View source for Talk:Fourier series

== Inconsistencies ==

I read it carefully and found several troubles.

1. In the definition we say f(x) is from L<sup>2</sup> and then, discussing convergence, we say "if f is from L<sup>2</sup> then..."

2. In the sample f(x) = x ~ ... the form is f(x) ~ a0 + ... instead of f(x) ~ a0/2 + ...

+idea: I am sure we have to emphasize the fact that the functions <math>\frac{1}{2\pi}e^{inx}</math> form an orthonormal basis and that the calculation of F<sub>n</sub> is done by simple projecting f onto the basis functions.

==Cis function and rewrite==
I haven't heard of a "cis" function since I was in high school. What's wrong with "exp"? It's by far the more common notation. -- [[User:Tim Starling|Tim Starling]]

I'm inclined to agree.  This article has a '''lot''' of room for improvement.  Maybe I'll look at it some time soon. [[User:Michael Hardy|Michael Hardy]] 19:50 Mar 21, 2003 (UTC)

I've done some rewriting.  The "cis" function is no longer called that after my revisions. Obviously far more can be said.  Maybe later I'll mention some of the vast array of applications, in physics, number theory, probability, cryptography, statistics, engineering, etc. [[User:Michael Hardy|Michael Hardy]] 22:17 Mar 21, 2003 (UTC)

==The fourier series of an arbitrary periodic continuous function need not converge to the function.==

A common error!  The fourier series of an arbitrary periodic continuous function need not converge to the function.

The article does not say that the Foureir series converges to the function; the article explicitly says otherwise!  I've seldom seen such sloppy reading. [[User:Michael Hardy|Michael Hardy]] 18:30, 27 May 2004 (UTC)

Michael,

I'm not the one who initiated this discussion, but I just took a look. I see above that you indeed understand that Fourier series of continuous functions do not converge pointwise. For the record I mention a nonconstructive proof (as you probably know, the constructive one is difficult.) Let S<sub>n</sub>f := &sum;<sub>k=-n</sub><sup>n</sup>f^(k), the value of the partial Fourier sum at 0.  The space C=C(T) is a Banach space in the uniform norm; consider F:={S<sub>n</sub>} as a family of operators on C(T). One may show that ||S<sub>n</sub>||=||D<sub>n</sub>||<sub>1</sub>->&infin; where D<sub>n</sub>:=&sum;<sub>k=-n</sub><sup>n</sup/>exp(ikx). Hence there is no uniform bound for F on C, and by the uniform boundedness principle, F is not pointwise bounded. Pick an f in C so that Ff is unbounded, in particular the Fourier series of f does not converge at 0 (not even to some value which is not f(0)). Since f is continuous, it is in particular piecewise continuous, and 0 is a point of continuity of f.

Now consider this paragraph:

:''That much was proved in the 19th century, as was the fact that if f is piecewise continuous then the series converges at each point of continuity.''

I too find it difficult to read that particular passage correctly. Perhaps you were talking about Cesaro or Abel means? Or piecewise Holder continuous functions with exponent &alpha;>1/2? Maybe I'm misunderstanding "point of continuity."

I would be grateful if you could explain that passage to me.

Thanks,

[[User:Loisel|Loisel]] 02:58, 29 May 2004 (UTC)

:I've excised the text for the time being. I'd still like to understand what you meant, Michael. [[User:Loisel|Loisel]] 19:53, 31 May 2004 (UTC)

==Constructiveness==

Since people don't seem to know, here is roughly how to construct a function (any use of Banach Steinhaus can be explicitized that way). Take a series of ''n<sub>k</sub>'' 's going to infinity very fast. Take continuous functions ''f<sub>k</sub>'' with <math>||f_k||\leq 1</math> such that 
: <math>\int f_k D_{n_k} > \frac{1}{2}||D_{n_k}||</math>
Basically, ''f<sub>k</sub>'' should be <math>\mathrm{sign} D_{n_k}</math>, but smoothed a little so that it would be continuous. Then take 
:<math>f=\sum_{k=1}^\infty 2^{-k} f_k</math>
If the ''n<sub>k</sub>'' increase sufficiently fast (you need to construct them inductively) then you get
:<math>\int f D_{n_k} > \frac{1}{2}\int 2^{-k}f_k D_{n_k} > 2^{-k-2}||D_{n_k}|| \to \infty </math>
And that's it.[[User:Gadykozma|Gadykozma]] 08:02, 21 Jul 2004 (UTC)

----
How about some example?

== Generalized to interval a-b ==

I have reverted the edits by 142.150.160.187 because there are just too many errors
and inconsistencies. The definition is wrong, because one of the exponents must be 
negative, and I believe the product of the normalizing constants must be 1/p not  
2/p. The orthogonality relationships are wrong, again, one of the exponents
must be negative, and the normalizing constant must be 1/p. The period of the
original was 2&pi;, but now there are at least three or four different periods,
variously b-a, p, L, and 2&pi;. I have no problem generalizing the Fourier series
to an arbitrary interval a-b, and I'm not famous for error free edits myself, 
but lets do it 90% right at least.
[[User:PAR|Paul Reiser]] 20:45, 17 Mar 2005 (UTC)


:on an interval [a,b), period is b-a, the normalizing constant is 2/(b-a), for example on interval [-Pi, Pi), the normalizing constant is 2/(Pi+Pi) = 1/Pi.  Or can be written as 1/L on an interval [-L, L).  If the interval is [0, 2*Pi), the constant would be 2/(2*pi-0). which is 1/Pi. So 2/p for a period p, and 1/L for interval [-L, L) are consistent statements. You are right about orthogonality, i made a typo.


What is the relationship between p, a, and b? That needs to be up front.
At present, the orthogonality in the continuous space is

:<math>\frac{2}{p}\int_a^b e^{\frac{2\pi inx}{p}}e^{-\frac{2\pi imx}{p}}dx=\delta_{mn}</math>

That means when m=n we must have

:<math>\frac{2}{p}\int_a^b dx=1</math>

which means

:<math>\frac{2(b-a)}{p}=1</math>

which means

:<math>p=2(b-a)</math>

Is this the relationship between p and a and b? If it is, then the definitions of the Fourier series and its inverse 
are inconsistent. If we use the definition of F<sub>n</sub> as it is
now, and substitue the definition of f(x) as it is now, from the
definition section, then we have:

<math>f(x')=\sum_{n=-\infty}^\infty \left[\frac{2}{p}\int_a^bf(x)e^{\frac{-2\pi inx}{p}}dx\right]
e^{\frac{2\pi inx'}{p}}</math>

<math>=\frac{2}{p}\int_a^bf(x)\sum_{n=-\infty}^\infty e^{\frac{-2\pi in(x'-x)}{p}}dx </math>

Using the orthogonality in the continuous space, the sum in the above equation is equal to

<math>\sum_{n=-\infty}^\infty \delta\left(\frac{2\pi}{p}(x-x')+2\pi n\right)</math>

which has a spike whenever x-x'=p.
If we use p=2(b-a), then that happens twice in the interval b-a and we have

<math>f(x')=2f(x')</math>

which is wrong.

Another problem is that we want to have the same period used in everything. The orthogonality relationships are used
to prove the validity of the inverse, the convolution theorems, Plancherel and Pareseval (as above). We should not be
using one period (p) for the definition and orthogonality in the discrete space, another period (2 &pi;) for orthogonality in the continuous space and yet another (L) in the convolution theorems. They may be correct but it makes a mess of the
article. [[User:PAR|PAR]] 17:09, 18 Mar 2005 (UTC)

: To the anonimous contributor: Please make yourself an account. That would be much helpful. Thanks. [[User:Oleg Alexandrov|Oleg Alexandrov]] 00:50, 19 Mar 2005 (UTC)

==How to compose good-looking non-[[TeX]] mathematical notation on Wikipedia== (See below. [[User:Michael Hardy|Michael Hardy]] 22:25, 17 Mar 2005 (UTC))

:period is b-a,
:period is ''b''&nbsp;&minus;&nbsp;''a'',

:on an interval [a,b),
:on an interval [''a'',&nbsp;''b''), 

:the normalizing constant is 2/(b-a),
:the normalizing constant is 2/(''b''&nbsp;&minus;&nbsp;''a''),

:the normalizing constant is 2/(Pi+Pi) = 1/Pi.
:the normalizing constant is 2/(&pi;&nbsp;+&nbsp;&pi;)&nbsp;=&nbsp;1/&pi;.

:interval is [0, 2*Pi), the
:interval is [0,&nbsp;2&pi;), the

:would be 2/(2*pi-0).
:would be 2/(2&pi;&nbsp;&minus;&nbsp;0).

:So 2/p for a period p,
:So 2/''p'' for a period ''p'',

:[-L, L)
:[&minus;''L'',&nbsp;''L'')

==Mistake==

I've calculated the continuous orthogonality relationship for a general interval [a,b]. For [-&pi;,&pi;] the orthogonality relationship is:

:<math>\frac{1}{2\pi}\int_{-\pi}^\pi e^{inx}e^{-imx}\,dx = \delta_{nm}</math>

Assume p=b-a and make the substitution

:<math>y=a+\left(\frac{x}{2\pi}+\frac{1}{2}\right)p</math>

so that y=[a,b] for x=[-&pi;,&pi;]. This gives for the orthogonality relationship:

:<math>\frac{1}{p}e^{2\pi i(m-n)(a/p+1/2)}
\int_{a}^b e^{2\pi iny/p}e^{-2\pi imy/p}\,dy = \delta_{mn}</math>

I haven't checked this, but that exponential mess before the integral will probably be showing up in the definitions, Plancherel, convolutions, etc. etc. so we need to make it disappear by using x=[-L,L] where L is anything, including possibly &pi;.  [[User:PAR|PAR]] 20:24, 22 Mar 2005 (UTC)


==Disagree with recent changes==

This page now is quite inconsistent, as it deals with Fourier series on [a, b], then on [-&pi;, &pi;], and then on [-L, L]. I suggest putting things back to [-&pi;, &pi;], as the generalization from there to an arbitrary interval is trivial. Other ideas? [[User:Oleg Alexandrov|Oleg Alexandrov]] 05:23, 20 Mar 2005 (UTC)

:I agree, this is silly.  It should just use [-&pi;, &pi;], [-L, L], or [-L/2, L/2], which are the most common conventions.  In any case, a ''single'' convention should be used throughout.  (At most, you can have a single subsection at the end that explains how to switch to different periods and intervals.) [[User:Stevenj|&mdash;Steven G. Johnson]] 05:47, Mar 20, 2005 (UTC)

: Its silly and worse - its wrong. Look at the second orthogonality relationship. Set m=n and very quickly you get 2=1. I tried to explain this to this person a few days ago, but it doesn't seem to sink in, and any repair gets re-damaged without discussion. [[User:PAR|PAR]] 17:31, 20 Mar 2005 (UTC)

Certainly, standardise to period 2&pi; at the outset. [[User:Charles Matthews|Charles Matthews]] 11:13, 20 Mar 2005 (UTC)

: I asked [[User:Chubby Chicken|Chubby Chicken]] (the recent contributor) to take a look here at this conversation. Let us not rush to revert things for now, but in a couple of days we definitly need to bring this article in good shape. [[User:Oleg Alexandrov|Oleg Alexandrov]] 20:11, 20 Mar 2005 (UTC)

fixed mistake, the normalizing constant is 1/(b-a) for a compelx fourier series, 2/(b-a) for a real fourier series ([[User:Chubby Chicken]] forgot to sign).

