications of wavelet transforms \* Uses knowledge of Fourier transforms, some elementary ideas of Hilbert spaces, and orthonormal systems to develop the theory and applications of wavelet analysis \* Offers detailed and clear explanations of every concept and method, accompanied by carefully selected worked examples, with special emphasis given to those topics in which students typically experience difficulty \* Includes carefully chosen end-of-chapter exercises directly associated with applications or formulated in terms of the mathematical, physical, and engineering context and provides answers to selected exercises for additional help Mathematicians, physicists, computer engineers, and electrical and mechanical engineers will find Wavelet Transforms and Their Applications an exceptionally complete and accessible text and reference. It is also suitable as a self-study or reference guide for practitioners and professionals. Preface to the Second Edition 6 Preface to the First Edition 10 Overview 10 Audience and Organization 11 Salient Features 12 Acknowledgments 13 Contents 14 1 Brief Historical Introduction 17 1.1 Fourier Series and Fourier Transforms 17 1.2 Gabor Transforms 20 1.3 The Wigner–Ville Distribution and Time–Frequency Signal Analysis 22 1.4 Wavelet Transforms 26 1.5 Wavelet Bases and Multiresolution Analysis 35 1.6 Applications of Wavelet Transforms 42 2 Hilbert Spaces and Orthonormal Systems 44 2.1 Introduction 44 2.2 Normed Spaces 45 2.3 The Lp Spaces 48 2.4 Generalized Functions with Examples 53 2.5 Definition and Examples of an Inner Product Space 63 2.6 Norm in an Inner Product Space 66 2.7 Definition and Examples of Hilbert Spaces 69 2.8 Strong and Weak Convergences 74 2.9 Orthogonal and Orthonormal Systems 76 2.10 Properties of Orthonormal Systems 81 2.11 Trigonometric Fourier Series 90 2.12 Orthogonal Complements and the Projection Theorem 94 2.13 Linear Functionals and the Riesz Representation Theorem 99 2.14 Separable Hilbert Spaces 101 2.15 Linear Operators on Hilbert Spaces 103 2.16 Eigenvalues and Eigenvectors of an Operator 121 2.17 Exercises 131 3 Fourier Transforms and Their Applications 143 3.1 Introduction 143 3.2 Fourier Transforms in L1(R) 144 3.3 Basic Properties of Fourier Transforms 149 3.4 Fourier Transforms in L2(R) 163 3.5 Discrete Fourier Transforms 178 3.6 Fast Fourier Transforms 183 3.7 Poisson's Summation Formula 187 3.8 The Shannon Sampling Theorem and Gibbs' Phenomenon 193 3.9 Heisenberg's Uncertainty Principle 204 3.10 Applications of Fourier Transforms in Mathematical Statistics 206 3.11 Applications of Fourier Transforms to Ordinary Differential Equations 213 3.12 Solutions of Integral Equations 217 3.13 Solutions of Partial Differential Equations 220 3.14 Applications of Multiple Fourier Transforms to Partial Differential Equations 232 3.15 Construction of Green's Functions by the Fourier Transform Method 237 3.16 Exercises 250 4 The Gabor Transform and Time–Frequency Signal Analysis 257 4.1 Introduction 257 4.2 Classification of Signals and the Joint Time–Frequency Analysis of Signals 258 4.3 Definition and Examples of the Gabor Transform 262 4.4 Basic Properties of Gabor Transforms 266 4.5 Frames and Frame Operators 271 4.6 Discrete Gabor Transforms and the Gabor Representation Problem 279 4.7 The Zak Transform and Time–Frequency Signal Analysis 282 4.8 Basic Properties of Zak Transforms 285 4.9 Applications of Zak Transforms and the Balian–Low Theorem 291 4.10 Exercises 298 5 The Wigner–Ville Distribution and Time–Frequency Signal Analysis 301 5.1 Introduction 301 5.2 Definition and Examples of the WVD 302 5.3 Basic Properties of the WVD 311 5.4 The WVD of Analytic Signals and Band-Limited Signals 319 5.5 Definitions and Examples of the Woodward AmbiguityFunctions 323 5.6 Basic Properties of Ambiguity Functions 330 5.7 The Ambiguity Transformation and Its Properties 336 5.8 Discrete WVDs 340 5.9 Cohen's Class of Time–Frequency Distributions 344 5.10 Exercises 347 6 The Wavelet Transforms and Their