FFT, Mathematica, Continuous Fourier Transform

**Anthony** · May14-09, 04:45 PM

Hi all,

First a warning: my Mathematica skills, and computery-type skills in general, are not very hot. My problem is thus: I have a function which I know:

$LaTeX Code: \\hat{f}(k)$

I'd like mathematica to approximate the inverse fourier transform of this function for me and plot the result. I've tried using the built-in function "NInverseFourierTransform", but it fails to produce meaningful results. My function oscillates quite rapidly, so NIntegrate doesn't work too well.

Now I'm aware that I could approximate the inverse Fourier transform using a discrete Fourier transform and the FFT algorithm - but I'm afraid I don't really know how to go about doing it. I can do the following:

Get as many sample points of $LaTeX Code: \\hat{f}(k)$ as you want. Call them $LaTeX Code: \\{\\hat{f}_n\\}$ .
I can make $LaTeX Code: \\hat{f}$ rapidly decreasing, so it's pretty much got compact support.
My $LaTeX Code: \\hat{f}$ is smooth.

I figure if I've got the above properties, there must be some way of approximating the inverse Fourier transform using the built in FFT functions in mathematica. I've tried using InverseFourier[list], where list contains the $LaTeX Code: \\hat{f}_n$ , and plotting the real part of it, but the answer is gibberish. I've proved lots of rigorous results regarding the function

, so I know (pretty much) what the plot of the inverse Fourier transform should look like!

If anyone could help me implement the built in mathematica functions to get a plot of this inverse Fourier transform, I'd be immensely grateful.

Thanks,
Ant

**Anthony** · May17-09, 03:16 AM

All sorted now - I rolled up my sleeves and got stuck into mathematica.