: But you are avoiding the bigger issue of the inconsisties you introduced with your changes (see above). And there is no gain in doing things on  [a, b] rather than the original [-&pi;, &pi;]. [[User:Oleg Alexandrov|Oleg Alexandrov]] 04:09, 21 Mar 2005 (UTC)

::Well, there might be a small gain in a list of mathematical formulae. On the other hand it is usual in an encyclopedia article to simplify in minor ways, for clarity. We should do that, here. [[User:Charles Matthews|Charles Matthews]] 09:16, 21 Mar 2005 (UTC)

the gain is it describes fourier series expansion for function defined on any intgerval [a,b], instead of restricting to functions that are defined on [-pi pi]. ([[User:Chubby Chicken]] forgot to sign).

::As I have just said, the greater generality goes against normal practice. [[User:Charles Matthews|Charles Matthews]]

: But the article is ''inconsistent'' now, see what I wrote above. It will be easier to do everything on [-pi, pi], and then only at the end put what you wrote as generalization. 

Also, please read the [[Fourier series]] article from beginning to end. Don't focus just on one section. [[User:Oleg Alexandrov|Oleg Alexandrov]] 18:59, 21 Mar 2005 (UTC)

== [-L,L] or [-pi,pi]==

It seems to me we now have to decide whether to keep the [-pi,pi] interval or go to the [-L,L]
interval. The [a,b] interval, when done correctly, is an abomination (please see [[Talk:Fourier series#Mistake|Mistake]] above). I'm in favor of the [-L,L] interval. It simply replaces pi with L in all the formulas, no sweat. Maybe Chubby Chicken's desire to generalize is not totally misplaced. I know some are in favor of [-pi,pi]. Can we get a consensus somehow? I would be glad to write the page with [-L,L] if thats
what we decide. [[User:PAR|PAR]] 20:24, 22 Mar 2005 (UTC) 

: [-L, L] is fine with me. [[User:Oleg Alexandrov|Oleg Alexandrov]] 20:50, 22 Mar 2005 (UTC)

: I'm in favor of [-pi, pi] here. Working with [-L, L] is going to throw a factor pi/L into the picture if I'm not mistaken; it's a slight complication but I really don't see a need for making this topic any more obscure than it needs to be. A section which describes how to go from [-pi, pi] to [-L, L] or [a, b] is OK by me but that shouldn't be front and center. For what it's worth, [[User:Wile E. Heresiarch|Wile E. Heresiarch]] 21:37, 22 Mar 2005 (UTC)

:: Now I would lean to agree with Wile. We can put some formulas for [-L, L] at the end. And the [a, b] case needs no treatment at all. Anybody knowing at least calculus will figure out how to do the shift from [-L, L]. [[User:Oleg Alexandrov|Oleg Alexandrov]] 23:51, 22 Mar 2005 (UTC)

::: Wile is right, exp(inx) becomes exp(i&pi;nx/L) so I'm less enthusiastic about [-L,L]. I still like [-L,L] because its easier to substitute L=&pi; into the equations than to make the substitution x=y&pi;/L in the equations, including the dx and then realize that y=xL/&pi; changes the limits of the integrals to [-L,L]. [[User:PAR|PAR]] 00:29, 23 Mar 2005 (UTC)

== a0 and 1/2 ==

I note in the example we say...

:<math>a_0 = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)\,dx=\frac{1}{2\pi}\int_{-\pi}^{\pi} x\, dx= 0,</math>

:<math>f(x)=x=a_0 + \sum_{n=1}^{\infty}(a_n\cos(nx)+b_n\sin(nx)) </math>

Whilst in the definition we say...

:<math>f(x) = \frac{1}{2}a_0 + \sum_{n=1}^\infty\left[a_n\cos(nx)+b_n\sin(nx)\right]</math>

:<math>a_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x)\cos(nx)\,dx</math> and <math>b_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x)\sin(nx)\,dx</math>

So, my question is: Am I right in thinking the "divide by two" has moved from "within a0" to "applied to a0" between definition and example? If so, should it be corrected and which is the correct way?

Just a thought. [[User:Mike1024|Mike1024]] [[User_talk:Mike1024|(talk]]/[[Special:Contributions/Mike1024|contribs)]] 23:47, 5 October 2005 (UTC)

:It is inconsistent.  The version of the top seems to be more standard, so I've fixed the example.  [[User:Stevenj|&mdash;Steven G. Johnson]] 05:14, 6 October 2005 (UTC)

== The first sentence ==

of an article is supposed to summarize the most important point about the subject. The first paragraph of this article contains no information at all. What mathematical concept is not useful and what mathematical concept is not about breaking up a problem into more manageable parts? 

This introduction is extremely poor!

== about "SQUARE WAVES" in Fourier expanding==

Can anyone tell me why the first term of Fourier series <math>a_0 = 0</math> when expanding a square wave

I have done an exercise about it for practicing after I read a page article..

thanks a lot
--[[User:HydrogenSu|HydrogenSu]] 15:36, 19 January 2006 (UTC)

: <math>a_0</math> is the area under the function. If the area under the square wave is zero, then <math>a_0=0</math>. That means if the  square wave goes up and down between -y and y, its zero, but if it goes up and down from something like 0 to y, then its not. [[User:PAR|PAR]] 16:26, 19 January 2006 (UTC)
::'''u'''nderstood,thx. and I've just now realized from
::<math>a_0 =\frac{1}{\sqrt{2\pi}}\int_{-\pi}^{\pi} x\, dx= 0</math> 
--[[User:HydrogenSu|HydrogenSu]] 18:18, 29 January 2006 (UTC)
::--[[User:HydrogenSu|HydrogenSu]] 18:22, 19 January 2006 (UTC)

To HydrogenSu - These questions should be asked at [[Wikipedia:Reference desk]], not on the talk page of an article. [[User:PAR|PAR]] 20:36, 29 January 2006 (UTC)

:: just making more clear that <math>a_0</math> is the ''average'' area under the function. [[User:Fresheneesz|Fresheneesz]] 07:57, 8 February 2006 (UTC)

== Transform:Fourier Integral ==

I have a question on[http:/upwiki/wikipedia/en/3/31/Math001.jpg]when doing an exercise about it after I read a article on the Wikipedia page..
Anyone please help me.
:(On which,why do for-all <math>\mathcal ,isinx=odd, </math> ,therefore they all equal to 0?)

--[[User:HydrogenSu|HydrogenSu]] 15:52, 3 February 2006 (UTC)


:Ask at [[Wikipedia:Reference desk]].  Do not ask here.  [[User:Pfalstad|Pfalstad]] 19:04, 3 February 2006 (UTC)
::You two may misunderstand..'''In fact'''I asked those for the newest changed version proposed here.--[[User:HydrogenSu|HydrogenSu]] 11:50, 5 February 2006 (UTC)

==More General Fourier series==

I didn't have time to read whether this is what was discussed up under the header '''Generalized to interval a-b''', but I have learned a more general fourier series for a function with an arbitrary finite period of T. This definition describes the coefficients as:

:<math>a_n = \frac{2}{T}\int_{-T/2}^{T/2} f(x)\cos(\frac{2nx}{T})\,dx</math> 
and 
:<math>b_n = \frac{2}{T}\int_{-T/2}^{T/2} f(x)\sin(\frac{2nx}{T})\,dx</math>

I think this definition is clear and fully general. Does anyone agree with putting this up? [[User:Fresheneesz|Fresheneesz]] 07:56, 8 February 2006 (UTC)
: I absolutely agree with that. Including a more general form, I think, will help people to understand more easily that 2&pi; is not some "magical" interval. Also these forms are used often in physics, in my experience. In fact the reason why I'm reading this talk page is because I couldn't understand why that definition wasn't already in the article and I was going to add it myself. Only your formulas for the coefficients are incorrect, you're missing a &pi;:
::<math>a_n = \frac{2}{T}\int_{-T/2}^{T/2} f(x)\cos(\frac{2\pi nx}{T})\,dx</math> 
::<math>b_n = \frac{2}{T}\int_{-T/2}^{T/2} f(x)\sin(\frac{2\pi nx}{T})\,dx</math>
::--[[User:Ptomato|Ptomato]] 09:39, 8 February 2006 (UTC)

::: Oh, you're right. Typed it in wrong. I'll add it in now I guess. [[User:Fresheneesz|Fresheneesz]] 20:29, 8 February 2006 (UTC)

::: I edited it, but I would assume the top part about the imaginary functions doesn't fit the new more general '''real''' fourier series definition. I hope someone can fix that, because I don't know how. [[User:Fresheneesz|Fresheneesz]] 21:07, 8 February 2006 (UTC)

== Inconsistencies in "definition of fourier series"==

Maybe i'm wrong, but I think that the first part under that header may not be consistant with the more general real fourier series definition. 

:<math>F_n =\frac{1}{2\pi}\int_{-\pi}^\pi f(x)\,e^{-inx}\,dx.</math>

In the above equation, should "1/2pi" be instead "2/T" ? And then the other two pi's would be T/2? 

::If you want to use a period ''T'' you need to change several things besides just the normalization in front of the integral. In general, the mixing of notation that was present made no sense (it mixed inconsistent formulas in a misleading manner), and I removed it.  A subsection on how to change the period from 2&pi; to ''T'' would be fine, though (assuming it were done correctly this time).

:<math>f(x) = \sum_{n=-\infty}^{\infty} F_n \,e^{inx}.</math>

Also, in the above equation, is it really supposed to be from negative infinity to positive infinity?
[[User:Fresheneesz|Fresheneesz]] 01:05, 13 February 2006 (UTC)

::Yes.  [[User:Stevenj|—Steven G. Johnson]] 02:37, 13 February 2006 (UTC)


==Restricting the period to 2&pi; versus 2''L'' versus no restriction ==

I was wondering why you changed the definitions on [[fourier series]] back to a less general form (with period being specified at 2 pi). Its true that some things were inconsistant, but the more general definition of fourier series' should be on the page - not a ''specific'' fourier series with a ''specific'' period. If you could help generalize the rest of the page, that would be very helpful. However, I don't think we should imply that all fourier series have a period of 2 &pi;. [[User:Fresheneesz|Fresheneesz]] 21:50, 14 February 2006 (UTC)

:It's better to be specific than to be incorrect.  The previous version mixed a general-period formula for the series coefficients with a 2&pi;-period formula for the series, which is confusing and inconsistent at best and wrong at worst.

:I have no objection to adding a short subsection on how to change variables to express the series as a function of a general period.  But it needs to be correct, and keeping it in a subsection (so that the distinction between it and the rest of the article is clear) is best.

:[[User:Stevenj|—Steven G. Johnson]] 05:36, 15 February 2006 (UTC)

:: I like it more with the period 2&pi;. Then it is most clear how things work, as that's the natural period of the sine and cosine. Extending things to an arbitrary period is something which is rather trivial but makes the notation much more clumsy. [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 16:07, 15 February 2006 (UTC)

::: You say its better to be specific than incorrect, however being inconsistant isn't incorrect, it might just be hard to understand. In any case, I don't know if defining fourier series to be *always* on a period of 2 &pi; is as correct as you say it is - my gut feeling is that its flat out wrong/incomplete. [[User: fresheneesz]]
:::: I taught Fourier series to undergrads last several weeks. It took me five or more lectures to deal with Fourier series on [-pi, pi], with how to find the coefficients, orthogonality of sines and cosines, how good the approximation is, etc. Then it took me half a lecture to extend those formulas to [-L, L]. In other words, not worth the trouble working on the general interval to start with. [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 16:38, 21 February 2006 (UTC)

:: but you did eventually teach how to extend the formulas. Here it doesn't. I think its much better to have at *least* a complete definition. It just bugs me that the stuff on this page isn't general. It makes it seem simply wrong. Sorta like defining the force of gravity to be mg rather than G m1 m2/r^2. It works.. sometimes... Besides, if its so trvial to extend it, why not extend it from the start? [[User:Fresheneesz|Fresheneesz]] 07:38, 1 March 2006 (UTC)

::: If you do it from the start from [-L, L], you need to rewrite the entire article. Besides, again, the Fourier series is trivial to extend once you know it on [-pi, pi], but it is pointless to add that extra difficulty right from the beginning. Why the heck work with <math>\cos \frac{xL}{\pi}</math> and complicate all the formulas from there on if you can work with <math>\cos x</math>? If you wish, you can add a section at the end of the article saying how things extend to [-L, L]. 

