FFT, Mathematica, Continuous Fourier Transform

[Anthony]

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:

$$\hat{f}(k)$$

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 $$\hat{f}(k)$$ as you want. Call them $$\{\hat{f}_n\}$$.
• I can make $$\hat{f}$$ rapidly decreasing, so it's pretty much got compact support.
• My $$\hat{f}$$ 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

• , where list contains the $$\hat{f}_n$$, and plotting the real part of it, but the answer is gibberish. I've proved lots of rigorous results regarding the function $$f$$, 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]

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

 

[gazi habiba]

i don't understand how fft algorithm works.
but i have to solve the problem by Mathematica code which i have attached.
can anybody help me to solve this function.
it will be if any explain by a simple function.

thanks,
Happy