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Re: Fourier and InverseFourier

Daniel Lichtblau wrote:
> 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:
>>>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.
>>>  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

Sir, thank you for the suggestion. I added the UnitStep[t] 
as below:

fs = FourierTransform[UnitStep[t]*Sin[t], t, w]
fe = FourierTransform[UnitStep[t]*Exp[-t], t, w]

x3[t_] := InverseFourierTransform[fs*fe, w, t]

Plot[x3[t], {t, 0, 15}, PlotRange -> All]

The x3 now computes and plots but gives a narrowband sine 
output, different from the original convolution which is a 
wide band pulse (looks like a Gaussian pulse with an 
undershoot). So at least I'm getting an answer. Any 
suggestion which might yield the same result as the original 
convolution -- which I repeat below for clarity?

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]

Thanks again, Rob

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