Fourier Transforms

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I 1 k limX→X−Xe ikx dx , I 2 k lim n→n k−nke ikx dx ,where X and n are a real variable and positive integer respectively. If Ik exists in theconventional sense, then Ik I 1 k I 2 k.For k ≠ 0, we obtainI 1 k limX→sinkXk, I 2 k lim n→sinnn .The function I 1 k does not exist for a fixed value of k, since its value varies continuouslywith X in an oscillatory manner and within the range −1/k,1/k. However, I 2 k does existand furthermore I 2 k 0. This is because the limit is attained via a sequence in whichevery value is zero.Thus the interpretation of Ik via I 2 k is consistent with the sequence approach for thedelta function, provided that k ≠ 0. If k 0, then integrals in (3.4.1) and (3.4.2) are bothinfinite and this matches our intuitive interpretation of the delta function. This is not aformal justification of the FT of x, nor is intended to be, but simply shows the type ofapproach necessary to employ generalised functions.3.4.2 ApplicationsConsider the equation of simple harmonic motiond 2 ydx 2 2 y 0,for which the solution is known to be given by yx Ccosx Ssinx, for realconstants C and S. An equivalent form is yx Aexpix Bexp−ix, for complexconstants A and B. Both forms of the solutions are bounded but neither is absolutelyintegrable and so standard FT theory may not seem to apply. However, this is a commonODE and often provides the LHS of a more complicated equation. It also arises quitecommonly in PDE systems after the FT has been taken and is associated especially withwavelike systems.Nevertheless, take the FT of the equation and assume that this can be justified,− k 2 − 2 yk 0 k k − yk 0.This tells us that yk 0 when k ≠ but what happens when k ?There is a relevant result in the theory of generalised functions: if gk is a generalised22
function and kgk 0, then gk is a constant multiple of k. Hence gk Qk, forsome constant Q. This means that yk is of the form yk 2A k 2B k − for constants A and B and where the 2 factor has been included for convenience.Inverting this givesyx 12 2 A k B k − e −ikx dk− Ae ix Be −ix ,in agreement with the earlier form.This illustrates clearly why generalised functions are required for the deployment of theFourier theory. The fundamental solutions of the equations cannot be obtained withoutthem.3.4.3 Fourier Transforms and Fourier SeriesThe complex FS for a real or complex-valued function fx, defined on −,, is given in§1.7 to befx ~ ∑n−c n e inx , where c n 12 fxe −inx dx−and generally the coefficients c n will be complex. We assume that the series converges.Suppose we wish to obtain the FT of this FS, how do we proceed? The function fxdescribes a function that is 2-periodic and if we attempt to integrate over the range−, we meet the same problems as encountered previously. However, if we deal withintervals that are multiples of 2, as with the quantity I 2 k above, then the limiting processand the integral are well-defined. Furthermore, the sequence is composed of goodfunctions.Assuming that such issues can be settled, take the FT of fx in the usual way23
Page 2 and 3: −xfx dx f0 (3.1.1)or equivalentl
Page 4 and 5: fx Hx a 1 − Hx − aSignum (Sig
Page 6 and 7: f̂k a−a1 e ikx dxa 1ike ikx −a
Page 9 and 10: fx fx −2 12−f̂k e −ikx dk
Page 11 and 12: −|e −a|x| | 2 dx 2 0e −2ax d
Page 13 and 14: −−d 2 ydx 2d 2 y− 2 y e ikx
Page 15 and 16: uk,y Ak e|k|y Bk e −|k|y .Now ap
Page 17 and 18: ecause of the convolution structure
Page 19 and 20: complex FS of f on −L ≤ ≤
Page 21: 3.4 Fourier Transforms and Generali

Fourier Transforms

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