       Re: Gaussian Fourier transform approximations?

• To: mathgroup at smc.vnet.net
• Subject: [mg120920] Re: Gaussian Fourier transform approximations?
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Tue, 16 Aug 2011 01:26:24 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201108150019.UAA07476@smc.vnet.net>

```On 08/14/2011 07:19 PM, AES wrote:
> [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}]

[I think you want s as the variable of integration.]

> 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.

This will be problematic if h2 is allowed to be positive.

> 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).
>

What type of approximation works well will really will depend on your
transfer function h. If a low order polynomial approximation is
reasonable, at least in the realm before the Gaussian decay swamps the
growth, then your second approach would seem to make sense.

It can be redone analytically as a sum of Gaussians convolved with
derivatives of Dirac delta functionals. That form might be useful from a
numerical evaluation point of view. If not, I'd go with NIntegrate.

Daniel Lichtblau
Wolfram Research

```

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