::: In different words: generalizing things from the very beginning it is just absolutely not worth it. Try to attack a problem or a subject at the optimal and natural level, which for expansions involving sines and cosines is [-pi, pi]. [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 16:14, 1 March 2006 (UTC)

:::: I guess I don't see the inherint complication that the L/pi makes. I can see how the generalization would be more tedious to write out, but once its written, its done. It seems like such a small addition to me. I would write how to extend it, but I don't know how, unless you mean simply writing the more general equation (which i might do). Even if I could write the general equation for real fourier series, I don't know enough about the complex version. I think this article would be 100 times better if it at least had the generalizations somewhere in it. [[User:Fresheneesz|Fresheneesz]] 21:10, 2 March 2006 (UTC)

:::: As said earlier, please make a new section, preferably towards the bottom, where you can put all the generalizations you wish. This will combine simplicity, inteligibility, and full generality, all in the same article; perfect. [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 02:26, 3 March 2006 (UTC)

Hypothetical:  Let's say that I am an undergraduate encountering FS for the first time. It's 2:00 in the morning, and I need to find the FS equivalent for a periodic function that has a period other than 2&pi; for a homework set that is due at 8 am tomorrow morning.  Unfortunately, I fell asleep during the lecture when the professor taught us how to compute the FS, and I don't have a clue how to do it.  So I turn to WP as my last hope, and I start reading the article on the FS.  Uh oh!  The article defines the FS for functions that are periodic in 2&pi; and there is some section that sort of explains that you can normalize a function of arbitrary period.  I am kind of tired, and I just don't see the connection....

IMO, the article should start with the general case of a periodic function of arbitrary period and explain how that translates to sines and cosines that are ''not'' periodic in 2&pi; but are periodic at frequencies that are integer multiples of the fundamental frequency.  The 2&pi; restriction certainly makes the analysis a lot easier, but it also eliminates about 90 percent of the richness of the topic and hides most of the key ideas that people need to know to really understand this topic.

I would recommend that we switch from using ''x'' as the domain to using ''t'' as the domain, which would make it easier to speak in physical terms such as time, period, frequency, and angular frequencey, and that we use ''T'' to represent the period of the original function.  Alternatively, if we continue to use ''x'' as the domain, then we can talk in terms of distance, wavelength, spatial frequency, and so on, and then we should use ''L'' (not 2''L'') as the wavelength or "period" of the original function.  IMHO. 

BTW, the interval then becoms either [-T/2, T/2] or [-L/2, L/2] or any interval of width ''T'' or width ''L'', depending on the domain of choice, ''t'' or ''x''.

-- [[User:Metacomet|Metacomet]] 22:40, 12 March 2006 (UTC)

== Trigonometric series ==

As I type this, "[[trigonometric series]]" is a redlink. I'll create it as a redirect to "[[Fourier series]]", with an explanation in the intro. Conceivably one could write separate articles, but I don't think it's necessary. [[User:Melchoir|Melchoir]] 00:33, 17 February 2006 (UTC)

==F<sup>*</sup> ?==

Under '''Definition of Fourier series''', it has an F<sup>*</sup>, its unclear to me what that means. What.. does it mean? [[User:Fresheneesz|Fresheneesz]] 08:53, 3 March 2006 (UTC)
:Complex conjugate? [[User:Melchoir|Melchoir]] 09:20, 3 March 2006 (UTC)

== My changes: March 9. ==

I may not have been logged in when I made some changes to the article on Fourier series. What I've done is just a start: I intend to go through and unify the thing. I'm taking the position that articles in Wikipedia should be introductory and designed to appeal to those without extensive background in the topic. The article should point to more sophisticated treatments, but not necessarily include them. [[User:Donludwig|Donludwig]] 00:45, 10 March 2006 (UTC)

==Recent changes: where did stuff disappear to?==

My generalization seems to have disappeared, and so did the example. I haven't had time to read over the new mess, but it would be infinitely helpful if the editor or editors of that humungous edit explained what they did.  [[User:Fresheneesz|Fresheneesz]] 10:15, 10 March 2006 (UTC)

It seems to me that much of those edits should be reverted, the explanations are now very hard to read, and make this page inaccessible to anyone but those that don't need it. [[User:Fresheneesz|Fresheneesz]] 10:17, 10 March 2006 (UTC)

: The sections on "Functions that vanish at one end-point of an interval", least squares, etc, are about applications. They don't belong at the beginning of the article, rather way at the bottom, after the theory is clarified. I don't like the current article reincarnation either. [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 20:06, 10 March 2006 (UTC)

:: Hi, I have no idea what least-squares has to do with this article: although the method (i.e. project a function on a basis of orthonormal functions) is common to Fourier series and to least-squares, I fail to understand why one would need to even mention least-squares in this article. [[User:Deimos 28|Deimos 28]] 21:39, 10 March 2006 (UTC)
::: There are reasons for mentioning them, as the Fourier coefficients can be obtained by doing least square. But it just does not belong there. I guess whoever rewrote the article was somebody very familiar with applications of Fourier series, but not with the theory. A revert is in order. [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 02:59, 11 March 2006 (UTC)

==My new concerns: x maps to somthing, new definition, etc==

First of all, it should be explained what it means for ''x'' to map to some ''f(x)''. 

Second of all, I'm skeptical as to whether:

:<math>f(x) = \sum_{n=0}^{\infty} \left[ A_n \cos \left( n x \right) + B_n \sin n x \right] </math>

is really the "original series". It seems more plausible that Fourier started out with a continuous integral, and then derived the discrete nature of the fourier series. But i'm really just guessing here. More work needs to be done to fix the intro up. [[User:Fresheneesz|Fresheneesz]] 11:19, 11 March 2006 (UTC)

: I don't quite understand what you mean by ''it should be explained what it means for x to map to some f(x)''. The original texts of Fourier are available [http://mathdoc.emath.fr/cgi-bin/oetoc?id=OE_FOURIER__2 here]. I read the article of 1807 entitled ''Mémoire sur la propagation de la chaleur''. This version was in fact written by Poisson but reproduces the steps of Fourier. One of the problems when reading this article is that the term ''integral'' was then used for the solution of a (partial) differential equation, which is confusing. It does seem, however, that Fourier used a formulation close to that of this Wikipedia article (although he was only considering the metal bar heating problem and probably didn't know the implications of his work):
::<math>\varphi(y)=a\cos\frac{\pi y}{2}+a'\cos 3\frac{\pi y}{2} + a''\cos 5\frac{\pi y}{2}</math>
:Fourier then multiplies by <math>\cos (2i+1)\frac{\pi y}{2}</math> and integrates from -1 to 1 to get:
::<math>a_i=\int_{-1}^1\varphi(y)\cos(2i+1)\frac{\pi y}{2} dy</math>
:[[User:Deimos 28|Deimos 28]] 19:10, 11 March 2006 (UTC)

== Recent edits ==

Hi,

I am the author of the latest edits in this article. I tried to give a succint but nonetheless accurate definition, referring the reader to the relevant (functional analytical) pages of the Wikipedia. I removed Wikilinks which, I thought, didn't bring much to the article (for example, linking to the birth-year of Joseph Fourier might be interesting, but I fail to see what it brings to the understanding of the article).

I did not edit as much as I wanted to, because there is a lot I would like to remove from this article (for example, the section on least-squares which seems completely out of place here). I do understand how these topics are related (projecting orthogonally on functions of a Hilbert space) and how one can see Fourier series as a special case of regression, but I don't think it is a central argument. This article is sufficiently long without straying to side-paths.

Although I think a lot should be remove, I also believe that many pictures should be included, in the example section for instance. Please tell me what you think.

Regards,
[[User:Deimos 28|Deimos 28]] 19:19, 11 March 2006 (UTC)

::This article is being thrown around, way the hell too fast. First of all, I think the definition of a fourier series should come first and formost after the initial intro. Second of all, I think people should keep in mind that this article needs to be ''understandable'', so that it can be read by people who are not mathematicians, and '''especially''' people who are not famliar with fourier series. Removing links to the year 18-whenever is fine, but removing links to people, is not fine. I find this page very cumbersome as a reference now. [[User:Fresheneesz|Fresheneesz]] 20:43, 11 March 2006 (UTC)

:::*I do not think it is a good idea to start straight off with the mathematical definition. The non-mathematically inclined will probably just need an overview which is what the introduction should do. I completely agree that the current introduction is ill-suited.
:::*That is the biggest problem. I think the definitions given should be mathematically exact. How to present them is a difficult problem. The definition I gave only requires under-graduate mathematical knowledge. People who are not familiar with Fourier series probably do not have the mathematical background to understand the definition. For them, the general idea (missing from the article), the applications and the history are probably sufficient. I think the mathematical parts of this article should be as accurate as possible.
:::*I only removed the links to the people who were linked twice. -- [[User:Deimos 28|Deimos 28]] 20:59, 11 March 2006 (UTC)

:::: I think these edits are moving way too fast. The problem with that is that people can't see the resemblance between the old article and the new article very well. Its very hard to see what has been improved, and what has been made less readible. I for one think that the new definition of fourier series is very VERY unreadable, and needs to be written so that it is understandable. Please explain all your changes on this talk page. [[User:Fresheneesz|Fresheneesz]] 06:23, 12 March 2006 (UTC)

::::: I'll try and improve the readability of the definition. My edits are mainly:
:::::*Change the introduction, with links to relevant applications (more links are more than welcome!). I think the new version by user Fresheneesz is an improvement on mine.
:::::*Grouping all mathematical sections to one section, so that the non-mathematical reader can skip this part. I saw that user Fresheneesz undid that part of my edits.
:::::*Grouping all examples in one section and '''thinking about''' putting in pictures (the most visible part of my edit ;))
:::::*Simplify the definition of ''f'' in the first example. I saw user Fresheneesz mostly undid that. I think the current definition is unecessarily complicated: it is equivcalent (and simpler) to say ''f'' is <math>2\pi</math>-periodic and <math>f(x)=x</math> for all ''x'' is <math>[-\pi,\pi]</math>.
:::::*Uniform notation (eg. <math>a_n, b_n</math> (like in Fourier's 1807 article) instead of <math>A_n, B_n</math>, <math>c_n(f)</math> instead of <math>F_n</math>,...) Incidentaly, I saw that user Fresheneesz thought it would be better to have to sets of notations instead of just one and reverted to <math>A_n, B_n</math> for '''one part''' of the article. Now I have nothing against using capitals against small script but I find the double notation confusing.
:::::*Adding and removing links. I added links on mathematical notions necessary to understand the formal definition and links to Fourier's original article. I removed the links that were made twice and the links to article which do not bring much to the understanding of Fourier series and their application. For example, the links to the years 18.. and the links to the mathematics or the physics article.
:::::I think presenting the complex coefficients after the real ones makes things harder to explain. Maybe one could quickly give the definition with <math>a_n</math> and <math>b_n</math> at the beginning of the article and then derive its expression in the mathematical part.
:::::Regards, --[[User:Deimos 28|Deimos 28]] 10:16, 12 March 2006 (UTC)


