# [SciPy-user] Fourier series

David Cournapeau david@ar.media.kyoto-u.ac...
Wed Oct 22 06:59:57 CDT 2008

```Joris De Ridder wrote:
>
> I guess you mean numerical, not analytical? As far as I know it's only
> possible if you can set an upper limit to its bandwidth.

Concretely, by having only some samples of your functions, there is an
implied bandwidth for your signal (otherwise, the sampling process would
have destroyed your signal through aliasing). Then, FFT coefficients can
approach the Fourier coefficients by refining the frequency "sampling"
(considering the FFT as a frequency sampled of the Fourier coefficients).

Also, numerically speaking, you can consider that any real signal has
finite bandwidth, or more exactly the coefficients are negligeable for
high frequency: the fourier coefficients are decreasing toward 0 for
smooth functions.  More precisely, the Fourier transform TF(n) "behaves
as" 1/n^k where k is bigger when the function is more regular: the sum
of (1+n^-k)^2 * F(n) is finite for the function k-times differentiable.
Incidentally, one mathematical object for the study of smooth functions
is Sobolev space, which definition is based on this property.

cheers,

David
```