Talk:Convergence of Fourier series

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Mathematics rating:	B Class	Mid Priority	Field: Analysis
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. More explanation needed, so as not to rely so much on links. Geometry guy 20:49, 9 June 2007 (UTC)

I think that convergence of Fourier series is not useless for engineers. In fact Fourier series and Fourier transform is used in signals compression (for example JPEG or MP3) in this contexts it's very important to know in which sense the information is reconstructed. Musical signals, for example, can be seen as functions with finite energy (L^1 and L^2)(is a function of the time which has a compact support and is in L^00) and a compact spectrum so it’s possible to represent their information (which has the power of the continuum) with discrete series. Moreover we can cut a part of this series and a part of the spectrum of the signal to obtain an MP3. (Using spectral analysis techniques). Gibbs phenomenon can be seen when we compress pictures made of color spots with sharp outlines (the ending titles of a DiVX) so to compress animated cartoons is better not to use a method related with Fourier series (for example we can use GIF).

... a function of the time which has a compact support ... and a compact spectrum is a contradiction. Any nonzero function which is compactly supported has necessarely unbounded spectrum and any function with compact spectrum has necessarely unbounded support. This is a very elementary version of the uncertainty principle in harmonic analysis. The Fourier transform for jpegs and mp3s can be viewed strictly in the discrete context, in which case the convergence is moot (since the Fourier series is a finite sum.) The remaining comments do not pertain to the convergence of Fourier series. Loisel 14:06, 24 Mar 2005 (UTC)

[edit] Mistake in formula?

I'm not confident enough to make this change myself. Someone who knows more please do this:

Under "Summability" the first summation says "summation of A sub n" but I believe that should be "A sub k."

[edit] Uniform convergence

Should we say something about when Fourier series converge uniformly? (I don't know the details off the top of my head, else I would do it.) -- Walt Pohl (talk) 02:35, 4 January 2008 (UTC)