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MathGroup Archive 2007

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Re: Re: Fourier and InverseFourier

  • To: mathgroup at smc.vnet.net
  • Subject: [mg75490] Re: [mg75482] Re: Fourier and InverseFourier
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Thu, 3 May 2007 03:40:48 -0400 (EDT)
  • References: <f0v61b$8u4$1@smc.vnet.net> <f11gpl$kph$1@smc.vnet.net> <200705020756.DAA05199@smc.vnet.net>

rob wrote:
> 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?


Your explicit convolution integrates from 0 to t. Your attempt with 
FT/IFT involves integrations from -infinity to infinity. In order to use 
  FT/IFT you'd need to have cutoff multipliers such as UnitStep, to get 
results comparable to the explicit code. Also for functions like Exp[-t] 
(with no cutoff) the FT does not exist because it grows too fast at 
-infinity.


Daniel Lichtblau
Wolfram Research



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