::Grouping the mathematical sections might not be such a good idea, given that this is an article about a mathematical construct. A good organization of sections is a small but important part of a good page, and by clumping all the math in one spot, you limit the ability to put together a good and intuitively flowing layout. 
::Also, I tried to incorporate your edits, but some parts I just copy reverted from the old edit because I thought it was more understandable. But I agree consistant notation should be used, I just missed the difference in capitolization, sorry about that. 
::About having complex fourier series first: one convention wikipedia uses is to have more complicated parts written first. And the complex definition of a fourier series is most definately more complex, pardon the pun. Not only that, but apparently the non-complex part was invented first, more reason for it to be ontop. Not only that, but ''most'' people, I think, learn about real fourier series first (or only). 
::Lastly, I fully disagree with having examples before the definition. People reading this won't know where the examples are comming from. Examples are useless without context - and it doesn't make sense to provide the context after the examples. Part of the real definition is in the intro, but .. well maybe it would be fine to have it like this. But since weird things often happen with a.0, its definition should be included there as well (for example if you try to derive a.0 exactly like a.1, sometimes you get divide by zero errors or something). 
::[[User:Fresheneesz|Fresheneesz]] 22:34, 12 March 2006 (UTC)

::Also, You said this is simpler: "Let f be periodic of period 2π, with f(x) = x for x from -π to π". Although it might be simpler, I think its harder to understand. But since lots and lots of fourier series problems are explained this way, it might be very helpful to explain what it means to define a non-periodic function as being with a period of 2&pi;. [[User:Fresheneesz|Fresheneesz]] 22:42, 12 March 2006 (UTC)

::One thing I dislike about part of the definition for real fourier series being at the top, and the rest being later is that I think people assume that the definition will be all in one place. They might come here, look at the definition at the top, and find that it doesn't include what they're looking for. They might just assume its not on the page. [[User:Fresheneesz|Fresheneesz]] 22:59, 12 March 2006 (UTC)

:::When creating a definition, it is better to make the definition as general and thus as simple as you possibly can, so that it encompasses as many special cases as possible.  That way you end up with many fewer definitions over all, and you don't need a separate definition to cover every little nit-picky special case that comes along.  In terms of this article on FS, the article should start by defining the FS in a very mathematically precise fashion, but with explanation in well-written prose so that non-experts can follow it.  The most general and not coincidentally the simplest definition of the FS is in terms of complex exponentials and for a function of ''arbitrary'' period ''T'' or ''L'', not the special case of a function that just so happens to have by some unknown miracle the period of 2&pi; (which is no function that I have ever seen in the real world).

:::Once the article establishes the ''general'' case, it then becomes possible, and quite useful, to ''use'' the general definition to illustrate a handful of interesting and relevant special cases, such as the sine/cosine form or a function of period 2&pi; or whatever strikes your fancy.  That is the beauty of starting general and then becoming specific.  You can actually help people to understand the world around them by first understanding ''general'' principles that have universal truth, and then applying those principles to specific situations.

:::This approach also prevents people from getting into trouble when they understand only a portion of a larger topic.  If you teach people by explaining special cases, then they will either not know what to do when they encounter a new unforseen situation, or worse yet, they will do the ''wrong'' thing because they will extrapolate from one specific situation to a different situation where some crucial assumption is no longer valid.  If, on the other hand, you present the information from first principles and for the most general case possible, then people will have the knowledge and skills they need to approach ''any'' situation that might come along, because they will know how the assumptions they are making will impact the specific details.

:::Okay, I will get off my soap box now.

::: -- [[User:Metacomet|Metacomet]] 23:33, 12 March 2006 (UTC)

::::I'm sorry if I seem not to change my position a lot, but this presentation is something I actually thought a lot about, so please forgive me for defending it so vigorously... Fresheneesz, I understand what you are saying: it is true that one should give the mathematical definition before the example. This is why the last lines of the introduction give the formula of the real Fourier coefficients which is sufficient to understand the example. I think it is a bit misleading to call this a definition: it's just a formula. The definition is given in the mathematical section, therefore I think this section should be renamed. The reason I didn't put the mathematical definition before the example is that I agree with Metacomet in saying that a definition should be stated once and for all in a general (and therefore abstract, i.e. abstracted from context) way. Such a formalism is probably not essential for someone with basic mathematical knowledge, which is why I chose to put it later on in the article. When you say "But since weird things often happen with a.0", you mean in computing <math>a_0</math>, because its definition is the same (given that <math>\cos(0.x)=1</math>). This is a computation problem, not a definition problem. I do not think this article should be a Fourier series how-to. I think it should provide an exact definition with the most important properties, give an idea of the applications, a few examples and a historical insight on the method. It is not a textbook. 

::::Fourier did not have all the mathematical tools that we now use and therefore could not formulate his definitions as precisely as we can today. Therefore, saying that Fourier started with the real coefficients is both true and irrelevant. If one just wants a ready-for-use formula like that of the introduction, then the real Fourier coefficients are probably simpler. However, if we want to know where this formula comes from, we see that it is the result of the very simple idea of projecting a function. The complex Fourier coefficients are simpler than the real ones if we see them as the coefficients of an orthogonal projection on the orthonormal basis <math>\{t\mapsto e^{int},n\in\mathbb{Z}\}</math>, which is the most general way of seeing Fourier series (I think).

::::This is also why the "<math>2\pi</math>-period version" is in fact the most natural one. It is beside the point that the functions can be of any period ''T'' because the "generality" lies not in the form of the function but rather in the method used (orthogonal projection, as stated quite rightfully in the least-squares section I would like to remove). Besides, the formula for other periods than <math>2\pi</math> is given in the article. Seeing Fourier series as the decomposition of a function on a basis of orthonormal functions allows a very simple expression and the Parseval equality can then be seen as a special case of Pythagoras' theorem. In other words, instead of developping a whole new theory, we simply apply what we know about orthonormal spaces to the special case of L² functions.

::::[[User:Deimos 28|Deimos 28]] 00:39, 13 March 2006 (UTC)

::I fully agree with Metacomet, especially on the general period of a fourier series. It has been said that a period of 2&pi; is more natural, and so it may be. But until either I understand why that is, or until its explained in this article, I disagree with the magical 2&pi; period popping up out of nowhere. And ideally, the complex definition should be the simplest and easiest, but unfortunately most people (including me) don't have the background in complex mathematics neccessary to understand it. So real fourier series needs to be given a first up.

::"I do not think this article should be a Fourier series how-to."
::Well, if a_0 isn't explicitely written on this page, a note should be made about the very potential divide by zero error you might get if you forget the 0 case. Most teaching completely ignores handling 0's correctly, and so far too many people will simply forget it when handling 0's matters.
::Although wikipedia is not meant to be a textbook, it should give all the information to be a perfect summary (ideally). It doesn't need to baby-step through everything, but it should give all the non-redundant information there is on fourier series, and enough links, so that one could learn straight from wikipedia if they were persistant. An exact definition, and examples is *not* sufficient for this page. Wikipedia is about compiling all human knowlege, not just part of it. That doesn't mean we have to fluff it up like a text book does, but we need to put all the information down.  [[User:Fresheneesz|Fresheneesz]] 10:28, 13 March 2006 (UTC)

== Convolution point-of-view==

I think the whole article could be made immensely simpler to understand if it were presented from a convolution point-of-view.  -- [[User:Metacomet|Metacomet]] 07:28, 12 March 2006 (UTC)

:I do not know this point of view: could you please explain? I think the geometrical interpretation (a projection on a space of trigonometric functions) is easy to understand, but I agree there are a lot of ways to look at Fourier series. [[User:Deimos 28|Deimos 28]] 10:16, 12 March 2006 (UTC)

A periodic function in the [[time-domain]] can be represented as the [[convolution]] of an [[Dirac comb|infinite unit impulse train]] with simple duration-limited [[kernel]] function, which represents a single period of the original function (which could be ''any'' period, as long as the original function repeats the kernel at interger multiples of a fundamental period ''T'', for ''all'' integers from negative infinity to positive infinity).  

One advantage here is that there is absolutely no restriction on the lower or upper bounds of the duration over which the kernel function is nonzero.  In other words, you don't have to restrict the analysis to the interval [-pi, pi] or [-L, L].  The convolution approach is completely general.

So what?  Well, convolution in the time domain corresponds to simple multiplication in the [[frequency domain]].  So the original function can be represented in the frequency domain as the product of another infinite unit impulse train times the [[Fourier Transform]] of the kernel function.  This result is nothing other than a [[sampling]] operation (in frequency), so the conclusion is that the frequency domain representation of the function is a discrete set of infinitely many values, and the interpretation is that each value in the set represents the coefficient of a [[complex number|complex]] sinusoid at a frequency that is an integer multiple of the fundamental frequency, which is 1/''T''.

Another advantage to this approach is that it makes explicit the connection between Fourier analysis and wave [[harmonics]], which are extremely important in physics, electrical engineering, [[acoustics]], and [[music theory]].  

One disadvantage of this approach is that it defines the Fourier Series in terms of the Fourier Transform rather than as an independent technique in its own right.  But of course that makes good sense, because the FT is a generalization of the FS.  So it shouldn't be a surprise that you can look at the FS as a special case of the more general FT.  On the other hand,  the historical development began with the FS, and it was only later that the idea developed into the FT (I think).

The other disadvantage, as I think about it, is that it is probably too deep and too advanced for an introductory article on the FS.  It might, however, make an interesting article that would complement the articles on Fourier analysis and signal processing.

-- [[User:Metacomet|Metacomet]] 20:11, 12 March 2006 (UTC)

:I think the idea of Fourier series comes more naturally than that of the Fourier transform. Indeed, Fourier series lend themselves to a clear geometrical interpretation (an orthogonal projection) whereas one looses this insight with Fourier transforms. Of course, this is just my point of view (I like to have a geometrical picture of the mathematical beings I deal with). Moreover, the Dirac impulse is a distribution, so if you adopt your point of view, as well as Fourier transforms, you add the theory of distributions to the prerequisits to this article. However, it is a good way of generalising Fourier series (Laplace transforms first, and the integral transformations with any kernel). By the way, what do you think of my suggestion to remove some sections of this article (see below)? Regards, --[[User:Deimos 28|Deimos 28]] 20:20, 12 March 2006 (UTC)

Thanks for your constructive and polite feedback.  That is certainly refreshing on WP.   I think you have made a strong case that this approach is, as I also suggested, too advanced for the baseline article on FS.  -- [[User:Metacomet|Metacomet]] 20:30, 12 March 2006 (UTC)

I will need some time to review your suggestions before expressing an opinion.  I don't like to judge things that I haven't given a fair shake one way or the other.  -- [[User:Metacomet|Metacomet]] 20:30, 12 March 2006 (UTC)

== Parts to remove ==

Hi,

I think the following parts should be removed from the article:
*'''Determination of the coefficients and the method of least squares''' (see previous comments by me to know why I think it should be deleted)
*'''Orthogonality''' is useless I think: in the first paragraph, I state that the functions <math>x\mapsto e^{inx}</math> form an orthonormal basis. A proof of this is, I think, beyond the scope of this article because if we start making proofs everywhere, this article will grow to be a hundred pages long... Moreover, it introduces the concept of distributions, which maybe is not necessary to understand Fourier series.
*'''Some positive consequences of the homomorphism properties of exp''' deals with the Fourier transform and not Fourier series.
*'''General formulation''' looks a lot like what is in the mathematical formulation paragraph.
I also think the plan of this article needs some work (fourteen sections seems a lot...)

