By David F. Walnut

This ebook offers a finished presentation of the conceptual foundation of wavelet research, together with the development and research of wavelet bases. It motivates the imperative rules of wavelet conception through supplying a close exposition of the Haar sequence, then exhibits how a extra summary process permits readers to generalize and enhance upon the Haar sequence. It then offers a few adaptations and extensions of Haar development.

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**Sample text**

If both of these series converge, then tliere is no problem. If C T00L =a, , = St and C:IP=, a p r 1= S ,then Cr=-, a,, = on+St+S- = S . In this case, we write also l i ~ n ~ , ~ ~ , ,r = a,,- =~S. ~ We will frrqllently denote the series a,, by C r L E artZ 01. simply by C,,a,, . xr=-, (g) If a doubly infinite series converges absolutely, then it converges regardless of the order in which the terms are surnmed. This is not tkic casc with series that do not convergc absolutely. Consider the series l / n ,where the n = 0 term is understood to be zero.

38(b) is false. To see this, let f,,(z) = J n X ( ~ , l / n(x). 2. 11. 36 shows that L2 convergence does not irnply pointwise convergence since the sequence defined there also converges to 0 in L~ on [0, 1). I . s and In,tegro,l,s A problerrl that we will encounter frequently in this book is the following. Slippose that a sequence of functions { f,,( x ) ) ,on~ an ~ interval ~ I converges in some serise described in one of the previous four subsections t o a fiirlction f (z)on I. Under wliat coiiditiorls is it true t,hat Silicc we call write f ' ( z ) = lim,,,, f , , ( .

Top Left: Graph of function f (x). Top Right: Graph of f (x) and f ( x - t ) where t = 1/16. Bottom: Graph of If (4 - f ( x - t)l. Area of shaded region is small for small t. 36. 37. ( a ) Let f (x) be L' o n R, and let R such that E > 0. T h e n there i s a function g(x), C: o n llf ( b ) Let f (x) be L~ o n R, and let R such that E - 9111 < E . > 0. T h e n there i s a function g(x), C: o n Ilf - 9/12 < t. 5, there is a compactly supported function h(x), L1 on R, such that 11 f - hill < €12. 32(b)).