Solid State Physics

Download An Introduction to Wavelet Analysis by David F. Walnut PDF

By David F. Walnut

This e-book offers a entire presentation of the conceptual foundation of wavelet research, together with the development and research of wavelet bases. It motivates the critical principles of wavelet idea via supplying a close exposition of the Haar sequence, then indicates 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 building.

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Piecewise dzfferentiable o n R. 8) converges t o f ( z ) i n L" o n R. What if the function f (x) is continuous but not piecewise differentiable? What can be said about the convergence of the Fourier series of such a function? It is by no means obvious that such functions exist, but they do. The most famous example is due to Weierstrass, who constructed a function continuous on R but not differentiable at any point of R. This function is defined by f (x) = CrLCN 3-n ~ o s ( 3 ~ xThe ) . 3 By the Weierstrass M-test, the Fourier series of the Weierstrass function converges uniformly on R.

1/72?. However, 011 average, f, (z) is close t o the lirnit function. 25(c) tells a different story. The sequence converges pointwise but not uniformly to f (z) Y 0 on [O. 11, but since the area under the graph of f,, (x) is always 1, f,(z) does not converge to f ( z ) in mean. The width of the triangle under the graph of f (z) decreases t o zero, but the height increases to infinity in such a way that the area of the triangle does not go t o zero. The above examples show that sometimes pointwise convergence and mean convergence go together and sometimes they do not.

The idea behind Fejbr's Thcorcrn is tlie following. 8), coilsider the arithmetic m e a n s of those part)ial suins; that is. coi~sicterthe sequence It is oftell the case that when the corivergence of a sequence fails due t o oscillatioli ill the terms of the sequence, the arithmetic rrleans of the sequence will have better convergerlee behavior. 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.

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