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Gaussian Fourier transform approximations?

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  • Subject: [mg120901] Gaussian Fourier transform approximations?
  • From: AES <siegman at>
  • Date: Sun, 14 Aug 2011 20:19:23 -0400 (EDT)
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[This is really a numerical analysis query, but since the results will 
be evaluated using Mathematica, perhaps I can post it here.]

I want to develop a closed-form approximation for the inverse 
transform  f[x]  of a gaussian spectrum  g[s] = Exp[ - s^2 / w^2 ]  
multiplied by a transfer function  h[s]  which is a reasonably smooth 
function about  s=0,  and use it to make reasonably accurate numerical 
plots of  f[x] .  (Think of  f[x] as a spatial profile, g[s] as its                  
spatial frequency transform.)

That is, I want a closed form approximation to

   f[x]  =  Integrate[ h[s] Exp[ - (s/w)^2 - I s x ],  
                                  {x, -Infinity, Infinity}]

where s, x and w are real; and f and h potentially complex-valued.

Evidently I can approximate the transfer function by expanding it as
h[s] ?  h0 Exp[ h1 s + h2 s^2] , in which case the integral above 
becomes closed form, giving me just a single complex gaussian with a 
somewhat complicated argument.

Or I can approximate  h[s]  by  h[s]  ?  h0 (1 + h11 s + h22 s^2 )  in 
which case I get three closed-form integrals involving a gaussian 
times Hermite polynomials of order n=0 up to n=2.

Either of these should be adequately fast for numerical calculations, 
but I somehow think the second should be more accurate (and maybe also 
give more physical insight).


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