Talk:Fourier series/Archive 2

Archive 1

Archive 2

Archive 3

Possible Error

In section 5.3, the one on convergence, the following setence appears:

"This is convergence in the norm of the space L2, which means that the series converges almost everywhere to f."

This doesn't seem correct.

— Preceding unsigned comment added by 87.69.5.52 (talk) 15:15, 28 November 2006 (UTC)

Format of equations

I find the equations difficult to read in their current format. Having a sentence, then a TeX png, then another sentence, etc, makes it hard for me to see what parts are associated with which equation, and where those parts end. Not only that, it makes the section unnecessarily long.

The format that I find very useful is this:

$equation\$

where

• some variable is this,
• some other variable is that, and
• some variable = this other stuff is this other thing.

Some people might find that ugly or whatever, but it makes it very easy to see what is associated with a single equation. Let me give my example in the format currently employed on this page:

The format that I don't find very useful is this:

$equation\$

where

some variable

is this,

some other variable

is that, and

some variable = this other stuff

is this other thing.

I find it especially confusing on this page, where more than one *separate* equation is written in one long string. I'm not saying we have to use my perferred format, but I do think that the format needs to be different to make it easier to read. Comments? Fresheneesz 17:41, 24 May 2006 (UTC)

I like more the formal format where everything is indented once. I don't think that the "staircasing" of formulas improves things that much, and is also nonstandard. I'd say we should be conservative and not invent new paradigms here. Oleg Alexandrov (talk) 01:04, 25 May 2006 (UTC)

The bulleting of variables in an equation is not a new paradigm, and I think it applies very nicely here. I started the staircasing of formulas, and so I won't push that if you don't like it. However, I disagree with keeping it the way it is now. If my format is "staircasing" then the current format is "laddering" - the variables, formulas, equations, and explanation, being strung out in a long, hard to read list.

I wonder though, why you removed the bullets. I kept the same indenting, but I feel that the bullets help a reader more clearly see what goes with what. I can't accept the current format, but I would like to hear whatever ideas you have to make those equations more readible.

Do you think its not useful for a reader to be able to understand the equations in a section without reading that whole section? Fresheneesz 02:51, 25 May 2006 (UTC)

I believe the equations are readable enough with just one indent, that's what used in math papers everywhere and people don't complain. :) However, if you do really positively want a star in front of a few equation for emphasis, well, it would not make sense for me to oppose that. Just not much staircasing. :) Oleg Alexandrov (talk) 15:12, 25 May 2006 (UTC)

Alright thanks! (although.. where would people complain about the format of math papers..?) Fresheneesz 04:18, 26 May 2006 (UTC)

I think the last equation (in the "Modern derivations of Fourier series") is wrong; if f is a Riemann-integrable function then the Lebesgue integral (left side) equals the Riemann integral (right side), there's no need to multiply by a constant (2pi in this case).

— Preceding unsigned comment added by 85.220.116.138 (talk) 22:08, 24 November 2006 (UTC)

relationship between real and complex forms

I was thinking about setting the real and complex forms equal, rather than writing them separately. I would guess people would think this way is "crowded" or "ugly". But I also have some related things I'd like to put up:

Some relationships between the variables in fourier series:

$c_{n}={\frac {a_{n}-jb_{n}}{2}}$

$c_{0}=a_{0}/2$

$a_{n}=c_{n}+c_{-n}=2Re[c_{n}]$

$b_{n}=c_{-n}-c_{n}=2Im[c_{n}]$

and for real functions:

$c_{-n}=c_{n}^{*}$

Fresheneesz 21:20, 26 May 2006 (UTC)

Still very convoluted

I'm not new to maths, and I find this article difficult to understand. Possibly it needs to be re-written in a more coherent fashion? Starting with a global definition, $f(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }[a_{n}cos({\frac {n\pi x}{L}})+b_{n}sin({\frac {n\pi x}{L}})]$
where
$a_{n}={\frac {1}{L}}\int _{-L}^{L}f(x)cos({\frac {2n\pi x}{L}})dx$
$b_{n}={\frac {1}{L}}\int _{-L}^{L}f(x)sin({\frac {2n\pi x}{L}})dx$

Should the function be odd (link to odd function definition) the fourier cosine series may be used. This simplifies to
$a_{n}={\frac {2}{L}}\int _{L_{1}}^{L_{2}}f(x)cos({\frac {2n\pi x}{L}})dx$
$b_{n}=0$

Should the function be even (link to even function definition) the fourier sine series may be used. This simplifies to
$a_{n}=0$
$b_{n}={\frac {2}{L}}\int _{L_{1}}^{L_{2}}f(x)sin({\frac {2n\pi x}{L}})dx$

Then explain the wave (string) equation, complex, real, properties, then historical.

If I have made mathematical mistakes, please forgive me, because I have apparently conflicting sources. :) I am only beginning fourier analysis, so I may have missed formulae which are important to more advanced parts.

Edit: oops, everything *is* in there - I must have skimmed past it these last 10 times I viewed the page. However, I still find the omega, t and T notation confusing - wouldn't f(x) be easier? Chrislewis.au

f(t) is used because in many (or most) cases, fourier series are implimented as functions of time. But you're right that f(x) might imply a more general use - however I think its a very minor thing, and might confuse people used to seeing f(t). Also, the use of t and T as period and time don't have an easily understood analog in other sets - like distance rather than time. So i really don't think that should be changed. I did make a note about what t1 and t2 are explicitely, rather than just compared to the period T. Fresheneesz 19:08, 2 July 2006 (UTC)

Chrislewis, please notice the difference between these:

$f(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }[a_{n}cos({\frac {n\pi x}{L}})+b_{n}sin({\frac {n\pi x}{L}})]$

$f(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }\left[a_{n}\cos \left({\frac {n\pi x}{L}}\right)+b_{n}\sin \left({\frac {n\pi x}{L}}\right)\right]$

• The square brackets and round parentheses are bigger in the second one.

• "sin" and "cos" are not italicized in the second one. The preceeding backslash not only prevents the letters from being italicized as if they were variables, but also provides proper spacing before and after "sin" and "cos" in some circumstances.

Michael Hardy 00:05, 3 July 2006 (UTC)

My apologies - I didnt notice the differences first time. (Chris 07:44, 13 August 2006 (UTC))

I think the original poster's idea is good because now it's really unclear why you can integrate on half of the function if it's odd or even and multiply this by 2. As it is now, there are no explanations and it creates confusion even with the editors. I'll try to come up with something. (LovaAndriamanjay 03:00, 28 November 2006 (UTC))

Error

if we say that $f(t)={\frac {1}{2}}a_{0}+\sum _{n=1}^{\infty }[a_{n}\cos(nt)+b_{n}\sin(nt)]$, then defining $a_{0}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(t)\,dt$ is incorrect, as we're trying to double correct for $a_{0}$ being twice as big.

It should be $a_{0}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(t)\,dt$. I reference Strauss Partial Differential Equations. 18.244.5.107 22:53, 13 May 2007 (UTC)

Thanks, you are right. Oleg Alexandrov (talk) 01:05, 14 May 2007 (UTC)

Trigonometric series redirect?

Not every Trigonometric series is a Fourier series. A trigonometric series has the form:

${\frac {1}{2}}A_{o}+\displaystyle \sum _{n=1}^{\infty }(A_{n}cosnx+B_{n}sinnx)$

It's called a Fourier series when the An and Bn terms have the form:

$A_{n}={\frac {1}{\pi }}\displaystyle \int _{0}^{2\pi }f(x)cosnxdx$ where $(n=0,1,2,...)$

$B_{n}={\frac {1}{\pi }}\displaystyle \int _{0}^{2\pi }f(x)sinnxdx$ where $(n=1,2,3,...)$

So, there are a whole class of trig series that are not Fourier series. Shall I change it? futurebird 13:29, 1 December 2007 (UTC)

I can't find where it says that "every Trigonometric series is a Fourier series".

--Bob K 15:12, 1 December 2007 (UTC)

to me the redirect implied that this was true, I've fixed it and made a new page.futurebird 15:29, 1 December 2007 (UTC)

"real Fourier coefficients"

In section: Fourier_series#Real_Fourier_coefficients it is misleading (at best) and incorrect (at worst) to refer to a_n and b_n as real Fourier coefficients, since they are not real-valued in general. They are the coefficients of real-valued functions.

