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MathGroup Archive 2007

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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?
> 


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