Download e-book for iPad: An Introduction to Wavelet Analysis by David F. Walnut

By David F. Walnut

ISBN-10: 0817639624

ISBN-13: 9780817639624

This e-book presents a entire presentation of the conceptual foundation of wavelet research, together with the development and research of wavelet bases. It motivates the imperative rules of wavelet thought through delivering an in depth exposition of the Haar sequence, then indicates how a extra summary technique permits readers to generalize and increase upon the Haar sequence. It then provides a couple of diversifications and extensions of Haar development.

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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. Specifically, for n E Z, and for n E N, and (c) A function that can be written as a finite linear combination of elements of the (period a) trigonornctric system is called a (period a ) t r i g o n o m e t ~ i c polynomial.

A) Any orthogonal system can be normalized so that it becomes an orthonormal system. x)) is a11 orthogonal system, then we may define the functions Then the system {&(x)) is an orthonormal system. 45 exists as a finite number. 45. 47. (a) Given any a > 0, the collection is an orthogonal system over [-a, a ] . I t is also orthogonal over [O,2a]and in fact over any interval I of length 2a. The collection is an orthonormal system over [-a, a ] . I t is also ort'honormal over [O, 2a] and in fact over any interval I of length 2a.

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.

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An Introduction to Wavelet Analysis by David F. Walnut

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