--Bob K 23:17, 3 December 2007 (UTC)

Making a mountain out of a molehill is not a good example.

The article says:

One application of this Fourier series is to compute the value of the Riemann zeta function at s = 2; by Parseval's theorem, we have:

${\frac {1}{2\pi }}\int _{-\pi }^{\pi }x^{2}dx={\frac {1}{2}}\sum _{n>0}\left[2{\frac {(-1)^{n}}{n}}\right]^{2}$

which yields: $\sum _{n>0}{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}$.

Several points:

1. The answer is incorrect. (My guess is that the writer overlooked the factor of 2 in front of the Re operator in Parseval's formula.)
2. The answer is much more easily obtained by direct integration: ${\frac {1}{2\pi }}\left[{\frac {x^{3}}{3}}\right]_{-\pi }^{\pi }={\frac {\pi ^{2}}{3}}.\,$
3. I don't think this example is worth keeping, especially with all the missing steps.

--Bob K 11:54, 4 December 2007 (UTC)

Dear Bob,

You are incorrect. Zeta(2) is indeed pi^2/6, which you can see in the Riemann zeta function article. This specific case is also known as the Basel problem and I apparently drove the math help desk at Temple U nuts in 2006 by giving this problem as a bonus question to Cal II students.

Although there are other methods for computing Zeta(2), using Parseval's identity is one of the only methods I can follow from beginning to end without going cross-eyed.

Sincerely,

Loisel (talk) 19:57, 8 January 2008 (UTC)

inconsistent use of "L"

Fourier_series#The_wave_equation says:

The wave equation

The wave equation governs the motion of a vibrating string, which may be fastened down at its endpoints. The solution of this problem requires the trigonometric expansion of a general function f that vanishes at the endpoints of an interval x=0 and x=L. The Fourier series for such a function takes the form

$f(x)=\sum _{n=1}^{\infty }b_{n}\sin \left({\frac {n\pi }{L}}x\right)$

where

$b_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\sin \left({\frac {n\pi }{L}}x\right)\,dx.$

I believe the formulas should be:

$f(x)=\sum _{n=1}^{\infty }b_{n}\sin \left({\frac {n2\pi }{L}}x\right)$

where

$b_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\sin \left({\frac {n2\pi }{L}}x\right)\,dx.$

And I suspect the subsequent paragraph is also wrong. This might be the correct version:

Vibrations of air in a pipe that is open at one end and closed at the other are also described by the wave equation. Its solution requires expansion of a function that vanishes at x = 0 and whose derivative vanishes at x=L. The Fourier series for such a function takes the form

$f(x)=\sum _{n=1}^{\infty }b_{n}\sin \left({\frac {(2n+1)\pi }{L}}x\right)$

where

$b_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\sin \left({\frac {(2n+1)\pi }{L}}x\right)\,dx.$

—Preceding unsigned comment added by Bob K (talk • contribs) 12:16, 4 December 2007 (UTC)

Interpretation?

I don't think the section Fourier_series#Interpretation:_decomposing_a_movement_in_rotations is a helpful "interpretation" of Fourier series. Perhaps it is misplaced.

--Bob K (talk) 00:41, 8 January 2008 (UTC)

General case

Regarding:

$\int _{G}f(g){\overline {\chi (g)}}\,dg={\frac {1}{2\pi }}\int _{2\pi }^{}f(t)e^{-int}\,dt$

in Fourier_series#General_case, where does the ${\frac {1}{2\pi }}$ factor come from?

--Bob K (talk) 15:25, 8 January 2008 (UTC)

I don't really get what G = R/2πZ means. But evidently it's our clue that:

$dg={\frac {1}{2\pi }}dt$

Is it just me, or is that too much of a stretch for this article?

--Bob K (talk) 18:25, 8 January 2008 (UTC)

Dear Bob,

There is always a normalizing coefficient that appears either in the computation of the Fourier coefficients, or in the summation formula (the Fourier series itself.) The easiest way to understand where it comes from is to think in terms of Hilbert spaces. The functions exp(ikx) form an orthogonal basis for L^2([0,2pi]), but in order for these basis functions to have unit norm (so that they form an orthonormal basis), you have to do something. You either have to change the "measure" of [0,2pi] so that it measures 1 (in which case, you get 1/2pi, as in the text), or you have to put a coefficient in front of the exp(ikx), it then becomes 1/sqrt(2pi)exp(ikx), and the term 1/sqrt(2pi) also appears in the Fourier series. There are another zillion ways of doing this.

Sincerely,

Loisel (talk) 20:04, 8 January 2008 (UTC)

Thanks, but you answered the wrong question. I was specifically questioning the sufficiency of "The General Case". The case for 1/2π has to made all over again.

--Bob K (talk) 21:48, 8 January 2008 (UTC)

Dear Bob,

I also had problems with that section. Please let me know what you think now. The characters are no longer discussed, because they are discussed in details in the Pontryagin duality article.

Cheers,

Loisel (talk) 00:02, 10 January 2008 (UTC)

Article is of very low quality.

After reading Bob's comments, I started trying to clean up the article. In so doing, I found a large number of errors and lots of nonsense. The job is not finished, but I have other things to do right now, so I am downgrading the quality of the article. This article is not to be trusted right now.

Loisel (talk) 21:20, 8 January 2008 (UTC)

For instance, you wrote:

Given a square-integrable function f(t) of the parameter t (sometimes referred to as time), the Fourier series of f(t) is

$f(t)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }[a_{n}\cos(nt)+b_{n}\sin(nt)]$

That's only true for functions that are 2π-periodic. Fourier series is more general than that.

--Bob K (talk) 22:45, 8 January 2008 (UTC)

Dear Bob,

As per your suggestion, I have emphasized that that particular section is talking about functions of the interval [0,2π]. Note that this is not exactly the same as being periodic.

If your function is periodic with period T, what you are interested in is "discussed" in the section Fourier series on a general interval [a,b], using, e.g., a=0 and b=T.

The question of which domain to use is a very interesting question, but using T-periodic functions instead of 2π-periodic functions is not the answer either because there are Fourier series for much more interesting domains, like Lie groups and differential manifolds. The basis functions are then no longer exponentials. See spherical harmonic for the spherical case.

In any case, I don't think the T-periodic case is unimportant, and I have a section for it. In a minute, I will add the necessary formulae to that section. Like I said yesterday, I haven't had time to do a thorough job of the article.

Loisel (talk) 22:11, 9 January 2008 (UTC)

Dear all,

I have just finished a first pass over the article. It's not perfect, but I hope that it has fewer errors and a better flow than before. Because there were so many topics, I have elected to relegate many of them to the daughter articles that are linked with the notation Main article: .... This is according to Wikipedia:Summary style.

That said, I am sure there is room for improvement, so please make comments and edits.

Sincerely,

Loisel (talk) 00:00, 10 January 2008 (UTC)

Here's what I think. The "general interval [a,b] and on a square" section should be split into two sections. And in the section "general interval [a,b]", the simplified formulas for [0,T] should be given in the form of an example. And we could also point out that setting T=2п produces the simplest-looking form (from your previous section).

--Bob K (talk) 13:06, 12 January 2008 (UTC)

I also think we're missing a fairly important point made by this 8-Jan excerpt:

$c_{n}={\frac {1}{T}}\int _{t_{0}}^{t_{0}+T}s(t)\cdot e^{-i\left(n{\frac {2\pi }{T}}\right)t}\,dt$     for all integer values of n,

where:

• $s(t)$ is periodic, with period $T$
• $t_{0}$ is an arbitrary instance of real argument $t.$

I.e., $c_{n}\,$ is invariant with respect to $t_{0}.\,$

--Bob K (talk) 13:22, 12 January 2008 (UTC)

Looking for an image of a vibrating drum

I am trying to improve the article with some images, and I would like to illustrate the 2d Fourier series with an image of a vibrating drum, like [1] or like [2] or like [3]. I haven't found any free images though.

