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Re: Fourier and InverseFourier

• To: mathgroup at smc.vnet.net
• Subject: [mg75482] Re: Fourier and InverseFourier
• From: rob <josh2499 at hotmail.com>
• Date: Wed, 2 May 2007 03:56:00 -0400 (EDT)
• Organization: Road Runner High Speed Online http://www.rr.com
• References: <f0v61b$8u4$1@smc.vnet.net> <f11gpl$kph$1@smc.vnet.net>

Hi, thanks for responding. No, I'm not sure it exists. I 
tried Exp[-t^2] and it doesn't work either. I haven't yet 
found a case where InverseFourierTransform[] works so I 
suspect I'm still doing something wrong.

Jens-Peer Kuska wrote:
> 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?
>>
> 
> 

• Follow-Ups:
  • Re: Re: Fourier and InverseFourier
    • From: Daniel Lichtblau <danl@wolfram.com>