By David F. Walnut

ISBN-10: 0817639624

ISBN-13: 9780817639624

This booklet presents a complete presentation of the conceptual foundation of wavelet research, together with the development and research of wavelet bases. It motivates the crucial rules of wavelet thought through providing an in depth exposition of the Haar sequence, then exhibits how a extra summary procedure permits readers to generalize and increase upon the Haar sequence. It then provides a couple of adaptations and extensions of Haar development.

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**Extra info for An Introduction to Wavelet Analysis**

**Example text**

34. If f,(x) -+ f (x) in Lm on n finite interval I , then f,(x) + f ( x ) in L' on I . 35. 34 is false if the interval I is infinite. Consider fur exarrlple the sequence f , (x) = ( l / n )X [ o , n (x). ~ Then f,(s) i 0 in L" on R but Jyx I f , (z) - 01 dx = 1 for all n, so that f,(z) does not converge to zero in L1. 33(b). In this example, f,,(x) convcrges to 0 in L' on [ O , 1 ] but does not converge to 0 in L" on [0, 11. (c) In all of the examples of mean convcrgcncc considered so far, the sequences have also converged pointwise.

29, If fn(x) + f (x) unzformly on the interval 1 , and if each f , , ( x ) is continzious on I , t h . sson I . 45. 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 ) .

Take the simple exariiple of the sequeiice { ( ~ ( n ) } ,where ~ ~ . a ( n ) = (- 1 ) " . Clearly lim,,, a ( n ) does not exist hecause the t,erms sirrlply oscillate back and forth between 1 and -1. Homrever. if mre coiisider the sequence of aritllrrletic means, { ~ ( T z ) } ~ ~ ~ . given 1 ) ~ . 25). If t,he original sequence { ~ ( n , ) } ,already , ~ ~ converges. taking the arithmetic means will not affect the convergence; that is, if lim,,,, a ( n ) = a. 26). (Ft:j&r'sTheorerri) Let f (x) be a f7~nctionwith period n > O con,tiri,vovs o n R.