1
$\begingroup$

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.

Improve this question
$\endgroup$

2 Answers 2

Reset to default
1
$\begingroup$

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

Improve this answer
$\endgroup$
0
$\begingroup$

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.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.