Hubbard, Barbara Burke

The World According to Wavelets : The Story of a Mathematical Technique in the Making

To the Reader xi(6) Acknowledgments xvii PART I The World According to Wavelets 1(110) Prologue 3(2) I Fourier Analysis: A Poem Transforms Our World 5(16) II Seeking New Tools 21(18) III A New Language Acquires a Grammar 39(18) IV Applications 57(34) V Beyond Wavelets 91(20) PART II Beyond Plain English 111(152) Apologia 113(4) 1 The Fourier Transform 117(8) 2 The Convergence of Fourier Series and the Stability of the Solar System 125(12) 3 Computing Fourier Coefficients with Integrals 137(4) 4 The Fast Fourier Transform 141(8) 5 The Continuous Wavelet Transform 149(4) 6 Orthogonality and Scalar Products 153(12) 7 Multiresolution 165(18) 8 The Fast Wavelet Transform 183(10) 9 Wavelets in Two Dimensions 193(6) 10 Pyramid Algorithms of Burt and Adelson 199(2) 11 Multiwavelets 201(2) 12 The Heisenberg Uncertainty Principle and Time-Frequency Decompositions 203(6) 13 Probability, Heisenberg, and Quantum Mechanics 209(14) 14 Traveling from One Function Space to Another: Wavelets and Pure Mathematics 223(8) 15 Wavelets and Vision: Another Perspective 231(8) 16 Which Wavelet? 239(10) 17 Different Transforms: A Summary 249(4) 18 Wavelets, Music, and Speech 253(4) 19 The Lifting Scheme 257(2) 20 Best Basis 259(4) PART III Appendices 263(32) A Mathematical Symbols 265(2) B A Review of Some Elementary Trigonometry 267(4) C Integrals 271(4) D The Fourier Transform: The Different Conventions 275(2) E The Sampling Theorem: A Proof 277(4) F The Heisenberg Uncertainty Principle: A Proof 281(4) G The Fourier Transform of a Periodic Function 285(6) H An Example of an Orthonormal Basis and a Proof of Fourier's Result 291(4) Further Reading 295(4) Wavelet Software and Electronic Resources 299(6) References 305(14) Index 319