Talk:Fourier series/Archive 1

message: Archived by Fresheneesz 02:13, 24 May 2006 (UTC)
replies: text: and User:Messedrocker 02:43, 25 July 2006 (UTC)

message: 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. Michael Hardy 18:30, 27 May 2004 (UTC)
replies: text: 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 α>1/2? Maybe I'm misunderstanding "point of continuity."
I would be grateful if you could explain that passage to me.
Thanks,
Loisel 02:58, 29 May 2004 (UTC)

text: I've excised the text for the time being. I'd still like to understand what you meant, Michael. Loisel 19:53, 31 May 2004 (UTC)
thread_title: The fourier series of an arbitrary periodic continuous function need not converge to the function.

message: 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 nk 's going to infinity very fast. Take continuous functions fk with ||f_k||\leq 1 such that 
: \int f_k D_{n_k} > \frac{1}{2}||D_{n_k}||
Basically, fk should be \mathrm{sign} D_{n_k}, but smoothed a little so that it would be continuous. Then take 
:f=\sum_{k=1}^\infty 2^{-k} f_k
If the nk increase sufficiently fast (you need to construct them inductively) then you get
:\int f D_{n_k} > \frac{1}{2}\int 2^{-k}f_k D_{n_k} > 2^{-k-2}||D_{n_k}|| \to \infty 
And that's it.Gadykozma 08:02, 21 Jul 2004 (UTC)
thread_title: Constructiveness

message: I've calculated the continuous orthogonality relationship for a general interval [a,b]. For [-π,π] the orthogonality relationship is:
:\frac{1}{2\pi}\int_{-\pi}^\pi e^{inx}e^{-imx}\,dx = \delta_{nm}
Assume p=b-a and make the substitution
:y=a+\left(\frac{x}{2\pi}+\frac{1}{2}\right)p
so that y=[a,b] for x=[-π,π]. This gives for the orthogonality relationship:
:\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}
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 π.  PAR 20:24, 22 Mar 2005 (UTC)
thread_title: Mistake

message: 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 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. PAR 20:24, 22 Mar 2005 (UTC)
replies: text: [-L, L] is fine with me. Oleg Alexandrov 20:50, 22 Mar 2005 (UTC)

text: 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, Wile E. Heresiarch 21:37, 22 Mar 2005 (UTC)
replies: text:  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]. Oleg Alexandrov 23:51, 22 Mar 2005 (UTC)
replies: text:  Wile is right, exp(inx) becomes exp(iπnx/L) so I'm less enthusiastic about [-L,L]. I still like [-L,L] because its easier to substitute L=π into the equations than to make the substitution x=yπ/L in the equations, including the dx and then realize that y=xL/π changes the limits of the integrals to [-L,L]. PAR 00:29, 23 Mar 2005 (UTC)
thread_title:  [-L,L] or [-pi,pi]

message: I note in the example we say...
:a_0 = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)\,dx=\frac{1}{2\pi}\int_{-\pi}^{\pi} x\, dx= 0,
:f(x)=x=a_0 + \sum_{n=1}^{\infty}(a_n\cos(nx)+b_n\sin(nx)) 
Whilst in the definition we say...
:f(x) = \frac{1}{2}a_0 + \sum_{n=1}^\infty\left[a_n\cos(nx)+b_n\sin(nx)\right]
:a_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x)\cos(nx)\,dx and b_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x)\sin(nx)\,dx
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. Mike1024 (talk/contribs) 23:47, 5 October 2005 (UTC)
replies: text: It is inconsistent.  The version of the top seems to be more standard, so I've fixed the example.  —Steven G. Johnson 05:14, 6 October 2005 (UTC)
thread_title:  a0 and 1/2 

message: I have a question onwhen doing an exercise about it after I read a article on the Wikipedia page..
Anyone please help me.
:(On which,why do for-all \mathcal ,isinx=odd,  ,therefore they all equal to 0?)
--HydrogenSu 15:52, 3 February 2006 (UTC)
replies: text: Ask at Wikipedia:Reference desk.  Do not ask here.  Pfalstad 19:04, 3 February 2006 (UTC)
replies: text: You two may misunderstand..In factI asked those for the newest changed version proposed here.--HydrogenSu 11:50, 5 February 2006 (UTC)
thread_title:  Transform:Fourier Integral 

message: 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. 
:F_n =\frac{1}{2\pi}\int_{-\pi}^\pi f(x)\,e^{-inx}\,dx.
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 2π to T would be fine, though (assuming it were done correctly this time).
:f(x) = \sum_{n=-\infty}^{\infty} F_n \,e^{inx}.
Also, in the above equation, is it really supposed to be from negative infinity to positive infinity?
Fresheneesz 01:05, 13 February 2006 (UTC)
replies: text: Yes.  —Steven G. Johnson 02:37, 13 February 2006 (UTC)
thread_title:  Inconsistencies in "definition of fourier series"

message: 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. Melchoir 00:33, 17 February 2006 (UTC)
thread_title:  Trigonometric series 

