A First Course on Wavelets

CRC Press, Sep 12, 1996 - Mathematics - 489 pages
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:
• Several bases generated by a single function via translations and dilations
• Multiresolution analysis, compactly supported wavelets, and spline wavelets
• Band-limited wavelets
• Unconditionality of wavelet bases
• Characterizations of many of the principal objects in the theory of wavelets, such as low-pass filters and scaling functions

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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Review: A First Course on Wavelets

User Review  - Zhitao Fan - Goodreads

An introductory level wavelet book. The author organize it in his own understanding and it basically includes all the elementary mathematical analysis of wavelet. Ch 1,2,3,4,7 gives the details of ... Read full review

Contents

 Bases for L²R 1 11 Preliminaries 2 12 Orthonormal bases generated by a single function the BalianLow theorem 7 13 Smooth projections on L²R 11 14 Local sine and cosine bases and the construction of some wavelets 20 15 The unitary folding operators and the smooth projections 31 16 Notes and references 40 Multiresolution analysis and the construction of wavelets 43
 61 Wavelets and sampling theorems 256 62 LittlewoodPaley theory 260 63 Necessary tools 267 64 The Lebesgue spaces LpR with 1 p oo 279 65 The Hardy space HlR 287 66 The Sobolev spaces LPSR 1 p oo s 123 296 67 The Lipschitz spaces AQ R 0 a 1 and the Zygmund class A R 313 68 Notes and references 323

 21 Multiresolution analysis 44 22 Construction of wavelets from a multiresolution analysis 52 23 The construction of compactly supported wavelets 68 24 Better estimates for the smoothness of compactly supported wavelets 93 25 Notes and references 98 Bandlimited wavelets 101 32 Completeness 105 33 The LemariéMeyer wavelets revisited 115 34 Characterization of some bandlimited wavelets 123 35 Notes and references 138 Other constructions of wavelets 141 42 Spline wavelets on the real line 152 43 Orthonormal bases of piecewise linear continuous functions for L2T 164 44 Orthonormal bases of periodic splines 176 45 Periodization of wavelets defined on the real line 186 46 Notes and references 193 Representation of functions by wavelets 201 52 Unconditional bases for Banach spaces 206 53 Convergence of wavelet expansions in V 217 54 Pointwise convergence of wavelet expansions 225 55 H¹ and BMO on R 230 56 Wavelets as unconditional bases for HR and V with 1 p oo 237 57 Notes and references 247 Characterizations of function spaces using wavelets 255
 Characterizations in the theory of wavelets 331 71 The basic equations 332 72 Some applications of the basic equations 348 73 The characterization of MRA wavelets 354 74 A characterization of lowpass filters 365 75 A characterization of scaling functions 381 76 Nonexistence of smooth wavelets in H²R 386 77 Notes and references 393 Frames 397 81 The reconstruction formula for frames 398 82 The BalianLow theorem for frames 403 83 Frames from translations and dilations 410 84 Smooth frames forH²l 418 85 Notes and references 419 Discrete transforms and algorithms 427 92 The discrete cosine transform DCT and the fast cosine transform FCT 432 93 The discrete version of the local sine and cosine bases 436 94 Decomposition and reconstruction algorithms for wavelets 442 95 Wavelet packets 449 96 Notes and references 464 References 467 Author index 479 Index 485 Copyright

References from web pages

JSTOR: A First Course on Wavelets
Just the right thing for a first course on wavelets. The subsequent chapters, nine in all, contain the essential information about or- thonormal wavelets. ...

The Wavelet Digest :: View topic - Book: A first course on wavelets
#1 Book: A first course on wavelets A First Course on Wavelets Eugenio Hernandez and Guido Weiss A Volume in the Studies in Advanced Mathematics Series ...
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mathnetbase: Mathematics Online
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JHU Libraries Catalog
A first course on wavelets / Eugenio Hernández, Guido Weiss. Author:. Hernández, Eugenio, 1954-. Subjects:. Wavelets (Mathematics). Publisher: ...
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A First Course on Wavelets (Studies in Advanced Mathematics)(页1 ...
[attach]2084[/attach]Title: A First Course on waveletsauthor: Eugenio hernandezpublisher: CRC-presspublication Date: 1996-09-12Number Of Pages: 489Wavelet ...
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Публичная Электронная Библиотека » Hernandez E., Weiss G. » A ...
A first course on wavelets. » Наука » Теория обработки сигналов. Начало « Наука « Математика « Теория обработки сигналов «...
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Biblioteca Ceride
A FIRST COURSE ON WAVELETS. -. HERNANDEZ, EUGENIO, WEISS, GUIDO. 489 p -24 cm x cm. 0849382742. QA403.3 H43 1996 E.2. No Posee ...
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Journal of Approximation Theory : Measures, densities and ...
E. Hernandez and G. Weiss, A First Course on Wavelets, CRC Press, Boca Raton, FL (1996). rl Long, Multidimensional Wavelet Analysis, ...
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Ja-seung's homepage
A First Course on WAVELETS Author : E. Hernandez and G. Weiss, CRC press, 1996 Wavelets in the Signal Processing A Wavelet Tour of Signal Processing ...
amath.kaist.ac.kr/ ~capricio/ wavelet/ wavelet.html