Talk:Convergence of Fourier series

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)

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."

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)

Dirichlet's proof?
Where does Dirichlet's line of proof fit into all this? The text of it is at http://arxiv.org/abs/0806.1294 --Ayacop (talk) 08:17, 22 June 2008 (UTC)


 * Dirichlet made an important historical contribution, as well as Jordan, Wilbraham, Riemann, Cantor, etc.. A more detailed history is available in the French version of the Fourier Series article, see http://fr.wikipedia.org/w/index.php?title=S%C3%A9rie_de_Fourier&oldid=30788202 and search for Dirichlet. Loisel (talk) 07:20, 23 June 2008 (UTC)

Periodicity of f?
The original definition of a trigonometric series does not require f to be periodic. This is reflected in the convergence criteria. For example, a function of bounded variation or a function with existing left and right derivatives does not need to be periodic. It would be good to add a paragraph that explains where this periodicity requirment comes from and when it is necessary. —Preceding unsigned comment added by 186.14.32.206 (talk) 15:13, 31 July 2009 (UTC)

Weak convergence
It might be nice to have a section on weak convergence of the Fourier series, i.e. in the distribution sense, and the fact that the series converges for every periodic distribution in this sense. (This is arguably the sense that is used in most practical contexts nowadays, as Fourier series usually come hand-in-hand with other distribution concepts such as delta functions. It also gives a concrete way to understand "almost everywhere" convergence, since finite errors on sets of measure zero obviously don't affect weak convergence.) — Steven G. Johnson (talk) 16:00, 23 April 2011 (UTC)

Deleted content from Fourier inversion theorem - suitable for here?
Fourier transforms of square-integrable functions

Plancherel theorem allows the Fourier transform to be extended to a unitary operator on the Hilbert space of all square-integrable functions, i.e., all functions satisfying


 * $$\int_{-\infty}^\infty\left|f(x)\right|^2\,dx<\infty.$$

Therefore it is invertible on L2.

In case f is a square-integrable periodic function on the interval $$[-\pi, \pi]$$, it has a Fourier series whose coefficients are


 * $$\widehat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^\pi f(x)\,e^{-inx}\,dx.$$

The Fourier inversion theorem might then say that


 * $$\sum_{n=-\infty}^{\infty} \widehat{f}(n)\,e^{inx}=f(x).$$

What kind of convergence is right? "Convergence in mean square" can be proved fairly easily:


 * $$\lim_{N\rightarrow\infty}\int_{-\pi}^\pi\left|f(x)-\sum_{n=-N}^{N} \widehat{f}(n)\,e^{inx}\right|^2\,dx=0.$$

What about convergence almost everywhere? That would say that if f is square-integrable, then for "almost every" value of x between 0 and 2π we have


 * $$f(x)=\lim_{N\rightarrow\infty}\sum_{n=-N}^{N} \widehat{f}(n)\,e^{inx}.$$

This was not proved until 1966 in (Carleson, 1966).

For strictly finitary discrete Fourier transforms, these delicate questions of convergence are avoided.


 * Lennart Carleson (1966). On the convergence and growth of partial sums of Fourier series. Acta Math. 116, 135–157.

Hi, I'm about to delete the above material from the Fourier inversion theorem article, because it's about Fourier series whereas the article is about Fourier transforms. If it belongs anywhere then it would be in this article, so I'm posting it here in case someone wants to integrate it into this article. By the way, feel free to join the discussion about the proposed changes to that article. Quietbritishjim (talk) 01:16, 31 December 2012 (UTC)

Uniform convergence condition wrong?
I do not have access to Jackson's book, so could someone else double-check the formula for the speed of uniform convergence? If f is a polynomial of degree p then $$f^{(p)}$$ has a modulus of continuity which is identically zero. But clearly the approximation error will not be zero for any finite N. The claimed formula is also wrong when N=1, since the RHS is zero, but maybe that can be fixed by stating it is valid for large N. 193.11.79.122 (talk) 13:38, 14 October 2014 (UTC)

Summability and Fejer
I think one should add in the section on summability that the uniform convergence by Fejer's theorem only holds for continuous, periodic functions. — Preceding unsigned comment added by 176.199.55.147 (talk) 14:54, 15 September 2015 (UTC)

Pretty sure uniform convergence bound is wrong
I looked up the convergence bound in https://en.wikipedia.org/wiki/Convergence_of_Fourier_series#Uniform_convergence which cites the book "Introduction To Classical Real Analysis". It's not stated explicitly in the book. Also, the claim can't be right. For instance, consider $$f(x) = 1000\cos(1000x)$$. The function satisfies the $$\alpha$$-Hölder condition for $$\alpha=1$$ but the inequality given is clearly false:
 * $$|f(x)-(S_Nf)(x)|\le K {\ln N\over N^\alpha}.$$

The value of $$K$$ is not stated anywhere. Whatever value $$K$$ supposed to take, change the coefficient 1000 above to 1,000,000 or 1,000,000,000 - it's bound to be wrong eventually. --Svennik (talk) 20:32, 29 June 2021 (UTC)

Discussion
The proposition that the family of absolutely converging Fourier series is a Banach Algebra is wrong, It is a vectorial space which is not complete. — Preceding unsigned comment added by Boutarfa Nafia (talk • contribs) 09:26, 12 April 2022 (UTC)


 * It is known as a Banach algebra with ordinary pointwise multiplication of functions, but its own norm which is not that of uniform convergence like for the space of continuous functions. Look in literature about the Wiener algebra, e.g. the article about it in wikipedia--UKe-CH (talk) 17:43, 22 April 2024 (UTC)

About the case of an absolutely continuous function
As Y. Katznelson shows in his textbook: when f is absolutely continuous, n times its n-th Fourier coefficient is not only bounded (as currently says this article with other words) but tends to 0 with n -> oo. Could someone change the article for this? UKe-CH (talk) 18:18, 22 April 2024 (UTC)