Basic Properties 351 6.1 Introduction 351 6.2 Continuous Wavelet Transforms and Examples 354 6.3 Basic Properties of Wavelet Transforms 365 6.4 The Discrete Wavelet Transforms 368 6.5 Orthonormal Wavelets 378 6.6 Exercises 384 7 Multiresolution Analysis and Construction of Wavelets 388 7.1 Introduction 388 7.2 Definition of MRA and Examples 389 7.3 Properties of Scaling Functions and Orthonormal Wavelet Bases 396 7.4 Construction of Orthonormal Wavelets 414 7.5 Daubechies' Wavelets and Algorithms 429 7.6 Discrete Wavelet Transforms and Mallat's Pyramid Algorithm 446 7.7 Exercises 450 8 Extensions of Multiresolution Analysis 454 8.1 Introduction 454 8.2 p-MRA on a Half-Line R+ 455 8.3 Nonuniform MRA 476 9 Newland's Harmonic Wavelets 487 9.1 Introduction 487 9.2 Harmonic Wavelets 487 9.3 Properties of Harmonic Scaling Functions 493 9.4 Wavelet Expansions and Parseval's Formula 496 9.5 Concluding Remarks 497 9.6 Exercises 498 10 Wavelet Transform Analysis of Turbulence 500 10.1 Introduction 500 10.2 Fourier Transforms in Turbulence and the Navier–Stokes Equations 503 10.3 Fractals, Multifractals, and Singularitiesin Turbulence 511 10.4 Farge's Wavelet Transform Analysis of Turbulence 517 10.5 Adaptive Wavelet Method for Analysis of Turbulent Flows 520 10.6 Meneveau's Wavelet Analysis of Turbulence 523 Answers and Hints for Selected Exercises 528 2.7 Exercises 528 3.16 Exercises 529 4.10 Exercises 531 5.10 Exercises 532 6.6 Exercises 533 7.7 Exercises 534 9.6 Exercises 537 Bibliography 541 Index 554 This textbook is an introduction to wavelet transforms and accessible to a larger audience with diverse backgrounds and interests in mathematics, science, and engineering. Emphasis is placed on the logical development of fundamental ideas and systematic treatment of wavelet analysis and its applications to a wide variety of problems as encountered in various interdisciplinary areas. Numerous standard and challenging topics, applications, and exercises are included in this edition, which will stimulate research interest among senior undergraduate and graduate students. The book contains a large number of examples, which are either directly associated with applications or formulated in terms of the mathematical, physical, and engineering context in which wavelet theory arises. Topics and Features of the Second Edition: · Expanded and revised the historical introduction by including many new topics such as the fractional Fourier transform, and the construction of wavelet bases in various spaces other than and several new extensions of the original multiresolution analysis. · Extensions of the classical theory of multiresolution analysis consisting of Ƥ-multiresolution analysis on the positive half-line and non-uniform multiresolution analysis. · Includes carefully chosen end-of-chapter exercises directly associated with applications or formulated in terms of the mathematical, physical, and engineering context and provides answers to selected exercises for additional help · Completely updated bibliography and enlarged index Mathematicians, physicists, computer engineers, and electrical and mechanical engineers will find Wavelet Transforms and Their Applications an exceptionally complete and accessible text and reference. It is also suitable as a self-study or reference guide for practitioners and professionals Front Matter....Pages i-xv Brief Historical Introduction....Pages 1-27 Hilbert Spaces and Orthonormal Systems....Pages 29-127 Fourier Transforms and Their Applications....Pages 129-242 The Gabor Transform and Time–Frequency Signal Analysis....Pages 243-286 The Wigner–Ville Distribution and Time–Frequency Signal Analysis....Pages 287-336 The Wavelet Transforms and Their Basic Properties....Pages 337-373 Multiresolution Analysis and Construction of Wavelets....Pages 375-440 Extensions of Multiresolution Analysis....Pages 441-473 Newland’s Harmonic Wavelets....Pages 475-487 Wavelet Transform Analysis of Turbulence....Pages 489-516 Back Matter....Pages 517-553