Loisel (talk) 21:53, 10 January 2008 (UTC)

I can make the images, but it is not completely clear the me the connection to the Fourier series. The modes of a vibrating membrane are solutions of the wave equation with zero Dirichlet boundary conditions. They modes are, if you wish, individual Fourier terms, but I don't think you can easily explain how you get a Fourier series out of there (do you add them up?) Oleg Alexandrov (talk) 05:28, 11 January 2008 (UTC)

See commons:Category:Drum vibration animations, although I think they would be more appropriate at other articles, e.g., at vibration. Oleg Alexandrov (talk) 06:37, 12 January 2008 (UTC)

Thanks a lot. I'm traveling right now, but I will look at it when I come back next week. Loisel (talk) 14:01, 13 January 2008 (UTC)

Contradiction

Here is the excerpt in question:

===Fourier's formula for 2π-periodic functions using sines and cosines===

Given a square-integrable function f(x) of the variable x whose domain is the interval $[0,2\pi ]$, the Fourier series of f(x) is

It's either periodic or the domain is $[0,2\pi ].$ But not both.

--Bob K (talk) 23:46, 12 January 2008 (UTC)

The new figure is not as good as the old one

That's my opinion. Judge for yourselves:

new: http://en.wikipedia.org/wiki/Image:Fourier-partial-sums-of-x.png

old: http://en.wikipedia.org/wiki/Image:Periodic_identity_function.gif

--Bob K (talk) 00:01, 13 January 2008 (UTC)

Dear Bob,

I have no problem with the animated image (although I personally prefer static images, they print better.) I have made the image box smaller, which is why I had changed the animated image in the first place. Please feel free to make further changes, but ideally I would like the images to be of this standard size. On a narrow screen, wide images are not good for the flow of the text, and even on a wide screen, the images were a bit too wide before.

Sincerely,

Loisel (talk) 15:10, 22 January 2008 (UTC)

incorrect use of "Fourier transform"

Regrading this excerpt:

${\begin{aligned}f(x)&=\sum _{n=-\infty }^{\infty }{\hat {f}}(n)\cdot e^{inx}\end{aligned}}$

The set of coefficients ${\hat {f}}(n)$ is called the Fourier transform of $f$. In various fields of science, it has different names and is also denoted differently, for example, $F[n]$ and the frequency domain in engineering, or the characteristic function in probability theory.

No. The Fourier transform is:

${\begin{aligned}\int _{-\infty }^{\infty }\left[\sum _{n=-\infty }^{\infty }{\hat {f}}(n)\cdot e^{inx}\right]\cdot e^{-i2\pi fx}dx&=\sum _{n=-\infty }^{\infty }{\hat {f}}(n)\cdot \int _{-\infty }^{\infty }e^{i(n-2\pi f)x}dx\\&=\sum _{n=-\infty }^{\infty }{\hat {f}}(n)\cdot \delta \left(f-{\frac {n}{2\pi }}\right)\end{aligned}}$

It is a Dirac comb modulated by the Fourier series coefficients.

Also, this shows why f(x) is not a particularly good choice of function names.

--Bob K (talk) 15:58, 24 January 2008 (UTC)

Dear Bob,

It turns out that in Harmonic analysis, whenever you map a function (either of R, or of the interval, or of some manifold or group) to its Fourier coefficients, that is called the Fourier transform. The reconstruction formula is then called the inverse Fourier transform, and takes the form of either a Fourier series or a Fourier integral. For instance, if you read Pontryagin duality, you could be forgiven for thinking that the Fourier transform only applies to integrals, because that's all you see in that article. However, the dual group of a compact group is discrete, and so that article hides some Fourier series under the guise of what looks like a Fourier integral!

This is somewhat frustrating for some real-world users like engineers, because often one thinks of the Fourier transform as the specific integral that occurs on R.

Sincerely,

Loisel (talk) 19:56, 24 January 2008 (UTC)

I can't say I'm surprised. We may need to disambiguate the term Fourier transform, because your usage conflicts with other articles in Wikipedia.

--Bob K (talk) 20:44, 24 January 2008 (UTC)

I've also seen harmonic analysis books call the relationship between a function and its Fourier series coefficients the finite Fourier transform (hence the disambiguations on that page) (somewhat confusingly, since the number of coefficients is not finite, although the domain of f(x) is). I don't think this usage is very widespread among most users of Fourier transformation and Fourier series, however (among whom pure mathematicians are a small minority). I would say that this isn't a task for a dedicated disambig. page per se, since the meanings are closely related...rather, just a note on the corresponding pages ("this is sometimes called the 'Fourier transform' in harmonic analysis, but in common usage the latter refers to ....") would be better, while the bulk of our articles sticks with the most widespread usage. —Steven G. Johnson (talk) 22:05, 26 January 2008 (UTC)

Simplified example

The way I see it, either you know how to integrate x sin nx, in which case you don't need the extra verbiage, or you don't, in which case the extra verbiage is useless anyway.

In any case, I like it much better where it is now. The text now reads: this is the Fourier series, and this is an example.

Loisel (talk) 03:44, 2 February 2008 (UTC)

Now sections are in the wrong order

The section Fourier_series#Example:_a_simple_Fourier_series uses the "general interval" concept. So it should follow section Fourier series on a general interval [a,b] .

--Bob K (talk) 19:24, 5 February 2008 (UTC)

An easy solution is to forget about the fixed interval. Just start with a general interval, like the article was just a few weeks ago (e.g. 8-Jan).

--Bob K (talk) 19:30, 5 February 2008 (UTC)

Dear Bob,

I thought you might bring that up, and one could go to arbitrary intervals, however, the article of 8 Jan had many errors in it.

What I'm going to do now instead is switch the rest of the article to [-pi,pi]. Please take a look (when I'm done) and tell me what you think.

Sincerely,

Loisel (talk) 21:59, 5 February 2008 (UTC)

Sure, I will wait and see. But before you go to that trouble, think about it... you are changing the "definition" to match one example. Wouldn't it make more sense to change the example to match the definition (which still isn't so great)? Better yet, solve the example as is, using the definition, as is. But best of all, just generalize the definition.

--Bob K (talk) 01:00, 6 February 2008 (UTC)

Dear Bob,

Well, you wrote that comment after I had changed the article.

Sincerely,

Loisel (talk) 07:33, 6 February 2008 (UTC)

This article lacks basic explanation

The math formulas are nice to be here, but one should put a chapter about what really Fourier series are and why do you need them. After that, you can put the formulas and other things. Do not forget tha this is Wikipedia not a math course for university. —Preceding unsigned comment added by 192.35.17.15 (talk) 12:29, 12 February 2008 (UTC)

Dear Anonymous,

Thank you for your helpful suggestion. Could you be more precise as to what information you would like to see? Maybe an example, "section XXX should start by explaining YYY".

We have been trying to make our article more useful to laypeople, as well as people who need to use the Fourier series. The introduction and the historical section are meant to be understandable to laypeople, except for the quote from Fourier, but that quote is historically significant.

I must also warn that there is no royal road to mathematics! (I love that quote.) There is bound to be some difficult material in a mathematical article.

Loisel (talk) 04:40, 13 February 2008 (UTC)

Dear Loisel,

My suggestion is not malicious, but my english is bad. Just to present you a quote, one of my math teacher, a bad one said once: "Math is beautiful, but people are making it look ugly." I don't know if he meant to say teachers by people.

There is a straight road in mathematics, but to show it to others you must have the abilities to see the road and to show it to others. If I say c * (a+b) = c * a + c * b and I ask what is this you can say, probably that I just wrote the expanded formula for common factor multiply. OK, but I can say that all I wrote are some letters from alphabet and a few extra signs :-) So, formula without enough explanations is just painting.

Encyclopedias do not replace textbooks and homework problems, hard as we might try. An encyclopedia is more like a reference book. Often, some basic background has to be assumed just to be efficient. The best thing about Wikipedia, in my opinion, is that one can often find the background they are missing by following the internal links. (Although that process may be hampered by inconsistent conventions & notations.) That doesn't mean I think this article can't still be improved or that I discourage anyone from trying. But I do think the introductory material is not too bad.

--Bob K (talk) 14:47, 15 February 2008 (UTC)

Dear Anonymous commenter,

I have tried to expand the article a little bit by adding expanding on the example and providing further motivation. I am guessing that this won't be enough to satisfy you, but perhaps it helps, or perhaps you can make some specific suggestions for improvements.

Sincerely,

Loisel (talk) 22:39, 15 February 2008 (UTC)

Dead link

The link "Une série de Fourier-Lebesgue divergente presque partout" does not seem to be working. —Preceding unsigned comment added by 129.241.138.61 (talk) 15:11, 24 April 2008 (UTC)

Introductory Sentence.