Regards,
--
[[User:Deimos 28|Deimos 28]] 21:47, 11 March 2006 (UTC)


== Complex exponentials are easier ==

Why does the article define the FS in terms of sines and cosines?  Although it may seem a bit odd, I think it is ''much'' easier to understand this topic if you start by defining the FS in terms of complex exponentials, and then use that to derive the sine and cosine form.  The exponential form also has the advantage that it makes explicit the connection between the FS and the [[Fourier transform]], and sets-up the transition to the more general theory much more quickly and much more easily. -- [[User:Metacomet|Metacomet]] 23:03, 12 March 2006 (UTC)

:I completely agree with you, but in order to do things properly, defining the Fourier series with complex exponentials requires a bit of math. For non-mathematically inclined people, it might be easier to just have the formula they will use the most often (the real coefficients) and they can refer to the mathematical section if they so wish. What do you think? [[User:Deimos 28|Deimos 28]] 00:41, 13 March 2006 (UTC)

You have a point, but I think you also have to write the article with the assumption that the majority of readers will have a certain minimum baseline of understanding, while also providing the means for those without the background to improve their understanding with some, perhaps not too strenuous, level of effort.  So, for example, it might make sense to define the FS using complex exponentials, but to also include [[Euler's formula]] immediately after the definition, as well as a link to the article on Euler's formula.  That way, you get the best of all worlds.  You have a general and sophisticated definition, you do not bore or insult the sophisticated reader, you provide a simple refresher for the reader who used to know this stuff but is a little bit rusty, and you provide a roadmap for the novice reader who is willing to make a small effort to learn something new. -- [[User:Metacomet|Metacomet]] 00:51, 13 March 2006 (UTC)

::Haha! Nice one... Unfortunately, I was referring to the theory of Hilbert spaces rather than the Euler formula and I think putting ''Hilbert'' or <math>L^2</math>-''space'' right at the beginning of the article might scare off the unsuspecting reader... I think the main problem is that Fourier series can be viewed in many different ways, not all of them as fertile from the mathematical point of view and therefore it is hard to find something that will both suit the physicists and the mathematicians... Going to bed now, sorry... -- [[User:Deimos 28|Deimos 28]] 00:59, 13 March 2006 (UTC)

The only thing that scares people off, IMO, is when articles use terminology without bothering to define it, and without providing enough support so that the uninitiated can get their foot in the door.  -- [[User:Metacomet|Metacomet]] 01:04, 13 March 2006 (UTC)

::Sure, I agree, and I'm unsure about what I did. Please tell me if you think sufficient support is provided for the mathematical definition. I '''really''' have to go to bed now, waaaaay too late... -- [[User:Deimos 28|Deimos 28]] 01:08, 13 March 2006 (UTC)

Not to worry, I wasn't saying (or implying) that ''you'' were guilty of such a hideous offense.  I was just trying to say that it is okay to include mathematical rigor as long as you also help us mere mortals scramble up the learning curve.

Oh, by the way, go to bed already!  

And check back tomorrow to see all of the massive editing that I have done to the article.

Just kidding!

-- [[User:Metacomet|Metacomet]] 01:13, 13 March 2006 (UTC)


== Proposed definition ==

The following definition, in my opinion, is much simpler, much more general, and much easier to understand than the way the article is currently written:

:Given a piecewise continuous, periodic function ''x''(''t'') of period ''T'', the Fourier series representation is

::<math> x(t) = \sum_{n=-\infty}^{\infty} X[n] \cdot e^{i 2 \pi nt/T} </math>

:where

::* <math> X[n] =  {1 \over T} \int_{-T/2}^{T/2} x(t) \cdot e^{-i 2 \pi nt/T}  \cdot dt </math>

:::is the ''n''th Fourier coefficient of ''x'',

::* <math> i \equiv \sqrt{-1} </math> 

:::is the [[imaginary unit]], and

::* <math> e^{i\phi} = \cos(\phi) + i \sin(\phi) \, </math>

:::is [[Euler's formula]].

It is also symetric, requires only one definition for the coefficients (instead of three), and looks eerily similar to the Fourier Transform (no accident that).

-- [[User:Metacomet|Metacomet]] 03:46, 13 March 2006 (UTC)

:I agree. By the way, nice work on the article. I wouldn't myself use the <math>X_t</math> notation, as it looks a bit "physical", and that must be bad ;) I am going to do the relevant changes to the article. Cheers, --[[User:Deimos 28|Deimos 28]] 08:56, 13 March 2006 (UTC)

A bit physical, eh?  Guilty as charged, your honor.  I tried to slip that one by all of you theoretical guys, but I guess you are too sharp!  ;-)   Can you tell that I learned this stuff in courses on system dynamics and signal processing?  -- [[User:Metacomet|Metacomet]] 13:55, 13 March 2006 (UTC)

:Yes, I was suspecting that! ;) I suppose you see from the sheer volume of the notations why I first chose to take <math>T=\pi</math>... What do you reckon should be done to the article now? -- [[User:Deimos 28|Deimos 28]] 14:34, 13 March 2006 (UTC)

I don't know.  You have made some good improvements, but the article overall still needs a lot more work to bring it up to an acceptable standard of quality.  No offense intended, I know you have said that you have given the overall approach to this article a lot of thought, but please realize that you are not the only one who has done so.  Try to listen more carefully to what others are saying, and maybe question some of your own assumptions about the best way to go.  No one person has all of the answers.  -- [[User:Metacomet|Metacomet]] 04:35, 14 March 2006 (UTC)

== Edits of Monday, March 13th 2006 ==

Hi all,

I've more or less finished what I had intended doing to this article. The mathematical definitions are accurate and "period-independant". There could be more examples perhaps, but I'm too lazy to write them. I removed the final sections which I considered to be irrelevant (see above) and I really do hope I haven't upseted anybody doing so. Please let me know what you think. Oh, and one other thing: I'm not sure that the second and third examples are best suited. They don't really illustrate the technique I think. Maybe a better idea would be to compute the Fourier series of the step function (<math>f(x)=-1\mbox{ on }[-\pi,0]</math>, <math>f(x)=1\mbox{ on }[0,\pi]</math> and ''f'' <math>2\pi</math>-periodic) to show that the extrema of the partial sums don't converge to what we would expect (Gibbs phenomenon illustration).

Regards,
--[[User:Deimos 28|Deimos 28]] 11:02, 13 March 2006 (UTC)


''I'm afraid the page is still a shambles. Whose bright idea was it to put in a definition of Fourier series 3/4 of the way down the page?! There are too many alternative definitions. And it is still inconsistent eg the a_0 vs a_0/2 question. PCM.''

:*The complete definition is after Fourier's original definition and the two examples because I figured starting straight off with Hilbert spaces and orthonormal systems wasn't the best way to go, as the vast majority of people who use Fourier series do not know (and do not need to know) how to derive them rigorously. The examples are quite understandable with the definition given just before them and for those wishing a more complete treatment, a rigorous definition is also available. I myself could very well do without Fourier's original definition, but I think it is interesting from a historical point of view and quicker to formulate. What are the "alternative definitions" you speak of? I only see the original historical one and the one used in current textbooks. Something I missed out?
:*What is inconsistent? Fourier's original formulation did not put <math>\frac{a_0}{2}</math> and there's nothing much I can do about that since his dead. However, as can be clearly seen in the mathematical part, unless we define <math>a_0</math> separately from the other <math>a_n</math> coefficients, it is necessary to seperate it from the other terms (in the sum <math>\sum_{n\in\mathbb{Z}}e^{inz}</math>, it is the only term to only appear once - therefore it has to be divided by two).
:*I'm afraid I don't see how these two things might make the article "a shambles", but I probably misunderstood what you said. By the way, please sign your comments with four tildes like this: <nowiki>~~~~</nowiki>. Regards, -- [[User:Deimos 28|Deimos 28]] 17:56, 13 March 2006 (UTC)

''Hi Deimos, there is an inconsistency in the equation after the heading Real Fourier coefficients where the first term is a_0 whereas at the top of the page it is a_0/2.  Overall, the page is too long and unnecessarily complicated. Unfortunately this applies to many of the Wikipedia mathematics pages. But I withdraw the word shambles :)  [[User:Paul Matthews|Paul Matthews]] 14:47, 14 March 2006 (UTC)''

:Hi Paul! No, sorry, there isn't: the section you're referring to starts with: "Historically, the original series used by Fourier to represent a function" this means it is a historical definition. The last formula in this section is the one to remember. But I agree the presentation is misleading and confusing. I think I should stick to what I said in the plan and first define the Fourier coefficients, then prove that they converge, not all at the same time. But it turns out it's not as easy as I thought - I have to think.
:Thank you very much for withdrawing the word "shambles", by the way: it doesn't look it, but it actually means a lot to me, as I spent a hell of a lot of time on this stupid article. I said I had finished my edits, but that's simply wrong: every time I take another look at the article, I see something else to correct, and as I now have written a lot of it myself, the only person I can blame is myself, which is very frustrating. Regards, -- [[User:Deimos 28|Deimos 28]] 16:05, 14 March 2006 (UTC)

''OK, you've removed the eqn I objected to, thanks. I just fixed a pi that should have been a T.''

==Period 2&pi; is absurd, arbitrary, and contrived ==

The definition of FS as currently shown in this article is absurd, arbitrary, and contrived.  It applies to functions that are periodic with period 2&pi;.  What should I do if I have a function that has a period of 7.925 picoseconds?  Or a wavelength of 632.8 nanometers?  The truth is that I have ''never'' encountered a function that has a period of 2&pi; in my life.  The fact is that limiting the definition to functions of period 2&pi; is artificial, of exceedingly limited utility (as in none), confusing, and totally unnecessary.  

I strongly recommend that we change all of the definitions of the FS in the article so that the function to be represented is either of period ''L'' for functions of argument ''x'', or of period ''T'' for functions of argument ''t''.

If no one strenuously objects to this change, I will make the necessary revisions in the near future.

-- [[User:Metacomet|Metacomet]] 05:18, 14 March 2006 (UTC)

: I believe dealing with period T makes things unnecessarily complicated. The true nature of fourier series is visible for T=2&pi;, and I don't see it necessary to start off with the most general case. [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 05:20, 14 March 2006 (UTC)

I disagree:  I don't think it makes things unnecessarily complicated.  To the contrary, I think it is unnecessarily abstruse to avoid allowing people to see how to deal with real functions that have real periods not 2&pi;.  Furthermore, it may be visible to you, and it may be visible to many people who are interested in FS as a theory, but it is absolutely not visible to me, and I would guess that it is not easily visible to people who use FS in real-world applications.  To me, the variable ''t'' represents time, and the period ''T'' is therefore also in the dimension of time.  As a result, ''T'' cannot equal 2&pi;, because 2&pi; is a pure number (and irrational at that, which only makes things worse).  It could equal 6.28318... seconds, but that would merely be a strange coincidence and it would have no particular meaning or significance.  -- [[User:Metacomet|Metacomet]] 05:37, 14 March 2006 (UTC)

You know, it is actually a bit ironic, because I was just reading a [http://www.memagazine.org/contents/current/webonly/wex80905.html web-page] that went into a great deal of detail about how Fourier was the first person to develop the idea that dimensions and units of measure are absolutely essential aspects of understanding the real world in terms of mathematical equations.  The site also referred to him as "the father of modern engineering" in part because of his very strong and at the time controversial stance on this issue.  -- [[User:Metacomet|Metacomet]] 05:42, 14 March 2006 (UTC)

A few excerpts:

:Fourier's contributions, many of which are presented in ''The Analytical Theory of Heat'' (1822), include:  The original and still globally accepted view of dimensional homogeneity—the view that natural phenomena can be rigorously described only by equations that are dimensionally homogeneous—i.e. equations that are dimensionally identical.

