Talk:Fourier series

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One of the 500 most frequently viewed mathematics articles.

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)
You don't have to teach concepts, but you should, if possible, give a lay understanding in the lead section so that the lay person can get a general understanding, even if it's incomplete. Right now, the image does a better job than the lead. — trlkly 22:26, 13 May 2014 (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)

This is a rather strange quote. Under the assumptions for pointwise convergence, the value of the Fourier series at a jump is the midpoint of the one-sided limits. Independent of the value of the function at that point. This gives a condition for pointwise convergence towards the original function, but it is not a common occurrence. It is at least as popular to have the function value be one of the one-sided limits.--LutzL (talk) 10:52, 14 May 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)

Eye-catching GIF

A Fourier series enables any periodic motion, such as the square wave motion of the blue circle at the bottom, to be represented as the sum of many (often infinitely many) sinusoidally-oscillating terms with different frequencies. The arrows show the six largest terms in the Fourier series, which sum to the purple circle.

It's cool, but IMO it's sideways. Amplitude is customarily vertical. Partly for that reason, I think the only people who will actually comprehend it are those who have already internalized File:Fourier_Series.svg and are able to perform the mental (or laptop) rotation. For them, it is just a novelty. For others, it might serve the purpose of attracting attention, like a blinking light. But hopefully they won't get "stuck" there. My 2 cents.

--Bob K (talk) 13:48, 7 March 2014 (UTC)

It's not vertical because tall and narrow figures tend to make wikipedia page layout very difficult, compared to short and wide figures. (Or is there a trick I'm not aware of?)
If you think the figure is doing more harm than good, you are entitled to delete it! --Steve (talk) 16:53, 7 March 2014 (UTC)
Incidentally, even disregarding page layout issues, I'm not convinced that vertical is really better. Fourier series the concept can appear in many superficially different settings: Vertical position as a function of horizontal position (your favorite), horizontal position as a function of time (my animation), vertical position as a function of time (your preference for my animation), color as a function of position (image processing), and on and on. Presenting the same concept in more than one superficial setting has a pedagogical value: It helps readers construct a more mature and refined conceptual understanding.
(Yes, I acknowledge that this philosophy should not be taken to an extreme, where the settings are so bizarre that they are distracting and frustrating. But I think that in this particular situation, horizontal is OK.)
Also, if it stays at the top of the article, it may be the first thing that people ever see about Fourier series, so they would not necessarily have the same preconception that you have, i.e. "x(t) is weird and y(t) is normal". --Steve (talk) 02:04, 8 March 2014 (UTC)

I should like to differ on the description of the Fourier series. First this definition of a Fourier series is applicable only to scientists and engineers. Hence is not really mathematics but engineering. In mathematics a Fourier series is defined within any inner product space within a maximal orthonormal set within that inner product space. The engineering version set out in this article is valid only for the inner product space L2[a,b]. — Preceding unsigned comment added by 2601:703:0:851E:91E5:CA2D:218:296C (talk) 16:27, 25 February 2016 (UTC)

there is an asterisk (*). What does it mean?

Dear Author, please accept my many thanks for your helpful article, and permit me a question: in "Definition" above "we can also write the function in these equivalent forms:" at the end of formula, there is an asterisk (*). What does it mean? Regards, Georges Theodosiou, 88.188.110.51 (talk) 08:59, 16 June 2016 (UTC)

It is a common notation for the Complex_conjugate#Notation operation. --Bob K (talk) 12:56, 16 June 2016 (UTC)
Mr Author, please let me express you my sincere thanks for your helpful answer. Georges Theodosiou 88.188.110.51 (talk) 13:48, 16 June 2016 (UTC)