The article begins "In mathematics, the Fourier series is a type of Fourier analysis, which is used on functions that might otherwise be difficult or impossible to analyze."

Would be much happier about a beginning like... "Fourier series decompose a periodic function into a sum of simple oscillating functions, namely sines and cosines. The subject of Fourier series is part of the general subject of Fourer analysis. Fourier series were introduced..."

I found the phrase "a type of Fourier Analysis" strange. I had encountered this usages like this when I studied signal processing, not not since switching to mathematics. Even in signal processing literature I would not have described it as very common. So I stoped and asked several graduate students in mathematics if they understood the first sentance. And they objected to the same phrase.

I have notice the phrase "type of Fourier Analysis" has become standard since some of the pages were merged a while back, and I think it might be nice open a discussion about it.

More importantly I find the phrase " ... which is used on functions that might otherwise be difficult or impossible to analyze" a bit misleading. Fourier series are used extensively on even very simple functions for various reasons. What exactly is meant by analyze in this part of the sentence?

Lastly, and perhaps least important, would be to move any discussion of complex exponentials a bit deeper into the article. I think it would be helpful to improve readability for non-technical audiences.

I would love some feedback on these ideas. Thenub314 (talk) 20:12, 24 April 2008 (UTC)

periodic functions as tempered distriutions.

In response to an edit comment let me explain my previous edit. As with any locally integrable function you can define a distribution by integrate against your function. Let f be a periodic function on R. For any Schwartz function φ the map

$\varphi \mapsto \int _{\mathbb {R} }f(x)\varphi (x)\,dx$

defines a bounded linear functional. Thus defines a distribution, since it defines a tempered distribution we can discuss it's Fourier transform on R. Thenub314 (talk) 02:30, 12 May 2008 (UTC)

OK, I'll not be challenging that, since I don't know what a Schwartz function is, or why that makes a bounded linear functional, or what a tempered distribution is. But maybe someone else will understand and tell me it's OK. Dicklyon (talk) 02:37, 12 May 2008 (UTC)

Let ${\mathcal {D}}$ be the set of infinitely differentiable functions $f(x)$ such that, for every k>0, $x^{k}f(x)$ tends to zero as x goes to infinity (the functions are rapidly decreasing). This space can be made into a metric space, and more precisely, ${\mathcal {D}}$ is an F-space (espace de type F), cf. topological vector space. The Schwarz space ${\mathcal {D}}$ is special because the Fourier transform of an infinitely differentiable, rapidly decreasing function is also infinitely differentiable and rapidly decreasing. A linear map $\phi$ on ${\mathcal {D}}$ which is also continuous in the metric is called a Schwarz tempered distribution. The space of all such linear maps ("functionals") is written ${\mathcal {D}}'$. Although ${\mathcal {D}}'$ is very abstract, one can squint and realize that it contains pretty much every function you can think of, as well as things that are not functions like measures and hairier things. The main reasons why ${\mathcal {D}}'$ is important is because it's the largest space of function-like things of $\mathbb {R}$ on which you can define differentiation and Fourier transform.

So, almost anything you can think of has a derivative and a Fourier transform.

Loisel (talk) 16:40, 12 May 2008 (UTC)

Addendum: if you ignore Fourier transform, there's a larger space of distributions (the ones that aren't necessarily tempered) where you can also differentiate.

Loisel (talk) 16:44, 12 May 2008 (UTC)

Thanks; I'll study up on Distribution (mathematics); it's actually good to know that there's some meaningful mathematics behind the handwaving engineering uses of Fourier transform that I always thought were suspect. I always thought the Fourier transform was defined only for square-integrable functions, and extended to delta functions and such only informally, but I see now in the article that it's not so. Dicklyon (talk) 17:42, 12 May 2008 (UTC)

Vibrating Drum

The solution to the wave equation on a disc produces Bessel functions, not Fourier waves. Those Bessel functions can of course be written as a Fourier series with a simple change of basis, but I don't believe that serves the pedagogical purpose for the diagram. I would suggest removing it or producing a new animation on a square. —Preceding unsigned comment added by Gmcastil (talk • contribs) 04:28, 25 May 2008 (UTC)

The Bessel functions you speak of (really, Bessel in the r variable, complex exponential in the angular variable) form an orthonormal basis for the Hilbert space of L^2 functions. The reconstruction formula in a Hilbert space is called the Fourier series. So all the examples (vibrating drum, spherical harmonics, etc...) are Fourier series in the sense of Hilbert spaces. Loisel (talk) 15:58, 26 May 2008 (UTC)

Loisel - Given your description of why this image is appropriate wouldn't the page Generalized Fourier series or perhaps even Fourier–Bessel series make a happier home for this example. Especially since for the purposes of this page " a Fourier series decomposes a periodic function into a sum of simple oscillating functions, namely sines and cosines." --Mposey82 (talk) 21:55, 27 May 2008 (UTC)

Ping-pong sections

We've had a lot of back-and-forth between the sections "A second example: trigonometric polynomials" and "Determinining the coefficients." I did some editing to fix a few nits and remove major duplication, but I must say I don't quite get it. The former section is a somewhat interesting if narrow example that shows that if you already have the coefficients of a trigonometric polynomial, then they agree with the formula given in "Fourier's formula for 2π-periodic functions using sines and cosines." The latter section shows that if you have any periodic function and find the coefficients that make a trigonometric polynomial equal to it, then they too obey the formula given before. If we accept Fourier's formula, then both of these sections are pretty redundant. The latter could be taken as a derivation of Fourier's formula, but it's not presented that way. What's the right way to develop this stuff? Dicklyon (talk) 04:35, 10 September 2008 (UTC)

The problems I have with the section on determining Fourier coefficients is that it is mathematically flawed. Most courses given point out this is what Fourier did, and that it doesn't work. If you want to precede this way you must end up at a narrow class of functions. I am happy to remove the example all together, but I thought if someone wanted to make explicit how orthogonality worked before the section on Hilbert spaces, then I would go along with it. I have removed the section on "Determining Fourier Coefficients", among other problems it gives the impression that every trigonometric series is a Fourier series, which is false. It also had explicitly stated that if the Fourier series converged to the function then this method would work, and this is also false. Thenub314 (talk) 06:50, 10 September 2008 (UTC)

Well, you must be a mathematician, because you've lost me. Where exactly can this method go wrong in the case of a function that the Fourier series converges to? Are you saying that dot products with sinusoids can give wrong Fourier series coefficients? Or that the reasoning to get there was flawed? Dicklyon (talk) 06:59, 10 September 2008 (UTC)

Mostly the concern is that the reasoning is flawed. Most of which come into play when you want to exchange the infinite sum and the integral. Thenub314 (talk) 17:42, 10 September 2008 (UTC)

As a PS to this conversation, I happend upon the example I was thinking of in Pinsky's book. It turns out that $\sum _{n=2}^{\infty }{\frac {\sin(nx)}{\log(n)}}$ converges, but it is not a Fourier series. But if you replace sin(nx) by cos(nx) then it is a Fourier series. Thenub314 (talk) 21:13, 19 September 2008 (UTC)

I've just taken a look at the article, I think we should remove Thenub's new section which relates the exponential formulation and the sine/cosine formulation. The crux of this new "example" is already a part of the subsequent sections like The modern version using complex exponentials and Hilbert space interpretation. Any new information, which is not included in those previously existing section, should rather be merged, instead of adding a new section which repeats information found elsewhere in the article and which does not flow well.

I am happy to delete the new section. I didn't want to completely delete RJFJRs additions, but I didn't want to leave a mathematical flawed argument. I wasn't trying to relate exponential and sine/cosine formulations, I don't even remember mentioning exponential functions. Those calculations are easy to deduce from the stuff in this article so I am fine with leaving them out. Thenub314 (talk)

As for Thenub's question of what is the difference between a Fourier series and a Trigonometric series, as far as I know, the two terms are used interchangeably in the literature.

