An Introduction to Wavelet Analysis by David F. Walnut PDF

By David F. Walnut

ISBN-10: 0817639624

ISBN-13: 9780817639624

This ebook offers a entire presentation of the conceptual foundation of wavelet research, together with the development and research of wavelet bases. It motivates the valuable principles of wavelet thought by means of delivering an in depth exposition of the Haar sequence, then indicates how a extra summary method permits readers to generalize and increase upon the Haar sequence. It then provides a few adaptations and extensions of Haar building.

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38(b),if f,,(x) + f (z) in LtX or L~ on [-R, R], then it also converges in L1 on [-R, R]. That is, Hence, there is an N such that if n LR1 R fn Therefore, if n > N, then > N, then (x) - f ( x i I dz < d3. 24 Chapter 1. 7) follows. 41. 42. Suppose that f o r every R > 0, f,,(x) o n [-R, R]. 41, except that we choose R > 0 and N E N such that for all n 2 N. 9) follows. 43. 25. 44. 28. 45. 29. 46. 31. 47. 34. 48. 38. 2. 49. 33(b),then f,(x) 4 0 in L~ on [ O , l ] . 50. (a) A sequence of functions { f , ( ~ ) ) ,defined ~ ~ on an interval I is said to be unzformly Cauchy on I if for every E > 0, there is an N > 0 such that if n, m 2 N then 11 f n - f,lI, < E .

Since as n + oo:f,,(z)+=O in mean on [o. 1). 18 Chapter 1. 25(b). The sequence converges pointwise but not uniformly t o f ( z ) 0 on [0, 11. Since the area under the graph of f,(z) is 1/2n for each n. the sequence also converges in mean t o f (z) on [0, 11. In this example, we can see the character of mean convergence. If n is large, the function f,(z) is close to the limit function f (z) 0 (in fact identical to it) on most of the interval [ O , l ] , specifically on [1/n, 11, and far away from it on the rest of the interval [O.

49). However, f,,(z) does not converge to 0 in L" on [o: 11. 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.

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