An Introduction to Wavelet Analysis - download pdf or read online

By David F. Walnut

ISBN-10: 0817639624

ISBN-13: 9780817639624

This ebook presents a accomplished presentation of the conceptual foundation of wavelet research, together with the development and research of wavelet bases. It motivates the critical rules of wavelet conception via delivering a close exposition of the Haar sequence, then exhibits how a extra summary method permits readers to generalize and enhance upon the Haar sequence. It then offers a few diversifications and extensions of Haar development.

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Extra resources for An Introduction to Wavelet Analysis

Sample text

F,(x) = C,LEz f ( x + np) = f (x) since all terms in the sum besides the n = 0 tern1 are zero. c) and placing thein side-by-side on the real line. 28 Chapter 2. 1. Top Left: Graph of f (x). Top Right: Graphs of f (z + n p ) for -2 n 2 and p = 1. Bottom: Graph of the 1-periodization of f (z). 5. 2) {e2711r'T'a )nt~ ss called the (period a ) trigonometric system. 6. ' = cos(x) i sin(x). 20). Therefore and it follows from this that each element in the trigonometric system has period a. 1. Trigonometric Series 29 linear combinations.

2. ~}:L~' (d) For each j . the collection of intervals forms a partlition of [O, 1);that is, the intervals are disjoint and cover all of [O; 1). - Now, define f , ( z ) = X I J I L , k r L( z ) . Then since I I j 3 , , k r L1 + O as j oo, f n ( z ) + O in mean on [O,l). However, f,(z) does not converge t o zero pointwise because for every x E [0, I), there are infinitely many n for which f,(z) = 1. Therefore, f,(z) does not converge to anything at any point of [0,1). 6. 6. 36 for 1 5 n 5 12. 37.

30. 5). 29 would be as follows. 29 irmplies that f (z)shou,ld also be continuous. Since this i s not the case, the convergence cannot be i n La. 31. (a) The sequence { x n I r L E N converges iiniformly to zero on [-a,a] for all O < a < 1 but does not converge uniformly t o zero on (-111). - ~ (b) The series 1 I" = - uniformly 1-x n=O on [-a, a] for all 0 < a < 1, but not on (- 1 , l ) . 30 (c) The series 1n! IC -= ex uniformly on every finite interval I, but not on n=O R. -test. n . 2. 5. r). 4 71.

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An Introduction to Wavelet Analysis by David F. Walnut

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