Fourier Transforms

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