I didn't really have a question. It is a theorem that not every trigonometric series is a Fourier series. The sets of convergence for trigonometric series is a difficult subject that I had looked into some time ago. Thenub314 (talk) 17:42, 10 September 2008 (UTC)

The article as I wrote it a while back imposed that $f$ be square-integrable, for simplicity, but User:Bob K attempted to generalize (although he went too far, diff is here: http://en.wikipedia.org/enwiki/w/index.php?title=Fourier_series&diff=183950639&oldid=183861651). I subsequently "fixed" it by not being too specific on the constraint on the Fourier coefficients (they must not increase too rapidly), here is the diff: http://en.wikipedia.org/enwiki/w/index.php?title=Fourier_series&diff=185638936&oldid=185638341. I must say I would still prefer to put a restriction on f, but I'm betting that the next editor who comes around and who may not necessarily know what the restrictions on the Fourier coefficients must be, will simply delete any restriction we place on f. Still, I'm all for going back to square-integrable functions for the earlier sections of the article.

Loisel (talk) 16:39, 10 September 2008 (UTC)

I absolutely agree you need a restriction on f. To my mind the most general condition that is easy to see makes sense is to require f to be integrable. Then by the triangle inequality you get:

$|c_{n}|=\left|\int _{0}^{2\pi }f(x)e^{-inx}\,dx\right|\leq \int _{0}^{2\pi }|f(x)||e^{-inx}|\,dx=\int _{0}^{2\pi }|f(x)|\,dx<\infty .$

Here I used of course that $|e^{-inx}|=1$. So all the Fourier coefficients are defined and finite. I'll try putting this in, we will see what happens. —Preceding unsigned comment added by Thenub314 (talk • contribs) 18:30, 10 September 2008 (UTC)

Example: a simple Fourier series

The coefficients are missing a factor of ${\frac {1}{\pi }}$. Or, conversely, the function should be periodic between -1 and 1, not $-\pi$ and $\pi$ (or normalized differently). Also,the heat-transfer example deals with $x\in \left(0,\pi \right)$, which would give different coefficients. Or am I mistaken? Squidgyhead (talk) 01:18, 17 November 2008 (UTC)

Sines and cosines more "accessible" than sinusoids?

I'm not sure what TheNub means in his edit summary. It always seemed to me that saying decompose into sines and cosines was a bit vague and confusing, since it really means sine waves or sinusoids. The term "sines" tends to suggest the sines of a few angles, as opposed to a few sinusoidal functions. Dicklyon (talk) 14:16, 17 December 2008 (UTC)

I just meant that sines and cosines are familiar terms from high school. Depending on the level of mathematics a student encounters before finishing high school they may (or may not) encounter the term sinusoid (or sine and cosine). So I thought keeping the lede as it was made kept it accessible to a wider group or readers. Thenub314 (talk) 16:06, 17 December 2008 (UTC)

Definition of Fourier series

I know that some readers are uncomfortable with the short sentence (in "Fourier's formula for 2π-periodic functions using sines and cosines")

The infinite sum

${\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }[a_{n}\cos(nx)+b_{n}\sin(nx)]$

is the Fourier series for ƒ on the interval [−π, π].

I understand that this must be interpreted as a formal series of functions of x, but people who learned to distinguish between a series and the sum of the series, between series of numbers and series of functions will find the text careless. Could it be possible to be more precise, without being too long? Bdmy (talk) 09:34, 28 December 2008 (UTC)

Well, one simple change may be to move the link from infinite sum to infinite sum (not that I am so happy with that section of the article). This way, if there is confusion we direct the reader to a point in the article where formal series are discussed and series whose elements are functions are mentioned. Thenub314 (talk) 10:48, 28 December 2008 (UTC)

Formula for a₀

What's the formula for $a_{0}$? —Preceding unsigned comment added by Danielsimonjr (talk • contribs) 16:59, 28 December 2008 (UTC)

The idea is to have the same formula for all $a_{n}$, namely

$a_{n}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(t)\cos(nt)\,\mathrm {d} t,$

so that

$a_{0}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(t)\,\mathrm {d} t.$

It is true that this formula is a little strange, but this is how mathematicians have done for almost 200 years, and we are not going to change it. It is a bit strange because $a_{0}$ is not the mean of the function, but twice the mean. This is reflected in the formula for the Fourier series, that contains $a_{0}/2$. When dealing with the complex Fourier coefficients $c_{n}$, you have that $c_{0}=a_{0}/2$.

--Bdmy (talk) 18:24, 28 December 2008 (UTC)

Merger proposal

Not sure if Fourier theorem has any information that this article doesn't have, but it's clearly subsumed in scope. --Taeshadow (talk) 18:33, 11 March 2009 (UTC)

• Support – Fourier theorem is little more than a stub, thinly edited, covered better at Fourier series. Dicklyon (talk) 05:24, 13 March 2009 (UTC)

I agree also. The Fourier series article is far better and encompassed everything from the short paragraph at Fourier theorem. --Paul Laroque (talk) 01:47, 16 April 2009 (UTC)

I agree. If the article titled Fourier theorem actually stated a theorem, maybe it would have a point. Michael Hardy (talk) 02:13, 16 April 2009 (UTC)

Merged. Dicklyon (talk) 02:17, 16 April 2009 (UTC)

Reviewers of the original Fourier article

In the section "Revolutionary Article" its is stated, "When Fourier submitted his paper in 1807, the committee (which included Lagrange, Laplace, Malus and Legendre, among others)". This statement is factually incorrect.

Fact: Four reviewers were appointed by the Academy. These were: Laplace, Lagrange, Monge and LaCroix.

T. N. Narasimhan, tnnarasimhan@LBL.gov 98.210.57.10 (talk) 02:25, 28 November 2009 (UTC)

Any sources for that? Because replacing a badly sourced statement by a badly sourced statement will, in all likelyhood, not happen.--LutzL (talk) 09:47, 30 November 2009 (UTC)

Vibrating Disc doesn't look right

it states that this is a variable when integer is constant —Preceding unsigned comment added by 117.196.34.205 (talk) 16:16, 14 December 2009 (UTC)

I would be surprised if the animation of the vibrating disc is correct. If it is a circular disk, as depicted, the solution should have circular symmetry. I understand that this kind of symmetry preservation doesn't always hold, but in this fairly simple case I think it should.

I learned that vibrating disks have solutions that look like radial Bessel functions anyhow. —Preceding unsigned comment added by 129.105.207.213 (talk) 04:12, 2 May 2008 (UTC)

Too technical

All this is too much math for a common user trying to understand or get a clue of what Fourier did. I know, we are not all math people, but more instroduction is necesary nad then you can start with formulas. 192.35.17.15 (talk) 16:00, 19 December 2007 (UTC)

The article as a whole is disjointed and probably too detailed, but the intro looks OK to me. If you don't understand it, the internal links, like Fourier analysis, provide background material which you might need. If you can't understand the background material either, you probably aren't in the target group for an article like this one. If you do understand the article, but think you can do better, you are free to show us the way.

--Bob K (talk) 22:47, 20 December 2007 (UTC)

I can imagine a more layperson-oriented intro, but it would certainly take some work. Michael Hardy (talk) 01:09, 21 December 2007 (UTC)

I took a stab at it. Loisel (talk) 20:32, 8 January 2008 (UTC)

Putting a lay person's introduction at the start of the article is fine. However you should not simplify the article at the expense of the technical user. Loisel: I think the edits you have made do just this. In particular the choice of a particular period and a particular integration regime instead of the more general $[t_{0},t_{0}+T]$ could (a) leave more technical users who don't know much about Fourier series with the impression that the integration region isn't movable by an arbitrary offset, (b) annoy those technical users who wanted the formula in the more general form and will now have to convert it themselves (which if they are intending to actually use the formulas will likely be quite a large number of such users, and (c) confuse intermediate users who wish to make the Fourier series of a function with a period of other than $2\pi$ but don't understand the details well enough to change the given formula.

If you really feel there is a class of user out there who will be unable to understand the more general form of the equations and yet will want more than a lay person's non-mathematical introduction to the topic, then maybe make a 'simple case' section where you give them in the form you have used in your edit.

Another criticism (sorry!) is that with respect to the lay person's introduction, I would have though that a simple modern description of the technique ranks higher than a historical account of its development --- so I'm not convinced about moving the 'Historical Development' section to the top of the article.

But really my largest gripe is just the removal of the more general form; as a technical user of Wikipedia I personally love the fact that you can get gritty detailed descriptions of mathematical techniques that you wouldn't find in other encyclopedias.

137.222.187.157 (talk) 12:25, 9 January 2008 (UTC)

Dear Anonymous,

You are very interested in the T-periodic case. I have made some notes to that effect in the T-periodic section, giving explicit formulae. However, please note that the T-periodic cse is not "the general" case. The general case is the Hilbert space orthonormal basis approach. However, without going to an abstract Hilbert space, there are interesting Fourier series which are not on intervals, see spherical harmonic.

