Talk:Fourier series

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Geometry guy 21:27, 9 June 2007 (UTC)

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[edit] Vibrating Disc doesn't look right

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)

[edit] 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.

[edit] 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).

[edit] 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:

$c_n = \frac{a_n - j b_n}{2}$

c 0 = a 0 / 2

a n = c n + c - n = 2 R e [c n]

b n = c - n - c n = 2 I m [c n]

and for real functions:

$c_{-n} = c_n^*$

Fresheneesz 21:20, 26 May 2006 (UTC)

[edit] 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
$a_n = \frac{1}{L} \int_{-L}^{L} f(x) cos(\frac {2 n \pi x}{L}) dx$
$b_n = \frac{1}{L} \int_{-L}^{L} f(x) sin(\frac {2 n \pi x}{L}) dx$

Should the function be odd (link to odd function definition) the fourier cosine series may be used. This simplifies to
$a_n = \frac {2}{L} \int_{L_1}^{L_2} f(x) cos(\frac {2 n \pi x}{L}) dx$
$b n = 0$

Should the function be even (link to even function definition) the fourier sine series may be used. This simplifies to
$a n = 0$
$b_n = \frac {2}{L} \int_{L_1}^{L_2} f(x) sin(\frac {2 n \pi x}{L}) dx$

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))

[edit] 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)

[edit] Trigonometric series redirect?

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

$\frac{1}{2}A_{o}+\displaystyle\sum_{n=1}^{\infty}(A_{n} cos nx + B_{n} sin nx)$

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

$A_{n}=\frac{1}{\pi}\displaystyle\int^{2 \pi}_0 f(x) cos nx dx$ where $(n = 0,1,2,...)$

$B_{n}=\frac{1}{\pi}\displaystyle\int^{2 \pi}_0 f(x) sin nx dx$ 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)

[edit] "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)

[edit] 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:

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.)
The answer is much more easily obtained by direct integration: $\frac{1}{2\pi}\left[\frac{x^3}{3}\right]_{-\pi}^{\pi} = \frac{\pi^2}{3}.\,$
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)

[edit] inconsistent use of "L"

Fourier_series#The_wave_equation says:

[edit] 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

$b_n = \frac{2}{L} \int_0^L f(x) \sin \left( \frac{(2n+1)\pi}{L} x\right)\, dx.$

[edit] 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

2π

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)

[edit] 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)

[edit] General case

Regarding:

$\int_G f(g)\overline{\chi(g)}\,dg = \frac{1}{2 \pi}\int_{2\pi}^{} f(t) e^{-i nt}\,dt$

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)

[edit] 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)

[edit] 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)

[edit] 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

[0,2π]

, the Fourier series of f(x) is

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

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

[edit] 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)

[edit] incorrect use of "Fourier transform"

Regrading this excerpt:

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

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:

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

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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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 $φ$ 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)

[edit] 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)

Talk:Fourier series

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Contents

[edit] Vibrating Disc doesn't look right

[edit] Possible Error

[edit] Format of equations

[edit] relationship between real and complex forms

[edit] Still very convoluted

[edit] Error

[edit] Trigonometric series redirect?

[edit] "real Fourier coefficients"

[edit] Making a mountain out of a molehill is not a good example.

[edit] inconsistent use of "L"

[edit] The wave equation

[edit] Too technical

[edit] Interpretation?

[edit] General case

[edit] Article is of very low quality.

[edit] Looking for an image of a vibrating drum

[edit] Contradiction

[edit] The new figure is not as good as the old one

[edit] incorrect use of "Fourier transform"

[edit] Simplified example

[edit] Now sections are in the wrong order

[edit] This article lacks basic explanation

[edit] Dead link

[edit] Introductory Sentence.

[edit] periodic functions as tempered distriutions.

[edit] Vibrating Drum

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