A First Course on Wavelets
Wavelet theory had its origin in quantum field theory, signal analysis, and function space theory. In these areas wavelet-like algorithms replace the classical Fourier-type expansion of a function. This unique new book is an excellent introduction to the basic properties of wavelets, from background math to powerful applications. The authors provide elementary methods for constructing wavelets, and illustrate several new classes of wavelets.
The text begins with a description of local sine and cosine bases that have been shown to be very effective in applications. Very little mathematical background is needed to follow this material. A complete treatment of band-limited wavelets follows. These are characterized by some elementary equations, allowing the authors to introduce many new wavelets. Next, the idea of multiresolution analysis (MRA) is developed, and the authors include simplified presentations of previous studies, particularly for compactly supported wavelets.
Some of the topics treated include:
The authors also present the basic philosophy that all orthonormal wavelets are completely characterized by two simple equations, and that most properties and constructions of wavelets can be developed using these two equations. Material related to applications is provided, and constructions of splines wavelets are presented.
Mathematicians, engineers, physicists, and anyone with a mathematical background will find this to be an important text for furthering their studies on wavelets.
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12 Orthonormal bases generated by a single function the BalianLow theorem
13 Smooth projections on LČR
14 Local sine and cosine bases and the construction of some wavelets
15 The unitary folding operators and the smooth projections
16 Notes and references
Multiresolution analysis and the construction of wavelets
61 Wavelets and sampling theorems
62 LittlewoodPaley theory
63 Necessary tools
64 The Lebesgue spaces LpR with 1 p oo
65 The Hardy space HlR
66 The Sobolev spaces LPSR 1 p oo s 123
67 The Lipschitz spaces AQ R 0 a 1 and the Zygmund class A R
21 Multiresolution analysis
22 Construction of wavelets from a multiresolution analysis
23 The construction of compactly supported wavelets
24 Better estimates for the smoothness of compactly supported wavelets
25 Notes and references
33 The LemariéMeyer wavelets revisited
34 Characterization of some bandlimited wavelets
35 Notes and references
Other constructions of wavelets
42 Spline wavelets on the real line
43 Orthonormal bases of piecewise linear continuous functions for L2T
44 Orthonormal bases of periodic splines
45 Periodization of wavelets defined on the real line
Representation of functions by wavelets
52 Unconditional bases for Banach spaces
53 Convergence of wavelet expansions in V
54 Pointwise convergence of wavelet expansions
55 Hč and BMO on R
56 Wavelets as unconditional bases for HR and V with 1 p oo
57 Notes and references
Characterizations of function spaces using wavelets
Characterizations in the theory of wavelets
71 The basic equations
72 Some applications of the basic equations
73 The characterization of MRA wavelets
74 A characterization of lowpass filters
75 A characterization of scaling functions
76 Nonexistence of smooth wavelets in HČR
81 The reconstruction formula for frames
82 The BalianLow theorem for frames
83 Frames from translations and dilations
84 Smooth frames forHČl
Discrete transforms and algorithms
92 The discrete cosine transform DCT and the fast cosine transform FCT
93 The discrete version of the local sine and cosine bases
94 Decomposition and reconstruction algorithms for wavelets
95 Wavelet packets
96 Notes and references
1-periodic 27r-periodic function algorithm apply associated assume atomic Balian-Low theorem Banach space band-limited basis for L2(R bell function belongs bounded Chapter characterization compactly supported compactly supported wavelets condition construct convergence Corollary cosine decomposition deduce definition dilations dyadic equality equations equivalent example exists a constant f 7r fact fcez formula Fourier series Fourier transform frame Franklin wavelet given graph Haar wavelet Hardy space Hence Hilbert space implies inequality integral interval ipj,k L2-norm La(R Lebesgue Lemarie-Meyer wavelets Lemma linear low-pass filter LP(R Moreover MSF wavelets multiresolution analysis norm Observe obtain operator orthogonal orthonormal basis orthonormal system Plancherel theorem polarities proof of Theorem properties Proposition 1.11 prove satisfies scaling function sequence Shannon wavelet smooth spline wavelets subspaces supp Suppose Theorem 4.1 translations trigonometric polynomial unconditional basis unitary wavelet ip wavelet packets write zero
Page ii - Steven R. Bell, The Cauchy Transform, Potential Theory, and Conformal Mapping John J. Benedetto, Harmonic Analysis and Applications John J. Benedetto and Michael W. Frazier, Wavelets: Mathematics and Applications Albert Boggess, CR Manifolds and the Tangential Cauchy-Riemann Complex Goong Chen and Jianxin Zhou, Vibration and Damping in Distributed Systems, Vol.
Page ii - John Ryan, Clifford Algebras in Analysis and Related Topics Xavier Saint Raymond, Elementary Introduction to the Theory of Pseudodifferential Operators Robert Strichartz, A Guide to Distribution Theory and Fourier Transforms Andre Unterberger and Harold Upmeier, Pseudodifferential Analysis on Symmetric Cones James S.
Page ii - Studies in Advanced Mathematics Series Editor STEVEN G. KRANTZ Washington University in St. Louis Editorial Board R. Michael Beats Gerald B.
Page ii - Kenneth L. Kuttler, Modern Analysis Michael Pedersen, Functional Analysis in Applied Mathematics and Engineering Clark Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd Edition John Ryan, Clifford Algebras in Analysis and Related Topics...
Page ii - Hernandez, Fernando Soria, and Jose-Luis Torrea, Fourier Analysis and Partial Differential Equations Peter B Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2nd Edition...
Page 2 - PRELIMINARIES We assume that the reader is familiar with the basic notions of formal languages and codes, cf . , eg,  or .
A Primer on Wavelets and Their Scientific Applications
James S. Walker
Limited preview - 2002
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