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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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    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](http://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem#Reconstruction), 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.2012-11-22

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