By David F. Walnut
This ebook offers a complete presentation of the conceptual foundation of wavelet research, together with the development and research of wavelet bases. It motivates the crucial principles of wavelet conception through supplying a close exposition of the Haar sequence, then indicates how a extra summary method permits readers to generalize and enhance upon the Haar sequence. It then offers a couple of adaptations and extensions of Haar building.
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Extra resources for An Introduction to Wavelet Analysis
Note that if K T ( x ) = (117)g ( x / r ) for some function g ( x ) , L1 on R and compactly supported, then the assumption that f (x) is L" on R is unnecessary. 42). Now let us consider uniform convergence. 34. T,et f(z) he L" and unzformly continuous on R . Suppose is an approximate identity o n R. I If (z) - f (X - t)l lKT(t)l df < E. 31(b). 33, for all r > 0. 2. Approximate Identities which converges to 0 as r that if 0 < r < T O , then 45 + 0+ for any 6 > 0. 21) follows. The condition that f (z)is uniformly continuous on R is satisfied if for example f (z) is C: on R.
Functions and Convergence 22 The following theorem gives several conditions under which interchanging the limit and the integral is permitted. 40. l;,(:c) + f (x) in L' 011 1 f (x)dr. I . 34, if f,,(z) -+f (z) in L" on I, then it also converges in L1. Then the result follows frorri part (a). 38(b), if f,,(z) + f ( z ) in L~ 011 I, then it also converges in L1. The11 the result follows from part (a). 39(c). In this example, f,,(z) + 0 both in LOC and L2 011 10, m). However, since JI f,, (x) d z = 1 for all n, However, in the case of infinite intervals.
20 Chapter 1. Furictions and Convergence T h e series Cr=,f,(x) = f ( x ) i n mean-square o n I zf the sequence of partial N f n ( 5 ) converges in mean-square to f ( x ) o n I . 48). 8). 38. (a) If fTL(:c) + f (x) i n La o n a finite interval I , then f n ( x ) + f ( x ) in L~ o n I. ( b ) If f,,(x) I. 39. 38(a) is false if Lm convergerice is replaced by poiritwise convergence. 33(c) shows a sequence that converges t o zero poirltwise on [O, 11 hiit not in L1 on [r), 11. 38(b) would imply that it also converged ill L1.
An Introduction to Wavelet Analysis by David F. Walnut
Categories: Solid State Physics