Talk:Fourier series

Fourier's theorem

It has been said

“ Fourier’s theorem is not only one of the most beautiful results of modern analysis, but it is said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics ” — Lord Kelvin [1]

but this article does explicitly mention "Fourier’s theorem". Does it go by another name? Several dictionaries define it as the theorem that states that under suitable conditions any periodic function can be represented by a Fourier series. Arbitrarily0 ^(talk) 14:22, 11 March 2013 (UTC)

:"'The Fourier theorem' consists not of one single theorem, but in several theorems all on a common theme."

— D.C.Champeney, A Handbook of Fourier Theorems, 1989, p 2, ISBN 0521366887

--Bob K (talk) 21:00, 12 March 2013 (UTC)

Fourier series on a general interval [a, a + τ]

This section needs some attention. It introduces in vague terms (the words "square integrable" are mentioned) two functions g and h, and claims that

• g(x) and h(x) are equal everywhere, except possibly at discontinuities

This contrasts with the more careful treatment in the section "Fourier's formula for 2π-periodic functions". Bdmy (talk) 20:40, 11 August 2013 (UTC)

Rather than repeat a lot of the same verbiage, perhaps we should simply generalize the "2π-periodic functions" section. --Bob K (talk) 13:58, 13 August 2013 (UTC)

Fourier's formulae for T-periodic functions

Unless I'm missing something, I'm a little confused as to why the formulae for $a_{n}$ and $b_{n}$ are given only for the specific case of 2π-periodic functions. Wouldn't it be better to state the more general T-periodic cases (e.g. $a_{n}={\frac {2}{T}}\int _{\tfrac {-T}{2}}^{\tfrac {T}{2}}f(x)\cos({\tfrac {2{\pi }nx}{T}})\,dx$  or  $a_{n}={\frac {2}{T}}\int _{0}^{T}f(x)\cos({\tfrac {2{\pi }nx}{T}})\,dx,$  from which the T=2π case would follow almost immediately?
A Thousand Doors (talk | contribs) 23:35, 22 August 2013 (UTC)

Yes, of course (in my humble opinion). But why stop there? Why not:

$a_{n}={\frac {2}{T}}\int _{\alpha }^{\alpha +T}f(x)\cos({\tfrac {2{\pi }nx}{T}})\,dx,$

where α is any arbitrary number?

--Bob K (talk) 03:39, 23 August 2013 (UTC)

Yep, I'd agree with that, it's probably a better idea. Happy editing, A Thousand Doors (talk | contribs) 10:42, 23 August 2013 (UTC)

Our answer (I assume) is that Fourier didn't do it that way. The "compromise" I settled for long ago is section Fourier series on a general interval [a, a + τ).

--Bob K (talk) 13:02, 23 August 2013 (UTC)

It just seems to me that we're missing a step when we're going from defining $a_{n}$ and $b_{n}$ for 2π-periodic functions with real-valued coefficients to defining $a_{n}$ and $b_{n}$ for τ-periodic functions with complex-valued coefficients. The equations that we've listed above at the sorts of things that I would expect to see in this article, as that was how I was always taught about Fourier series. I think it would be best to include them in the article somewhere, even if it's just something like this. A Thousand Doors (talk | contribs) 11:08, 27 August 2013 (UTC)

My preference is to keep that section as it was, even if we have to add a similar section for real-valued coefficients. However, there is no rule that says this article must preserve the chronology of historical events. I.e. we are free to begin with the general interval approach for both real and complex coefficients. Then simply point out that the special case τ=2π, and a=-π was the historical starting point for Fourier.   Simple, clean, and effective.

--Bob K (talk) 12:18, 27 August 2013 (UTC)

Sounds like a good idea to me, I'd be happy with that. I just think that it's important to list those definitions for $a_{n}$ and $b_{n}$ somewhere in the article. Beginning with the general interval approach and then describing Fourier's special case would be my preference too. A Thousand Doors (talk | contribs) 12:27, 27 August 2013 (UTC)

Quite like the new structure of the article, nice work. I think it's a smart idea to introduce the concepts of Fourier series and Fourier coefficients as early as possible. A Thousand Doors (talk | contribs) 22:16, 30 August 2013 (UTC)

translation of paper's title?

Under divergence we have "Une série de Fourier-Lebesgue divergente presque partout". Should we include the translation of this title into English? RJFJR (talk) 19:42, 10 September 2013 (UTC)

layperson readability

First we must ask who is the audience. In this case it is the general public in my opinion, or the layperson.

This page is a long way from a layperson finding a Fourier series coefficient of y(t). It isn't above the ability of someone who passed high school to do. It is too obscure though.

The subscripts and symbols are hurdles for laypeople in my opinion. People could be referred to half a dozen other pages to learn the symbols. I suspect most would give up.

Somewhere along the line in mathematics, someone's shorthand became standard, and mathematics became another language. Bajatmerc (talk) 19:58, 17 September 2013 (UTC)

It is not the purview of every Wikipedia article that relies on mathematics to re-teach standard concepts and notation to the general public. Yes, mathematics has its own language... no way around that.

FWIW, the missing concept, IMO, is that in my 45 years of experience with "Fourier analysis", I have never knowingly "found" a Fourier series coefficient, and I don't know anyone who has. What we actually do is analyze "data" with tools such as DFTs. And our ability to interpret those DFTs depends on our understanding of how they are related to the underlying continuous transforms and inverse transforms.

--Bob K (talk) 23:26, 17 September 2013 (UTC)

The coefficient in "2.1 Example 1: a simple Fourier series"

I think there is an error or typo in the Fourier coefficient $b_{n}$ of Example 1. It should be:

${\begin{aligned}b_{n}&{}={\frac {1}{\pi }}\int _{-\pi }^{\pi }s(x)\sin(nx)\,dx\\&=-{\frac {2}{n}}\cos(n\pi )+{\frac {2}{\pi n^{2}}}\sin(n\pi )\\&={\frac {2\,(-1)^{n+1}}{n}},\quad n\geq 1.\end{aligned}}$

That is, there is an extra $\pi$ in the denominator of the current article.

Here is one reference: http://watkins.cs.queensu.ca/~jstewart/861/sampling.pdf

Can someone confirm that this is an error? — Preceding unsigned comment added by Wangguansong (talk • contribs) 16:42, 6 November 2013 (UTC)

Perhaps you forgot that s(x) = x/π (not just x). Otherwise see this link.

--Bob K (talk) 22:44, 6 November 2013 (UTC)

Thank you for clearing that for me! I missed that pi. Wangguansong (talk) 14:35, 7 November 2013 (UTC)