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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 $|x| \to \infty$?

Thanks in advance.

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2 Answers 2

This is a subject of a very nice paper by Charlie Epstein (2004)

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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.

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