# Talk:Fourier series

## Fourier's formulae for T-periodic functions

Unless I'm missing something, I'm a little confused as to why the formulae for ${\displaystyle a_{n}}$ and ${\displaystyle b_{n}}$ 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. ${\displaystyle a_{n}={\frac {2}{T}}\int _{\tfrac {-T}{2}}^{\tfrac {T}{2}}f(x)\cos({\tfrac {2{\pi }nx}{T}})\,dx}$  or  ${\displaystyle a_{n}={\frac {2}{T}}\int _{0}^{T}f(x)\cos({\tfrac {2{\pi }nx}{T}})\,dx,}$  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:
${\displaystyle a_{n}={\frac {2}{T}}\int _{\alpha }^{\alpha +T}f(x)\cos({\tfrac {2{\pi }nx}{T}})\,dx,}$
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 ${\displaystyle a_{n}}$ and ${\displaystyle b_{n}}$ for 2π-periodic functions with real-valued coefficients to defining ${\displaystyle a_{n}}$ and ${\displaystyle b_{n}}$ 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 ${\displaystyle a_{n}}$ and ${\displaystyle b_{n}}$ 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)

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 ${\displaystyle b_{n}}$ of Example 1. It should be:

{\displaystyle {\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 ${\displaystyle \pi }$ 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 (talkcontribs) 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)

## 1/2 [f(x0+)+f(x0-)]

If f were continuous at x0, f(x0)=f(x0+)=f(x0-). At a jump however, there is no prior relation between f(x0) and f(x0+-), but it is fairly common for the value of f at the jump x0 to be precisely at the midpoint of the jump. That is f(x0)= 1/2 [f(x0+)+f(x0-)].

[1] Dalba 08:38, 2 February 2014 (UTC)

## Why are we using a finite series in the definition?

The definition section uses a summation from n=1 to N, and then takes the limit as N approaches infinity. I don't see the point of introducing the variable N. If you're trying to learn fourier series, you presumably already understand the idea of infinite sums being a limit. — Preceding unsigned comment added by 174.3.243.185 (talk) 02:15, 10 February 2014 (UTC)

I've never seen it done that way either. But three of the figures reflect that approach. And it leads nicely into the discussion of convergence. The term "partial sum", seen in one of the figure captions, used to be in the prose as well. It got dropped (by me) when I did some work on the Definition section a while ago. It just wasn't a good fit anywhere. But perhaps that was a mistake.(?)
--Bob K (talk) 05:14, 10 February 2014 (UTC)

Maybe we can make the definition an infinite sum, and then include something akin to "the infinite sum can often be approximated with a finite sum to a high degree of accuracy, sometimes called a partial sum of the Fourier series." This can lead into the approximation and convergence section.198.73.178.11 (talk) 17:40, 10 February 2014 (UTC)

Replace "can often be approximated" with "can always be approximated". The ability to approximate with arbitrary accuracy by a (sufficiently large) finite partial sum is the definition of convergence. — Steven G. Johnson (talk) 18:33, 10 February 2014 (UTC)