:'''Fourier's view of homogeneity expressed in modern terms'''

:White [3] expresses Fourier's view of homogeneity in modern terms:

::(Dimensional homogeneity) is almost a self-evident axiom in physics. . . (It) can be stated as follows:

::If an equation truly expresses a proper relationship between variables in a physical process, it will be dimensionally homogeneous; i.e. each of its additive terms will have the same dimension.

::In other words, scientifically rigorous equations cannot call for the addition or subtraction of terms that have different dimension. For example, ft./sec. cannot be added to lbs./ft.


:::::Fourier series is easier to teach on 2&pi; period. But OK, it is not me doing the work here, so if you really want it on arbitrary period straight from the beginning, I can't object. [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 05:59, 14 March 2006 (UTC)

Rhetorical question:  why is it that no one is willing to have an honest and meaningful discussion about ''anything'' on WP?  Either people resort to viscious personal attacks, or they totally withdraw from the conversation instead of trying to formulate a valid argument in support of their position.  Or, better yet, to try to work out a win-win situation that will make everyone happy ''and'' improve the article at the same time?  I am just about at the point where I think I will walk away.  Screw it.  It is simply not worth the time or effort that it takes to try to resolve important issues.  And everyone is going off in a different direction with no consensus whatsoever about what distinguishes a good article from an also-ran.

To be frank, I think this article ''sucks''.  But I don't think I have the time or the energy to worry about it anymore.  Oh well

-- [[User:Metacomet|Metacomet]] 06:17, 14 March 2006 (UTC)

: If you want an honest discussion why do you behave like a pissed-off nuisance? See the title of the section heading you put. You came here you assumed you were damn right, and if people happen to disagree with you, you just turn mad.  Eh. :) [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 06:19, 14 March 2006 (UTC)

You disagree but you never say why you disagree, and you never provide any kind of a rationale for your point of view other than that it is your point of view.  So it is impossible to have any kind of discussion that will lead anywhere other than that we disagree.  I am not pissed off at you because you disagree with me, I am pissed off because there is no discussion or debate, just assertion of differing opinions.   I made the wording of the header strong because I wanted to take a strong position.  Also, that header is my honest opinion.  I guess I should hold back and keep my opinions to myself.   I have been watching this page without comment for many weeks, while I formulated some opinions about things.  In the last few days, I finally decided to get involved.  Now I am starting to regret it (again).  -- [[User:Metacomet|Metacomet]] 06:31, 14 March 2006 (UTC)

Oleg, I don't want to do this.  You seem like a very reasonable person, and very bright.  I know you have made and continue to make a lot of excellent contributions to WP.  You and I don't see eye-to-eye on certain things, and there is nothing wrong with that.  Sometimes I come across too strong when I am expressing my opinion.  Maybe it is the way I was taught to approach a discussion of a controversial subject.  Maybe it works okay out there when I am face-to-face, but it doesn't translate well to this form of communication.  Anyway, I am going to back off for now.

If I offended you or came on too strong, then I apologize.  It was not my intention to attack you or anyone else personally.  I just like to assert my opinions and ideas as strongly as possible, and I guess it comes out the wrong way in written form.  Sorry.

-- [[User:Metacomet|Metacomet]] 06:37, 14 March 2006 (UTC)

:I had the impression that I had changed the definition of the Fourier coefficients to <math>2T</math>-periodic functions, but maybe I am mistaken. What passage of the text are you referring to? [[User:Deimos 28|Deimos 28]] 08:14, 14 March 2006 (UTC)


'''The Definitive 2π-period Guide'''

I'm not sure everybody understands why some of us refer to <math>2\pi</math> as the ''natural'' period. In advance, I'm '''really sorry''' if all of this is perfectly clear to everybody -- I apologize and cover my head with ashes and all that. Well anyway, saying that <math>2\pi</math> is the natural period is not correct -- well not quite, that is. <math>2\pi</math> is the natural [[angular frequency]]. What have angles to do with Fourier series? Well this is what I am going to (try to) explain and it should give one of the possible justifications for using <math>2\pi</math> instead ''T''. Of course, as can now be seen in the article in the section about Fourier coefficients, another very simple justification is the fact that setting <math>T=\pi</math> in the formula makes it much more compact. How about a more ''physical'' reason?

The functions <math>t\mapsto e^{it}</math> can be viewed as a [[rotation]] on the [[unit circle]]. Therefore, Fourier series are a way of decomposing a movement as a sum of rotations of different radii. If you will, the original function (with, in general, complex values) can be seen as describing a movement on a plane (if the function takes only real values, then the movement is restricted to a line, which can be the case when something [[Harmonic oscillator|oscillates]] for example). This movement can be described by a vector of <math>\mathbb{R}^2</math>: <math>f(t)=\left[u(t),v(t)\right]</math> or, equivalently, in <math>\mathbb{C}</math> by <math>f(t)=u(t)+iv(t)</math> because there exists a bijection between <math>\mathbb{R}^2</math> and <math>\mathbb{C}</math>. In effect, this is the same as describing the movement as a sum of translations. What Fourier series do is that they present this movement as a sum of rotations instead. Why? Because rotations are much easier to manipulate when dealing with periodic movements. The Fourier transform uses this idea and generalizes it to non-periodic movements (or movements of "infinite" period, whatever that might mean) but that's another story.

In what is often called the '''time domain''', the function is simply <math>t\mapsto f(t)</math>. Now if we are wiewing it as a sum rotations, having the angular frequency (i.e. the number of radians per second) as a variable makes much more sense (well at least ''physically'' -- not that anybody would care about that ;)). Therefore, we map <math>f:t\mapsto g\left(\frac{2\pi}{T}t\right)</math>, where ''g'' is the sum of rotations. In other words, we project ''f'' on a space of rotations <math>e_n:t\mapsto e^{i\omega_n t}</math>, where <math>\omega_n=\frac{2\pi}{T}n</math>. If you choose <math>T=2\pi</math>, then <math>\omega_n</math> is simply the number of ''turns'' (or ''revolutions'' if you want to sound smart) per second. In other words, <math>e_n</math> would then be the ''n''-turns per second rotation. But if you choose another period ''T'', well then everything's a mess. It's the same, but it's harder to interpret "physically", and hence it's a mess :). OK so I chose <math>T=\pi</math> which makes it the number of '''half''' turns per second, big deal. The reason is that I don't like putting fractions everywhere and as we integrate arround 0, we divide the period by two every time so it makes life easier if the period is already the double of something. Actually, I'm surprised no one came up the idea of centering arround <math>e</math>, say -- just kidding.

Now don't get me wrong here: if ''f'' is a function of period ''2T'', then <math>x\mapsto f\left(\frac{T}{\pi}x\right)</math> is <math>2\pi</math>-periodic. Therefore, both formulations (i.e. with ''2T'' and with <math>2\pi</math>) are completely equivalent. For me it's just the same as choosing another number than ''e'' for basis for an exponential function, but nobody thinks twice about that. In other words, I don't give a shit one way or the other, and I don't understand what all the fuss is about but there are very good reasons for choosing <math>T=\pi</math> in the formula. If it can make people happy, I even propose myself to change the period to anything anybody wants, single-handedly ;).

Regards,
--[[User:Deimos 28|Deimos 28]] 09:29, 14 March 2006 (UTC)

''I agree with Oleg and Deimos. <math>2 \pi</math> is sensible, simple and natural. Have you never worked with a function in polar coordinates? Or a function defined on a circle? This is one of the key applications of Fourier series, and the natural period is <math>2 \pi</math>. An arbitrary period is just a rescaling. It is not something to get upset about! [[User:Paul Matthews|Paul Matthews]] 14:57, 14 March 2006 (UTC)''

:Deimos and Paul -- Thank you for presenting a well-written and detailed explanation for your rationale on 2&pi;.  I more or less understand a lot of what you are saying (not all of it mind you but a lot of it).  The problem that I have is that if you present FS using 2&pi; as the period of the function, then you immediately obscure or cover over all of the very important and very interesting issues that you have just discussed.  The whole problem with using 2&pi; as the period of the function is that it so overly simplifies the topic that, although the essence of FS is still preserved, very little other than the essence is preserved.  Moreover, there are a number of issues related to engineering and physics that are impossible to discuss when you normalize away all of the interesting dimensional information, ugly things like nanometers and picoseconds and GigaHertz and yes radians per second.  So it may be just fine for the core purpose of explaining the essence of FS, but it is absolutely horrible for explaining anything other than the essence of FS, things that are very closely related and very important.  -- [[User:Metacomet|Metacomet]] 15:08, 14 March 2006 (UTC)

:BTW, with a bit of cleaning up, a lot of what Deimos just wrote as explanation would make an excellent section either in this article or in a related article on Fourier decomposition.  -- [[User:Metacomet|Metacomet]] 15:08, 14 March 2006 (UTC)

::Thanks a lots, Metacomet! Please feel free to ask me about anything I explained badly because I think that what I said shouldn't be too complicated if I explain it correctly. Including my explanation in the article might be a good idea, but it'll set another set of problems (the plan, for example and the length because it would be adding a substantial amount to this already humongus article) I '''will''' convert you to the <math>2\pi</math>-faith, if it's the last thing I do! ;) Cheers, -- [[User:Deimos 28|Deimos 28]] 16:22, 14 March 2006 (UTC)

== What's still missing? ==

Hi!

Just wanted to know: what do you think's still missing from the article? I for one consider it finished, but if you disagree, please say what you think should be added/removed/changed. If you agree, maybe we could remove the tag:

<nowiki>{{attention_see_talk}}</nowiki>

Cheers,
--
[[User:Deimos 28|Deimos 28]] 10:48, 16 March 2006 (UTC)

''Nothing is missing. It is much improved. In my opinion there is a bit too much stuff already (personally I would drop the wave eqn, Plancherel and the cumbersome eqn in the complex section) so please dont add to it. Yes, I agree, remove the needs_attention tag. [[User:Paul Matthews|Paul Matthews]] 13:10, 16 March 2006 (UTC)''

:Thanks for your reply. I too would drop the wave equations, as I've said before, but maybe put a better example in its place. Plancherel's equation is pretty important - as is Bessel's equation, which is not in the article. I guess it might be sufficient to link to external articles. What cumbersome equation(s) do you refer to? -- [[User:Deimos 28|Deimos 28]] 16:24, 16 March 2006 (UTC)

::It's not that anything is missing, and I don't mean to burst your bubble, but in my opinion, except for the introductory section (the three paragraphs above the Table of Contents) and the Examples, the article is virtually unreadable for anyone without a Ph.D. in mathematics.  Sorry to be blunt, but I am tired of beating around the bush.  Like I said, that is my opinion, for what it's worth (which probably isn't much).  -- [[User:Metacomet|Metacomet]] 20:29, 16 March 2006 (UTC)

:::I don't think it's particularly offensive to say that the article is unreadable to the nonmathematician; there's certainly no need for 2 1/2 sentences worth of apology.  :)  I don't have a Ph.D. in math, but could probably figure out most of it.  The introduction and the first three sections are fairly readable.  The mathematical derivation section is harder, with the stuff about Hilbert spaces and L^2(mu); it would be nice if that could be simplified or moved, because the material on complex fourier expansions and properties of the coefficients is useful to the non-PhD, and you shouldn't need to know what an orthonormal basis is to understand it.  [[User:Pfalstad|Pfalstad]] 21:22, 16 March 2006 (UTC)