Sincerely,

Loisel (talk) 22:29, 9 January 2008 (UTC)

I agree, the 2pi period constraint threw me off quite a bit. It's the reason for the equations, the explanation of what's being done, that's missing. Simplifying shouldn't even be necessary, except to show what happens when you plug in certain circumstances. In other words, it's not the math that's wrong, it's the article. 97.123.87.61 (talk) 18:16, 17 October 2009 (UTC)

In regards to a comment above, awhile back: you should not simplify the article at the expense of the technical user ... I understand your general points about article quality, but an encyclopedia can't be all things to everyone. (There's an old chestnut about "Try to make everyone happy, you make no one happy.") Besides, the more technical readers can always skim/skip the intro. Wikilinks are fine for further detail or branching topics, but an article should stand on its own. I think the intro reads well at the moment but could use some tweaking. For instance, what general field of math does this fall under? I haven't studied diffy-q's since college, but I always thought this was the domain of calculus -- yet that term does not appear in the article. -PrBeacon (talk) 07:46, 6 October 2010 (UTC)

Would it be appropriate to add CAT:Multivariable Calculus, Differential equations and/or Partial differential equations? -PrBeacon (talk) 07:56, 6 October 2010 (UTC)

No, No and No. Fourier series are in one variable, ODEs and PDEs are analyzed using the Lagrange tranform or the Fourier transform, not the series. There is a connection from the series to Sturm-Liouville, but that does not deserve a category.--LutzL (talk) 13:33, 6 October 2010 (UTC)

Ok, what branch of mathematics would you suggest? -PrBeacon (talk) 01:03, 7 October 2010 (UTC)

A great many of the scientific articles in Wikipedia seem to be addressed primarily to fellow specialists, or at any rate students already well-versed in the subject matter. That's great for professionals, but why cannot the articles of an enyclopedia for the general public unpack the technical terms as it goes along? I don't see why such unpacking would have to come at the expense of technical precision or comprehensiveness. —Preceding unsigned comment added by 72.225.204.182 (talk) 06:55, 9 December 2010 (UTC)

I think that's what wiki-links are for. If you find terms that you don't understand, and they're not linked to an article that explains them, let us know here. Dicklyon (talk) 07:50, 9 December 2010 (UTC)

Very first example

There will be no π in the denominator in the final result since it's canceled out.
look at http://planetmath.org/?op=getobj&from=objects&id=4718 Thanks. —Preceding unsigned comment added by 70.119.103.136 (talk) 04:14, 24 May 2010 (UTC)

NOTE now fixed; this error was introduced by a very recent, incorrect edit. 128.95.41.29 (talk) 02:51, 26 May 2010 (UTC)

Quote isn't actually Fourier's?

We have a quote under the heading Revolutionary Article attributed to Fourier. However, if you check up the reference with the link provided, you'll see that the words aren't actually his. Despite the article being listed as "par M. Fourier", it's written in the third person with constant references to "M. Fourier" did this/that etc. On the first page, there is a footnote:

Cet Article, que nous avons déjà signalé dans l'Avant-Propos du Tome I, n'est pas de Fourier. Signé de l'initialo P, il a été écrit par Poisson, qui était un des rédacteurs du Bulletin des Sciences pour la partie mathématique. A raison de l'intérêt historique qu'il présente comme étant le premier écrit où l'on ait fait connaître la théorie de Fourier, nous avons eru devoir le reproduire intégralement.

In my own loose, and possibly mildly inaccurate translation:

This article, as we have already indicated in the "Avant-Propos" of Book 1, is not by Fourier. Signed with the initial P, it has been written by Poisson, who was one of the editors of the Bulletin des Sciences for the mathematical part. For reason of historical interest, it is presented as though it were the first writing where the theory of Fourier was made known, [we have had to reproduce it in full?] (is "eru" supposed to be "eu"?)

Not quite sure what to make of this - sounds like a rather bizarre practice, but I believe that the original memoire may have been lost. Anybody got any ideas on how this should be correctly attributed? —Preceding unsigned comment added by Tcnuk (talk • contribs) 15:56, 9 June 2010 (UTC)

I've put a footnote to this effect into the article. If anybody can shine any light on this mystery, I'd be delighted to hear from them... Tcnuk (talk) 12:03, 10 June 2010 (UTC)

The last sentence: Because of the historic interest that it presents as being the first .... —Tamfang (talk) 16:22, 9 December 2010 (UTC)

Not "eru, not "eu", it's "nous avons cru devoir le reproduire intégralement", "we believed it to be our duty to reproduce it in full".--LutzL (talk) 19:20, 9 December 2010 (UTC)

And the translation of the last sentence is slightly wrong: "For reasons of historical interest, as it appears to be the first text to publicate the theory of Fourier, we deemed it indispensable to publish it in full."--LutzL (talk) 19:31, 9 December 2010 (UTC)

And perhaps it should be "in its original form" or similar for "integralement".--LutzL (talk) 20:26, 9 December 2010 (UTC)

On decembre 21st 1807, Fourier presented his current work in something like a seminar at the "Academie des Sciences". The base of this presentation was a "memoire", perhaps hand-written, that was probably given to some mathematicians, but not published. Obviously, Poisson found it interesting enough to publish his notes of this memoire in 1808 on 4 pages in the "Nouveau Bulletin des Sciences, par la Societe Philomathique", <a href="http://www.archive.org/stream/nouveaubulletind11807soci#page/112/mode/2up"> thanks to google available online</a>. This was reprinted as the earliest published occurence of the computation of (Fourier)trigonometric series coefficients by integrals and the notorious claim that the series converges to the given function. The computation of coefficients of trigonometric polynomials by something like the discrete Fourier transform was known some decades earlier.--LutzL (talk) 21:18, 14 January 2011 (UTC)

Poisson's 1808 publication of a 4-page brief summary and comment on Fourier's memoir/presentation to the Institut doesn't qualify as a first publication of the memoir, although it did help protect Fourier from anyone who might have wished to publish before Fourier was able to, and thereby compete for priority. Although Fourier included a version of his earlier 1807 and 1811 work in his 1922 book, the first real publication of the 1807 memoir manuscript is given below, and yes, the 1807 manuscript did go missing for several decades. Fourier, Jean-Baptiste Joseph. "1811 mathematics prize paper." In Joseph Fourier, 1768-1830; a survey of his life and work, based on a critical edition of his monograph on the propagation of heat, presented to the Institut de France i 1807., by Ivor Grattan-Guinness, 30-440. Cambridge, MA: The MIT Press, 1972. Fourier did turn in a 234-page hand-written copy of his 1807 memoir to the Institute. However, Grattan-Guinness thinks that Fourier may have presented without the aid of the usual extract, since none survives [For new information and discussion on this point, please see note below by AJR]. To the previous contributor, the notoriousness of Fourier's claim faded as and when it became known that all practical data satisfies the requirements for convergence. Even Fourier himself, in using the words "arbitrary function" also said that the function had to be finite in extent, thereby presaging Dirichlet's conditions. In effect Fourier was saying that the function could be arbitrary, as long as it was integrable. Fourier's daring gave mathematics the necessary kick in the pants, but it took over 100 years for the bruise to fully go away.  ;-)

AJR: John Herivel's book, Joseph Fourier: The man and the physicist, published in 1975, in other words 3 years after I. Grattan-Guinness' landmark publication of Fourier's 1807 memoir, has this:

J Herivel, Page 318: Chapt XXI Fourier to an unknown correspondent, around 1808-9

"I have the honour to send you two notes concerning the memoir on heat. The first(Note 1) is the one that was read at the Institut in place of the reading of the memoir... [presumably due to the excessive length of the actual memoir, comprising 234 sides!, in other words Fourier is referring to the extract. Therefore Fourier *did* read an extract, rather than presenting without using one].

J Herivel, Page 320: Note 1. This was the abstract (extrait) of Fourier's 1807 memoir which has been retained in the MS. 1851 of the library of the Nationale École des Ponts et Chaussées, Paris. It must therefore be dated 1807, and not 1809 as suggested by Grattan-Guiness (3), p. 497.

