I'm currently facing the following problem: I want to approximate the Fourier transform $F(\omega)$ of a (let's say, $L^2(\mathbb R)$) function $f(x)$ by calculating the discrete Fourier transform, using only a fixed number of sampling points. These points are allowed to lie arbitrarily far away from the origin. In addition, we don't care about computational costs, so the grid can be chosen completely free.
What is the optimal (with respect to minimizing the error) choice for the grid?
I guess that this choice is dependent on the bandwidth $B$ of $f$. Does this mean that Nyquist-Shannon forces the grid to being broader than $2*B$?
Also, I found literature about sparse Fourier transform sampling in case of known sparsity in $F(\omega)$ (compressed sensing). But since my signal does not appear to have sparse support in the frequency domain, I don't see how these methods would be applicable in my case.