::::The properties are quite understandable without the <math>L^2(\mu)</math>-spaces section. However, the mathematical basis of this theory '''is''' <math>L^2(\mu)</math>-spaces, which is why I mentioned them. The whole plan of this article was designed so that people didn't have to read the whole article to get what they wanted. If you go into the math section, then you're bound to find math: I don't see anything unusual about that.
::::Sure, the article (well, the math section at least) is unreadable if you don't have enough math. Mind, you don't need to be an expert: I don't have a phD in math for instance. I'm not sure it's a good idea to turn this article into some kind of hand-wavy explanation on Fourier series. I didn't write the article to please anybody: I just wrote what I know to be true, i.e. the mathematical definitions. If I can't find the exact mathematical definition of a mathematical being in the Wikipedia, then I think the Wikipedia is useless for me and that I misunderstood its purpose. I don't necessarily want to buy a textbook every time I want to learn about something. Sometimes just having a well-written Wikipedia article would be sufficient. If you read the article on wavelets for example, you'll see that it's much less accessible than this one, but it's a good article because apart from being a good reference, you actually '''learn''' a lot from it.
::::In this Fourier series article you can find a practical definition which can be used in most applications '''and''' a derivation of this definition (i.e. where it comes from). Both seem to me to be indispensable and I think it's a bit selfish to imply that this article will only be read by non-mathematicians and that therefore the more technical sections should be removed. I believe that there should be something in there for everyone to enjoy, the aim being that everyone can learn a little something from it.
::::Regards,
::::--
::::[[User:Deimos 28|Deimos 28]] 22:42, 16 March 2006 (UTC)

:::If it can't be simplified, that's fine.  I thought I'd seen derivations of fourier series in physics books that didn't assume linear algebra, but I'm not sure how rigorous they were.  I'm not saying we should take those ideas out of the article, because they're important.  Mainly, I thought it would be nice if there were a more elementary presentation of the complex fourier series, higher up in the article, that does not rely on orthonormal bases and scalar products and such.  [[User:Pfalstad|Pfalstad]] 00:01, 17 March 2006 (UTC)

::::There probably exists such a presentation. I'll have a look. [[User:Deimos 28|Deimos 28]] 07:50, 17 March 2006 (UTC)

==Factually incorrect==

I believe that the definition of Fourier series as presented in the second section of this article is factually incorrect.  According to the definition as presented, Fourier series is defined only for functions that are periodic with period 2&pi;.  That is quite simply and demonstrably false.  -- [[User:Metacomet|Metacomet]] 20:53, 16 March 2006 (UTC)

:Did you actually '''read''' what I wrote in the <math>2\pi</math> guide or have I just being speaking in a void for the past week? I chose period <math>2T</math> now, so as to put in the nice little fraction <math>\frac{T}{\pi}</math> for everyone to strain their eyes on. I would very much appreciate if you could provide me with a demonstration to prove I cannot choose <math>T=\pi</math> in this formula... I never said the Fourier series was '''only''' defined for <math>2\pi</math>-periodic functions: I said "given a <math>2\pi</math>-periodic function, its Fourier series expansion is..." which is not at all the same thing.
:On the one hand you want things to be rigorous and on the other you complain when you can't understand them: I feel lost.
:[[User:Deimos 28|Deimos 28]] 22:48, 16 March 2006 (UTC)

The definition in the the article that is given immediately following the Table of Contents under the header "Definition" defines FS for a function of period 2&pi;.  It does not define FS for functions of any period other than 2&pi;.  Therefore, either the definition is incomplete, in that it ignores the vast majority of periodic functions, or the definition is complete as presented, in which case it is factually incorrect.  -- [[User:Metacomet|Metacomet]] 23:28, 16 March 2006 (UTC)

I never said that you cannot choose T = &pi;.  What I said is that if you ''define'' the FS only for functions of period 2&pi; then you have ''not'' defined FS for functions of any other period.  That is fine if FS works ''only'' for functions of period 2&pi;.  But you and I both know that it works for periodic functions of ''any'' period, and is in no way limited to period 2&pi;.  If the definition of something is in terms of a special case, then either the definition can apply only to that special case, or the definition must be incomplete.  -- [[User:Metacomet|Metacomet]] 23:34, 16 March 2006 (UTC)

:He just fixed it; go read it again.  [[User:Pfalstad|Pfalstad]] 23:32, 16 March 2006 (UTC)

I can see that he fixed it.  Thank you.  Now it is no longer factually incorrect.  -- [[User:Metacomet|Metacomet]] 23:36, 16 March 2006 (UTC)

== Interpretations ==

Look, I know you'll all hate what I wrote in the Interpretation paragraph. The thing is, this section needs a lot of drawings that I haven't had the time or the energy to make yet. It would be nice, I think, if there were more (physical) interpretations than just this one.

In advance, sorry...
[[User:Deimos 28|Deimos 28]] 20:48, 17 March 2006 (UTC)

==Period: 2T, 2Pi, T, etc==

I think that the definition should contain an arbitrary period '''T''', rather than 2T, 2pi or anything else. One argument against a period of 2T is that (while it may make the equations have less 2's in them) you know that ''someone'' or lots of someones will miss seeing the "period of 2T" thing, and take T to mean what it usually means: ''period''. For more clarity, a period of simply ''T'' should be used. As for a period of 2pi, while it might be easier to explain, the definition should be as general as possible. In the explanation, one can start with a period 2pi to make things simple, and then go on to expand it. Does anyone agree or disagree with this proposal? [[User:Fresheneesz|Fresheneesz]] 01:52, 19 March 2006 (UTC)

:Hi there! I completely understand what you mean: being general (i.e. not a "special" period) and simple (T instead of 2T). I disagree however. A period of <math>2\pi</math> can be thought of as the '''canonical form''' of Fourier series rather than just an arbitrary period. Indeed, given any periodic function ''f'' of period T, <math>x\mapsto f\left(\frac{Tx}{2\pi}\right)</math> is <math>2\pi</math>-periodic. As I stated before, it's just the same with exponentials: <math>a^x= e^{a\ln(x)}</math>, but everyone studies <math>x\mapsto e^x</math> and not <math>x\mapsto a^x</math>, without giving it even as much as a second thought. I think that the small effort involved by changing variables is more than compensated by the gain in clarity and simplicity. As for choosing <math>T</math> instead of <math>2T</math>, well it makes the formulae even more unreadable, I think, and doesn't bring much to the understanding. You can't force people to pay attention and read <math>2T</math> instead of <math>T</math>: whatever way you look at it, there'll always be someone to think his or her idea was better...
:I made the changes to period ''T'' to try to keep everybody happy (so that people would be more indulgent with my other edits... :cough:) but the more I think of it, the more I disagree.
:Regards,
:--
:[[User:Deimos 28|Deimos 28]] 07:56, 19 March 2006 (UTC)

I agree 100 percent with Fresheneesz.  -- [[User:24.218.218.28|24.218.218.28]] 20:26, 19 March 2006 (UTC)

:: One thing I don't understand: ''"function ''f'' of period T, <math>x\mapsto f\left(\frac{Tx}{2\pi}\right)</math> is <math>2\pi</math>-periodic"''. I don't understand how a ''"function of period T"'' could be neccessarily 2-pi periodic. 

:: I'll argue for a period T so that you might be happier about your edit : ) . One quote that caught my attention, "''everyone studies <math>x\mapsto e^x</math>'' ... ''without giving'' ... ''a second thought''". That seems like a bad thing in my point of view. First of all, most people do study only e^x, so having a more general equation on wikipedia would enlighten many more people to the general case. Also, the "without a second thought" part calls to mind people doing things blindly, which i'm sure happens a lot. The point of these articles is to most clearly allow people to understand, without becomming a textbook. 

::The equation might be slightly uglier, but it really isn't that bad to read. Also, you say that using 2pi is clearer, but it simply isn't to me. The only difference to me is that the 2pi is in there, but since pi is only a number, it becomes unclear whether some pi's aren't due to the period being 2pi. But i'm curious, in what way is a period of 2pi clearer to you? also.. <math>a^x= e^{x\ln(a)}</math>, right?

::I'll completely agree with a period of 2pi if it's made clear how it helps understand an arbitrary period. Because listing the equations for both periods of 2pi and T seems redundant if there isn't more to it. [[User:Fresheneesz|Fresheneesz]] 09:05, 20 March 2006 (UTC)

:::The quote you made does not say that a T-periodic function is a <math>2\pi</math>-periodic function which, unless <math>T=2\pi</math>, is wrong. What it says is that using a technique called '''change of variables''', you can limit yourself to the study of <math>2\pi</math>-periodic functions. Indeed, using the notations of above, if you define <math>g:x\mapsto f\left(\frac{Tx}{2\pi}\right)</math>, then <math>g(x+2\pi)=f\left(\frac{T(x+2\pi)}{2\pi}\right)=f\left(\frac{Tx}{2\pi}+T\right)=f\left(\frac{Tx}{2\pi}\right)=g(x)</math>.
:::This complicated formula simply means that you get exactly the same graph if you plot ''f'' with axis labeled 0, T, 2T, 3T or if you plot ''g'' with labels <math>0, \pi, 2\pi, 3\pi</math>. Therefore, it is adding an unnecessary complication to study T-periodic functions instead of <math>2\pi</math>-periodic ones. For the coefficients, just use the change of variable defined by <math>u:=\frac{Tx}{2\pi}</math>. Hence I completely agree with you that it is redundant to put both periods <math>2\pi</math> and ''T'', but it should be done nonetheless for people who aren't familiar with the change of variable technique.
:::The theoretical part becomes completely ridiculous if you consider ''T'' instead of 2π. I wrote my reasons earlier on this page, in '''The Definitive 2π-period Guide''', which apparently no one has read, and which gives a very good motivation for choosing a period of 2π. However, I'm pretty sure that my idea will not prevail in the end, not because it's a bad one, but because people won't be bothered to try to understand it or ask me for an explanation for what they didn't understand. Putting T instead of π gives a sense of false generality and completely hides the elegance of the idea (decomposing a movement in rotations).
:::As for the exponential, <math>a^x=e^{x\log(a)}</math> is precisely the reason why we don't study individually each of the <math>x\mapsto a^x</math> functions. If you strip these functions from the unessential to just get the basic mathematical idea (transforming sums into products) you get <math>x\mapsto e^x</math>. I'm not saying you should never mention <math>a^x=e^{x\log(a)}</math> but rather derive all the main properties with <math>x\mapsto e^x</math> and then say that for the other exponentials it's the same (or say it before, whatever).
:::[[User:Deimos 28|Deimos 28]] 10:29, 20 March 2006 (UTC)

::::Alright, I understand what you're saying now. It would be very very helpful to add a bit about the change of variables used to do that. If the theoretical part is bad with an arbitrary period, then by all means use 2&pi; instead. Why do you say that having an arbitrary period gives a false sense of generality? If we do provide both periods 2&pi; and ''T'', which do you suggest we put at the top? [[User:Fresheneesz|Fresheneesz]] 12:06, 20 March 2006 (UTC)

:::::Hi, sorry for the delay. I only saw your message just now because I have trouble understanding the selection process of the watchlist as it doesn't show all last modifications but <b>the</b> last modification... Anyway, it was a bit clumsy to say that having a period of ''T'' gives a false sense of generality. What I really meant was that the fact that we can Fourier-decompose a function of any period is quite independant from the method we are using: the reason period doesn't matter is because of the change of variable technique, not because of the general idea behind Fourier series which is to project a periodic function on a space of rotations. That that function has such and such a period is beside the point: of course we can do it to any function, but that's a technical detail. Therefore, I would suggest putting the <math>2\pi</math> version (canonical version) first, and then say something in the line of "by using the change of variable defined by <math>t=\frac{Tx}{2\pi}</math>, we get the formula for any period ''T''" and then write the formula for ''T''. I suggested that to Metacomet (see below), but he doesn't seem to agree. Thanks a lot for your feedback, anyway. --[[User:Deimos 28|Deimos 28]] 21:32, 21 March 2006 (UTC)

==small issue: plural vs singular fourier series==

The singular vs plural thing is becoming more of an issue than it should be. Since its been changed a couple times, I just wanted to resolve it. First of all, its gramatical fact that "X are a Y" is incorrect. So if fourier series is plural, then "Fourier series are mathematical techniques" should be the way to write it. 