When I looked in Grattan-Guinness, I immediately saw that this would not be a simple correction, since discussion of the extrait in q uestion appears in multiple locations, and is quite embedded in the discussion. For example, see

Grattan-Guinness, page 24 (lines 14-18, 22-24); page 26 (lines 6 and 24).

Grattan-Guinness, Page xii Abbreviations of Titles of Key Works by Fourier

4. "Extrait," for the paper sent by Fourier to the Institut de France in October 1808. [I had long wondered about this, since as far as I knew there was only the 1807 extract -AJR]

5. "Notes," for the set of footnotes to the Extrait.

I have not seen a correction published by Ivor Grattan-Guinness, nor any acknowledgement of the problem. I also have yet to find any further discussion by John Herivel. One of them must be more correct than the other in this matter.

Formula incorrect

I believe the formula giving the conversion of coefficients into exponential form is incorrect (and the equivalent problem exists in the other direction as well). it suggests that c₀ = a₀/2. This should be c₀ = a₀. since a₀ cos(0x) + b₀ sin(0x) = a₀, and c₀ e^i0x = c₀.

99.52.193.237 (talk) 16:09, 13 January 2011 (UTC)

You appear to be assuming something like a_n cos(nx) + b_n sin(nx) = c_n e^inx.  Try proving it.

--Bob K (talk) 18:39, 14 January 2011 (UTC)

What I'm suggesting (in more detail) is that a₀ cos(0x) + b₀ sin(0x) = c₀ cos(0x) + c₀ i sin(0x), by Euler's formula. The sine terms drop out so: a₀ cos(0x) = c₀ cos(0x) and c₀ = a₀. —Preceding unsigned comment added by 99.52.193.237 (talk) 17:04, 16 January 2011 (UTC)

And that is totally correct, except that the constant term of the real form of the series is given as ${\frac {a_{0}}{2}}$. This is done to have a unified integral formula for the real coefficients. And to be consistent with the conversion in the opposite direction given above the discussed one. Write that down explicitely for the case n=0.--LutzL (talk) 17:12, 16 January 2011 (UTC)

Might it be useful to include this formula: $a_{0}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)dx$ in the article as the general form for a₀? —Preceding unsigned comment added by Brvman (talk • contribs) 23:49, 4 May 2011 (UTC)

Formula and graph axis labelling

Sorry to ask such a stupid question, but shouldn't the graph axes in the "Plot of a periodic identity function" associated with Example 1 be the same as the formula states? I.e. instead of x(t) and t (which appear nowhere in the formula or the coefficient calcs), shouldn't it be labelled f(x) and x? Or am I missing something obvious? (I'm no mathematician, as you can probably tell) Cephas Borg (talk) 14:52, 7 December 2011 (UTC)

You're not missing anything. I'd guess the figure was originally created for a different purpose or different text... not an unusual occurrence at Wikipedia. It's not a perfect fit, but it's close enough that nobody has bothered to fix it. I don't think anyone would complain if you want to fix it yourself.

- I would if I could but I do not know how to fix a fig. It is confusing enough that it should be fixed 172.190.77.188 (talk) 02:11, 10 February 2013 (UTC) aburr@aol.com

--Bob K (talk) 18:16, 7 December 2011 (UTC)

Thank Fourier it wasn't just me! I'm currently learning to modify the original source. Thanks for the encouragement, Bob. (This is my first real Wikipedia contribution, woo hoo! :). Cephas Borg (talk) 02:49, 8 December 2011 (UTC)

One thing I should have checked and mentioned is the section File usage on other wikis at http://commons.wikimedia.org/wiki/File:Sawtooth_pi.svg. Luckily, all it lists is this article. (Whew!) Thanks for your help, and good luck with your first "adventure".

--Bob K (talk) 01:20, 9 December 2011 (UTC)

I changed the axis titles in Inkscape to f(x) and x which are OK for all Wikipedia articles where the image is used. Olli Niemitalo (talk) 06:42, 10 February 2013 (UTC)

Request for comment by readers of this article

A request for comment regarding a reference to this article is open for discussion. Readers of this article are invited to contribute. Brews ohare (talk) 17:07, 21 April 2012 (UTC)

Riemann paper and 1811 essay competition

I have an idea for a new reference for Riemann. I think it might be this : Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe. (Part of the) Habilitationsschrift, 1854.

Help from German speakers, please? Also what was the name of the prize competition that Fourier entered in 1811? — Preceding unsigned comment added by 5.151.82.6 (talk) 04:10, 10 November 2012 (UTC)

• Here are two important potential reference sources that I think could help with the issues I raised. [4] and Connections in Mathematical Analysis: The Case of Fourier Series, Enrique A. Gonzalez-Velasco. American Mathematical Monthly, Volume 99, Issue 5 (May, 1992), 427-441. [5] — Preceding unsigned comment added by 5.151.82.6 (talk) 04:45, 10 November 2012 (UTC)

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 [6]

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)

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)

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)

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)

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)

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-)].

[7] 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)

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)

Fourier Series definition and other problems

There is a problem with the first equation on the page. The Fourier Series is defined as a series with amplitude and phase angle for each harmonic then the cosine function should be used. This is consistent with the idea of "phasor". The consistent Fourier Series definition follows as:

$s_{N}(x)={\frac {A_{0}}{2}}+\sum _{n=1}^{N}A_{n}\times \cos \left({\tfrac {2\pi nx}{P}}+\phi _{n}\right),\quad {\text{for integer}}\ N\ \geq \ 1.$

Note, I have also used the multiplication symbol $\times$ instead of the dot product symbol $\cdot$. From an aesthetic perspective, I would even prefer to remove the multiplication symbol completely.

Another problem is in the definition of the equivalent forms. The identity only holds if the sine part is complex, as shown in the following equation:

${\begin{aligned}s_{N}(x)&=\overbrace {a_{0}} ^{A_{0}}/2+\sum _{n=1}^{N}\left(\overbrace {a_{n}} ^{A_{n}\sin(\phi _{n})}\cos \left({\tfrac {2\pi nx}{P}}\right)+\overbrace {b_{n}} ^{A_{n}\cos(\phi _{n})}i\,\sin \left({\tfrac {2\pi nx}{P}}\right)\right)\\&=\sum _{n=-N}^{N}c_{n}\times e^{i{\tfrac {2\pi nx}{P}}},\end{aligned}}$

Please discuss and comment.

The generalization to complex-valued s(x) happens in section Fourier_series#Complex-valued_functions.

As clearly stated in the article, the identity you don't understand is just basic trigonometry:

$\sin \left({\tfrac {2\pi nx}{P}}+\phi _{n}\right)\equiv \sin(\phi _{n})\cos \left({\tfrac {2\pi nx}{P}}\right)+\cos(\phi _{n})\sin \left({\tfrac {2\pi nx}{P}}\right).$

Therefore the formula:

${\begin{aligned}s_{N}(x)&=\overbrace {a_{0}} ^{A_{0}}/2+\sum _{n=1}^{N}\left(\overbrace {a_{n}} ^{A_{n}\sin(\phi _{n})}\cos \left({\tfrac {2\pi nx}{P}}\right)+\overbrace {b_{n}} ^{A_{n}\cos(\phi _{n})}\sin \left({\tfrac {2\pi nx}{P}}\right)\right)\\&=\sum _{n=-N}^{N}c_{n}\cdot e^{i{\tfrac {2\pi nx}{P}}}\end{aligned}}$

is intentionally real-valued. And in case you missed it:  $c_{n}\cdot e^{i{\tfrac {2\pi nx}{P}}}$  and  $c_{-n}\cdot e^{i{\tfrac {2\pi (-n)x}{P}}}$  are complex conjugates. Their sum is real.

The  •  operator is commonly used to represent the product of 2 operands, which technically is a one-dimensional dot product.  So, at worst, you can call it overkill, if you insist on thinking in those terms. But I'm confident that most readers just see it as an alternative to ×, just as / is an alternative to ÷.

