# Numerical calculation of Fourier transform with a nice error bound

I'd like to have an algorithm for a numerical calculation of Fourier transform with a nice error bound. To be precise, if $f$ is a function from $L_1(R)$, $F[f]$ is it's exact Fourier transform and $F_a[f]$ is the "numerical" Fourier transform, I'd like to have the inequality $$\int_{R} Err(x) dx < \infty,$$ where $Err(x):=|F[f](x) - F_a[f](x)|$, or at least $$Err(x) < G(x),$$ where $G(x) \to 0$ as $|x| \to \infty$. Do such algorithms exist (at least for functions $f$ from some narrow classes)? Or maybe someone may give concrete examples of algorithms and functions $f$ for which the error $Err(x)$ tends to zero as $|x| \to \infty$?

If a numerical algorithm is only allowed to evaluate $f$ at a finite set of points, you're going to have a problem: $f$ might happen to be $0$ at all the points where you evaluate it. Clearly no error bound is possible under such circumstances.