I think that maybe this sentence makes more sense, and might appease this issue:
"Fourier series are the result of a mathematical technique for ....."

Does that sound good or... does anyone have any better ideas? [[User:Fresheneesz|Fresheneesz]] 08:39, 20 March 2006 (UTC)

:I agree for the grammar part: sorry for the mistake. The term "Fourier series" denotes a mathematical being which, if convergent, can be used to represent a periodic, piecewise continuous function. I don't think the term "technique" is correct (I'm not sure who wrote it - if it's me, well I made a mistake & I apologize): it's a mathematical tool, not an engineering technique, just like integration is a tool and not a technique. I don't think it is the result of a mathematical technique: what technique? Projection on Hilbert spaces? I would prefer the formulation: "Fourier series is a mathematical tool used..." What do you think?
:[[User:Deimos 28|Deimos 28]] 10:01, 20 March 2006 (UTC)

== Definition section ==

Thanks, Metacommet, for putting in "canonical definition". I think this definition paragraph could perhaps be made even better by regrouping both definitions, for example by putting something in the line of:
:<blockquote style="background: white; border: 1px solid black; padding: 1em;"><center><pre><nowiki>[insert canonical definition here]</nowiki></pre></center>This canonical definition can easily be extended to an arbitrary perdiod: if ''f'' is <math>T</math>-periodic function, then the change of variables defined by <math>u=\frac{T}{2\pi}x</math> gives: <center><pre><nowiki>[insert formula for </nowiki><math>a_n, b_n, c_n</math><nowiki> of your first paragraph here</nowiki></pre></center>which is the form most commonly used by engineers.</blockquote>

What say you?
--
[[User:Deimos 28|Deimos 28]] 10:56, 20 March 2006 (UTC)

:I prefer the presentation with the general definition first, and then the "Canonical form" second as a special case.  First, it is logically correct and consistent to start with a general definition, and then to show how you can narrow the defintion by limiting the value of a parameter or relaxing a key assumption  Second, the article flows better this way, since the very first example in the "Examples" section, just below the "Definition" section, is based on a function of period 2&pi;.  If you reverse the order in the definition section, then the flow becomes back-and-forth, rather than straight-line.  Third, the current presentation is really actually better for those of you in the 2&pi; camp.  Once you get past the ugliness of the general definition, you don't ever have to look at it again if you don't want to!  Just substitute ''T''= 2&pi; and all of the ugliness goes away.  Moreover, since the article does exactly that in the "Canonical form" section, the work is already done for you.  Just use the "Canonical form" and ignore the "General form". -- [[User:Metacomet|Metacomet]] 13:53, 20 March 2006 (UTC)

::<b>(1)</b> Not necessarily: you can also choose to go from the simple to the complex, i.e. generalize the basic idea to any period.
::<b>(2)</b> Agreed. &nbsp; &nbsp; <span style="color: darkgreen">Wow &ndash; there is a first for everything! &nbsp; ;-) &nbsp; -- [[User:Metacomet|Metacomet]]</span>
::<b>(3)</b> OK, so if I understood what you said correctly, I can change the math section back to <math>2\pi</math>? <i>:hopeful:</i>
::--[[User:Deimos 28|Deimos 28]] 14:42, 20 March 2006 (UTC)

:On '''(3)''':  If you are referring to Section 4 Mathematical derivation of Fourier coefficients, it looks to me like it is already written in terms of ''T'' = 2&pi;.  In any event, I cannot really follow Section 4 because there is too much techno-math jargon that is beyond my limited experience, so I really don't much care what you do with it.  My own preference would be for someone to re-write it so that it would be accessible to the non-mathematician, but failing that, do whatever you want with Section 4, as long as you make it clear that you are deriving the results for the special case of ''T'' = 2&pi;, and that the results can be generalized easily to functions of arbitrary ''T''.  -- [[User:Metacomet|Metacomet]] 22:53, 20 March 2006 (UTC)

::Hey Metacomet, I think It'd be nice if we could agree on some sort of compromise or something. I like the arbitrary period ''T'', and it would be most useful for me personally if that was above the 2&pi; thing, but many people especially mathemeticians seem to find the ''change of variables'' technique to be much simpler way to express it. I would vote for keeping it with arbitrary ''T'' first, and then the "canonical form" below it like it is. But a note about the change of variables neccessary to generalize it, I think, should be added. My only issue with change of variables is that many people don't understand it (including myself, fully). I think there are two things that we can compromise, either 1) having canonical form first, then explaining the change of variables needed to change it to to the general case, which it should also have near the top, or 2) as it is now, with the addition of the change of variables explanation. [[User:Fresheneesz|Fresheneesz]] 00:12, 22 March 2006 (UTC)

:It seems to me like we have already achieved a compromise that in theory should make everyone reasonably happy if not entirely ecstatic.  The article now shows '''both''' the arbitrary period ''T'' and the normalized perios 2&pi; in the definition section.  It also shows the connection between the two, and it relates the angular frequency to the concept of harmonics (which is sort of important). Moreover, it also shows the change of variables approach in the mathematical derivation section.  I am not sure what else it needs, but if you think that there is something missing, feel free to make the revisions, or write up some ideas here on the discussion page. -- [[User:Metacomet|Metacomet]] 02:17, 22 March 2006 (UTC)

:BTW, at a certain point, I think it might make sense to leave the article alone for a while, stop making additional revisions, and maybe take some time to evaluate the article to see if it makes any sense at all.  IMO  -- [[User:Metacomet|Metacomet]] 02:17, 22 March 2006 (UTC)

::I didn't make the corrections to the introduction that you reverted, but I thought they made sense: what is the difference in phase between <math>\cos(nt)</math> and <math>\cos(pt)</math>? If you agree that there is none, why state in the introduction that the original function can be written as a sum of periodic functions of different phase? Moreover, Fourier being a French mathematician and his papers being written in French, I think the French title at least has its place somewhere beside the English one (it's nice to be patriotic from time to time!) I do agree that the article doesn't need many more additions, except maybe for the sine and cosine transform: I'll have a look, but I suspect it doesn't really belong here. While I'm not entirely happy with the definition section, if I can't convince people then I'm perfectly OK to leave it the way it is: compromising is always difficult. On both sides. Regards, -- [[User:Deimos 28|Deimos 28]] 07:41, 22 March 2006 (UTC)

==Phase angle==

===Example 1===

Suppose the following:

:<math>x(t) = A \cos( \omega_0 t) \,</math>
:<math>y(t) = B \sin( \omega_0 t) \,</math>
:<math> A \ne B </math>
:<math> A, \ B, \ \omega_0 \, </math> are constants
:<math> A, \ B, \ \omega_0 \in \mathbb{R} </math>

Question:  Do the functions ''x''(''t'') and ''y''(''t'') have the same or different phase angles?

Hint:

:<math>\sin(\theta) = \cos(\theta - \pi/2) \,</math>

-- [[User:Metacomet|Metacomet]] 23:35, 22 March 2006 (UTC)

=== Example 2 ===

Suppose the following:

:<math>x(t) = A \cos( \omega_1 t) \,</math>
:<math>y(t) = B \cos( \omega_2 t) \,</math>
:<math> A = A_0 e^{i \pi/6} \,</math>
:<math>B = B_0 e^{i 3\pi/8} \,</math>
:<math>  A_0, \ B_0 \in \mathbb{R} </math>are constants.
:<math> \omega_1, \ \omega_2 \in \mathbb{R} </math>are constants.


Question:  Do the functions ''x''(''t'') and ''y''(''t'') have the same or different phase angles?

Hint:  ''A'' and ''B'' are complex numbers with different phase angles.

-- [[User:Metacomet|Metacomet]] 23:58, 22 March 2006 (UTC)

===Response===

:This is a very good illustration of why we choose the same phase for all cosine functions and the same phase for all sine functions. If we chose, say, sines of different phases, then the Fourier series decomposition would no longer be unique and the functions would no longer be orthogonal (sines and cosines are orthogonal precisely because they are dephased versions of each other). As you seem to agree with my remark, I don't understand why you haven't reverted the edit yet. Regards, -- [[User:Deimos 28|Deimos 28]] 08:34, 23 March 2006 (UTC)

I haven't reverted it for several reasons.  First, the statement is true.  All it says is that any two sinusoids ''can'' differ from each other in one or more of precisely three ways:  amplitude, frequency, and phase.  It also implies, correctly, that there are no other possible ways that any two sinusoids can differ from one another.  Second, if you look at the revision history, you will see that I added the word "phase" in response to an edit that [[User:MFH|MFH]] made.  In that edit, MFH implied that sinusoids can differ from one another in ways other than amplitude and frequency, and so he removed the entire phrase about how sinusoids can differ.  On reflection, I realized that he had made a good point, but that his revision went in the wrong direction.  Instead of adding the word "phase", which would have made the sentence correct, he removed the entire notion that all sinusoids are identical except for a very small number (three) of characteristics.  Third, you have said more than once that the cosine functions in FS all have the same phase (true) and that the sine functions also all have the same phase (also true).  The point, however, is that each sine function is out-of-phase (by 90 degrees) with each cosine function (see Example 1, above).  As you know, a sine function is nothing more than a cosine function delayed by 90 degrees, which is by definition, a phase shift.  Fourth, even if that were not true, that is not the point of the first paragraph of the article.  The point was to take notice of the fact that all sinusoids are identical except for three very specific characteristics, as I have already said.  It was not to say that the sinusoids in FS necessarily differ by all three, but that they ''can'' differ by ''no more than'' those three.  Fifth, as Example 2 illustrates, the Fourier coefficients may themselves be complex numbers, which means that each coefficient has not only a magnitude, but also a phase.  So it is quite possible that the Fourier coefficients may themselves be out-of-phase with each other, even if all of the sines and cosines have the same phase.  As a result, each term in the Fourier expansion can have a different phase angle from any of the other terms.  -- [[User:Metacomet|Metacomet]] 13:04, 23 March 2006 (UTC)

All that being said, I really don't think this issue is all that important.  I was perfectly happy with the first paragraph before [[User:MFH|MFH]] removed the phrase about amplitude and frequency, and I don't think the word "phase" is actually necessary to the meaning of the first paragraph (although it is correct).  But I do think that it is absolutely key to include the notion that the sinusoids in the FS are identical to one another except for amplitude and frequency.  So IMO, if you want to remove the word "phase", that is fine with me, but I would prefere to leave the rest of the idea about amplitude and frequency differences alone.  -- [[User:Metacomet|Metacomet]] 14:11, 23 March 2006 (UTC)

:"I was perfectly happy with the first paragraph before [[User:MFH|MFH]] removed the phrase about amplitude and frequency": so was I: therefore I'm deleting the word "phase" and leaving the rest alone. Man, if we start making issues of stuff like this, we're nowhere near finished... -- [[User:Deimos 28|Deimos 28]] 14:59, 23 March 2006 (UTC)

I didn't make an issue of item, MFH did.  Be careful what you ask for.  If someone wants to nit-pick every little item, then they better be prepared for others to go the whole distance.  -- [[User:Metacomet|Metacomet]] 07:18, 26 March 2006 (UTC)

==sine and cosine series==

the terms "fourier cosine series" and "fourier sine series" link here. There should be some discussion on what those mean. Maybe i'll add something about them when I have time. [[User:Fresheneesz|Fresheneesz]] 03:13, 22 March 2006 (UTC)