--Bob K (talk) 20:33, 17 October 2017 (UTC)

I think you're both right. That is, the derivation seems correct, but starting with the sine instead of the cosine for defining the sinusoid with a phase angle seems a bit unconventional, and makes it a little messier, like needing to divide by i to get c_n. Dicklyon (talk) 05:24, 25 April 2019 (UTC)

"Conventional" is the sine-cosine form:

${\frac {1}{2}}a_{0}+\sum _{n=1}^{\infty }[a_{n}\cos(2\pi \ n\ x)+b_{n}\sin(2\pi \ n\ x)].$

Unconventional would be:

${\frac {1}{2}}a_{0}+\sum _{n=1}^{\infty }[a_{n}\cos(2\pi \ n\ x)-b_{n}\sin(2\pi \ n\ x)].$

But the former suggests:

$\sin(2\pi nx+\varphi _{n})\equiv \sin(\varphi _{n})\cos(2\pi nx)+\cos(\varphi _{n})\sin(2\pi nx),$

because of consistently "+" signs. For sign consistency, the latter suggests:

$\cos(2\pi nx+\varphi _{n})\equiv \cos(\varphi _{n})\cos(2\pi nx)-\sin(\varphi _{n})\sin(2\pi nx),$

which is also OK with me.

--Bob K (talk) 23:33, 26 April 2019 (UTC)

Label for the cool gif is unclear.

The gif showing the synthesis is fabulous, but perhaps it should be stated explicitly that it is the SUM of the four sine waves that creates it. — Preceding unsigned comment added by 2001:8003:E448:D401:7D27:BA42:5CB1:E259 (talk) 08:23, 25 August 2019 (UTC)

Coefficient of $a_{0}$

Sorry if I'm just being stupid here, which is probably the case.

Shouldn't the coefficent of $a_{0}$ (underneath the first equation box in the definition section) be ${\frac {2}{P}}$ rather than ${\frac {1}{P}}$. This feels like an easy mistake to make when $P$ is normally going to be $2\pi$. — Preceding unsigned comment added by Thellamabotherer (talk • contribs) 19:16, 31 May 2020 (UTC)

I believe you are correct. If you substitute n=0 into the defintion of $a_{n}$ you arrive at $a_{0}={\frac {2}{P}}\int _{P}s(x)dx$.

In fact, there are two equivalent ways to define $a_{0}$ and your formula for $s_{N}(x)$ is affected by this choice. If you choose $a_{0}$ to have a consistent formula with that of $a_{n}$ (ie. the formula with ${\frac {2}{P}}$), your $s_{N}(x)$ is defined as follows

$s_{N}(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{N}a_{n}\cos \left({\frac {2\pi nx}{P}}\right)+b_{n}\sin \left({\frac {2\pi nx}{P}}\right)$ (Equation 2 in the article)

However you can choose $a_{0}$ to be half of this value (ie. the formula with ${\frac {1}{P}}$) so that you don't have to later divide by two in the formula for $s_{N}(x)$.

Whoever made this edit was unaware of that difference in formula for $s_{N}(x)$ and made a bad edit. Moreover, the rest of the article makes use of the consistent formula for $a_{0}$. So in order to ensure the consistency of the article, it was best that I undid his edit. The formulas should now be correct.

--Hqurve (talk) 15:56, 5 June 2020 (UTC)

No such topic yet

See also: Fourier analysis § History 104.162.239.6 (talk) 23:15, 29 October 2020 (UTC)

The first few paragraphs should be moved

From the engineering and the maths perspective, those 3 paragraphs aren't helpful for the reader where they are placed :

       1  History           2 Definition           2.1 Complex-valued functions

We need to start with the definition of the series and the coefficients, the visual gif plus the problem of the convergence, the whole in less than 10 lines. Take a look at Convergence of Fourier series. Only after that, we can go into disgression, the intuitive ideas, the theorems, the history, the real vs complex notation. Assuming the reader doesn't know what is a series and a complex number is ridiculous, as well as beginning the article with the partial sums. Reuns (talk)

From the perspective of a high school student who's interested in this, I agree with Reuns. The history should be lower, then the definition. Cahmad25 (talk) 00:09, 28 October 2021 (UTC)

Thank you, and I am inclined to agree. So I would propose a reorganization something like this:

• Definition (excluding complex-valued functions, for now)
• Convergence & Example 1 (A simple Fourier series)
• History, Beginnings, Birth of harmonic analysis, Example 2 (Fourier's motivation)
• Table of common Fourier series
• Complex-valued functions & Other common notations
• Table of basic properties (section 7.1)
• Other properties (sections 7.2 - 7.10)
• Extensions & Example 3 (complex Fourier series animation)
• Approximation and convergence of Fourier series

Note that this leaves two sections on the topic of convergence, but they are too dissimilar to be merged.

--Bob K (talk) 13:52, 28 October 2021 (UTC)

Should this article definition start with a scary correlation function?

I have honestly never seen correlation functions be used as an introductory starting point to Fourier series. Is invoking correlation even necessary in the definition? This article doesn't even have a clear transition from the correlation function into the Fourier series amplitude-phase form — Preceding unsigned comment added by Em3rgent0rdr (talk • contribs) 07:36, 15 March 2022 (UTC)

well I moved the correlation function down a bit so it wasn't the first thing in definition. I'm still not entirely sure it even belongs in definition. Em3rgent0rdr (talk) 12:12, 15 March 2022 (UTC)

I believe I have transitioned into talking about the correlation function better now, by saying that this correlation can be used to determine the coefficients A_n and phi_n. So I think I am ok now with this correlation function being here. Em3rgent0rdr (talk) 11:39, 16 March 2022 (UTC)

Fourier's Theorem is necessary but not in Wikipedia

Fourier's Theorem underpins the concept of Fourier Series being able to represent any reasonable periodic function. However there is no mention of this crucial theorem. searching Wikipedia for it results in a redirect to Fourier Series, which is honestly a bit of circular logic. I believe it should be somewhere. maybe the Fourier theorem deserves its own page, or at least a section in this article. Em3rgent0rdr (talk) 12:17, 15 March 2022 (UTC)

I think I have resolved this by renaming a section at the bottom of the article to be called "Fourier theorem proving convergence of Fourier series" and now I can refer to that section earlier in the article to reference "the Fourier Theorem". Em3rgent0rdr (talk) 11:41, 16 March 2022 (UTC)

I brought the animations to be fixed in between paragraphs of the intro (using align = none).

I think these animations are great for understand the basics. Unfortunately, they previously would appear on the right-bottom corner of my browser and were sortof missed. I think I have reworded the introduction so that those animations can be included directly inside the intro. After invoking "Fourier synthesis" I am immediately showing the square wave being added up. And after invoking "Fourier transform" I am immediately showing the 6 sine waves being broken up into frequency domain representation. — Preceding unsigned comment added by Em3rgent0rdr (talk • contribs) 12:42, 16 March 2022 (UTC)

Maybe use 'ξ' instead of 'f' for frequency to avoid conflict with 'f' for function

This would also make this article more consistent with the Fourier transform article. I guess the downside would be that people scared of the letter 'ξ' and not knowing how to pronounce it. — Preceding unsigned comment added by Em3rgent0rdr (talk • contribs) 07:31, 15 March 2022 (UTC)

My preference is to keep $f$ for Ordinary frequency, which is consistent with other articles. And I agree that $\xi$ is unnecessarily intimidating. So what we've pretty much settled on is to avoid using $f$ for both things in the same article. It's the same thing in the literature, different authors, different preferences.

$x$ is another one. Many articles use it instead of $f$. So we have both  $x(t)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ X(f)$  and  $f(x)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ {\hat {f}}(\xi )$  expressing the same concept.

Since I don't like using $x$ for a function, and I do like using $f$ for frequency, my preference is:

$s(t)\ ({\text{or}}\ s(x))\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ S(f),$   ($s$ for "signal" and $S$ for "spectrum")

--Bob K (talk) 19:52, 15 March 2022 (UTC)

Yeah, I understand. What I have now done is added a sentence at the start of "Table of common Fourier series" which says

"Readers be aware that up until this point in the article, f had previously represented frequency, but no longer does."

So I suppose I can accept this. Em3rgent0rdr (talk) 11:03, 16 March 2022 (UTC)

Thank you. Actually, as I wrote the above I had forgotten that this article an exception to the "avoid using $f$ for both things in the same article" mantra. (I knew there was at least one exception, but did not remember this is it.) Anyhow, I think your work-around is sufficient. If not, I'd be willing to replace f(x) with s(x) in the table, including the figures.
--Bob K (talk) 12:04, 16 March 2022 (UTC)

If you could replace f(x) with s(x) in the table, that would be great and avoid any possible confusion. Em3rgent0rdr (talk) 12:37, 16 March 2022 (UTC)

Happy to give it a try. --Bob K (talk) 11:56, 17 March 2022 (UTC)