# Understanding Fourier Transform and FFT

First off, I'm sorry if this is a repost. I am currently writing my thesis, and I've been thrown into some Fourier analysis, which I know nothing of. So, even if this question has been posted before, I wouldn't know where to look.

The situation is this: I am given a function $f$ (lets say its defined on $\mathbb{R}$), and its Fourier transform $\mathcal{F}[f]$, and I want to know the value of $f$ in some points $x_1,\ldots,x_N$, which are equally spaced (e.g. by $\Delta x$). I need to do this on a computer, in MATLAB. However, it would be way faster to use ifft on $\mathcal{F}[f](\xi_1),\ldots,\mathcal{F}[f](\xi_N)$ for some $\xi_1,\ldots,\xi_N$, instead of calculating $f(x_1),\ldots,f(x_N)$ directly. The reason for this is that $f$ contains some horrible infinite integrals, which, in my experience, MATLAB does not handle very well (and by well I mean fast, among other things).

So my question is this: how do I know which values $\xi_i$ to use?

As I said, I'm completely new in this, and therefore my knowledge about it is basically non-existing.

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So you have a formula for $f$, which you could use directly to compute $f(x_1),\ldots, f(x_N)$, but these computations would be expensive. And you have a formula for $\mathcal F[f]$, which is considerably simpler/cheaper than your formula for $f$? Is that the situation? –  littleO Nov 22 '12 at 11:03
That is exactly the situation, yes! –  cimrg.joe Nov 22 '12 at 11:09
Somebody who's better at signal processing can give you a better answer. But if I understand correctly, this situation has been studied carefully, and the correct procedure is specified in the Nyquist sampling theorem. Check out the wikipedia article, in particular the sections on the sampling process and reconstruction. Grad students in electrical engineering, focusing on signals and systems, should be good at this. –  littleO Nov 22 '12 at 11:34
Thanks - I will take a look! –  cimrg.joe Nov 22 '12 at 11:58