Re: Fourier and InverseFourier

*To*: mathgroup at smc.vnet.net*Subject*: [mg75421] Re: Fourier and InverseFourier*From*: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>*Date*: Sun, 29 Apr 2007 03:13:23 -0400 (EDT)*Organization*: Uni Leipzig*References*: <f0v61b$8u4$1@smc.vnet.net>*Reply-to*: kuska at informatik.uni-leipzig.de

Hi, and you are sure that FourierTransform[Exp[-t], t, w] is exist ? Because Integrate[Exp[-t]*Exp[I*w*t], {t, -Infinity, Infinity}]/Sqrt[2Pi] gives the correct error message that the integral does not converge and in general Fourier transforms are only defined for quadratic integrable functions and Exp[-t] is not quadratic integrable. Regards Jens rob wrote: > I kind person on this ng (Gulliet) recently contributed a > convolution scheme which works nicely to plot x2 below: > > conv[f1_, f2_] := Module[{u}, Evaluate[Integrate[f1[u] f2[# > - u], {u, 0, #}]] &] > > x2[t_] := convolve[Sin[t], Exp[-t]][t] > > Plot[x2[t], {t, 0, 15}, PlotRange -> All] > > Wondering if I could achieve the same thing in the freq. > domain, I tried what I thought should give the same result > in x3: > > fs = FourierTransform[Sin[t], t, w] > fe = FourierTransform[Exp[-t], t, w] > > x3[t_] := InverseFourierTransform[fs*fe, w, t] > > Plot[x3[t], {t, 0, 15}, PlotRange -> All] > > I find this does not work, getting this err message and Mathematica > (v.5.1) didn't stop in over 30 minutes. > > NIntegrate::ploss: Numerical integration stopping due to > loss of precision. Achieved neither the requested > PrecisionGoal nor AccuracyGoal; suspect one of the > following: highly oscillatory integrand or the true value of > the integral is 0. If your integrand is oscillatory on a > (semi-)infinite interval try using the option > Method->Oscillatory in NIntegrate. > > Since I'm using the internal integrals of > InverseFourierTransform I don't know how to try the > suggestion of Method->Oscillatory as the message suggests. > > I changed the Sin[t] to t and the process gave no err > messages and finished in just a few minutes. The plot had > axes but nothing on it. > > Can someone give me any hints as what might work? >