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

message: 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. Donludwig 00:45, 10 March 2006 (UTC)
thread_title:  My changes: March 9. 

message: 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.  Fresheneesz 10:15, 10 March 2006 (UTC)
replies: text: 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. Oleg Alexandrov (talk) 20:06, 10 March 2006 (UTC)
replies: text:  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. Deimos 28 21:39, 10 March 2006 (UTC)
replies: text:  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. Oleg Alexandrov (talk) 02:59, 11 March 2006 (UTC)
thread_title: Recent changes: where did stuff disappear to?

message: I think the whole article could be made immensely simpler to understand if it were presented from a convolution point-of-view.  -- Metacomet 07:28, 12 March 2006 (UTC)
replies: text: 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. Deimos 28 10:16, 12 March 2006 (UTC)

text: 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, --Deimos 28 20:20, 12 March 2006 (UTC)
thread_title:  Convolution point-of-view

message: 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 x\mapsto e^{inx} 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,
--
Deimos 28 21:47, 11 March 2006 (UTC)
thread_title:  Parts to remove 

message: 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. -- Metacomet 23:03, 12 March 2006 (UTC)
replies: text: 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? Deimos 28 00:41, 13 March 2006 (UTC)
replies: text: Haha! Nice one... Unfortunately, I was referring to the theory of Hilbert spaces rather than the Euler formula and I think putting Hilbert or L^2-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... -- Deimos 28 00:59, 13 March 2006 (UTC)

text: 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... -- Deimos 28 01:08, 13 March 2006 (UTC)
thread_title:  Complex exponentials are easier 

message: 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
:: x(t) = \sum_{n=-\infty}^{\infty} X[n] \cdot e^{i 2 \pi nt/T} 
:where
::*  X[n] =  {1 \over T} \int_{-T/2}^{T/2} x(t) \cdot e^{-i 2 \pi nt/T}  \cdot dt 
:::is the nth Fourier coefficient of x,
::*  i \equiv \sqrt{-1}  
:::is the imaginary unit, and
::*  e^{i\phi} = \cos(\phi) + i \sin(\phi) \, 
:::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).
-- Metacomet 03:46, 13 March 2006 (UTC)
replies: text: I agree. By the way, nice work on the article. I wouldn't myself use the X_t notation, as it looks a bit "physical", and that must be bad ;) I am going to do the relevant changes to the article. Cheers, --Deimos 28 08:56, 13 March 2006 (UTC)

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

message: 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...
Deimos 28 20:48, 17 March 2006 (UTC)
thread_title:  Interpretations 

message: Suppose the following:
:x(t) = A \cos( \omega_0 t) \,
:y(t) = B \sin( \omega_0 t) \,
: A \ne B 
: A, \ B, \ \omega_0 \,  are constants
: A, \ B, \ \omega_0 \in \mathbb{R} 
Question:  Do the functions x(t) and y(t) have the same or different phase angles?
Hint:
:\sin(\theta) = \cos(\theta - \pi/2) \,
-- Metacomet 23:35, 22 March 2006 (UTC)
thread_title: Example 1

message: Suppose the following:
:x(t) = A \cos( \omega_1 t) \,
:y(t) = B \cos( \omega_2 t) \,
: A = A_0 e^{i \pi/6} \,
:B = B_0 e^{i 3\pi/8} \,
:  A_0, \ B_0 \in \mathbb{R} are constants.
: \omega_1, \ \omega_2 \in \mathbb{R} 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.
-- Metacomet 23:58, 22 March 2006 (UTC)
thread_title:  Example 2 

message: 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. Fresheneesz 03:13, 22 March 2006 (UTC)
replies: text: I took the liberty of adding a section on the cosine and sine series. Perhaps someone will find it a little elementary and redundant with later sections, but until then, its there and (hopefully) accurate. --Entangledphotons 23:40, 27 April 2006 (UTC)
replies: text: Hi! Thanks for the edit. I don't really understand what new information this section brings: in the "Properties" section, I mentioned the fact that a_n=0 for odd functions and b_n=0 for even functions. It's a bit lost in the middle of the other properties and the layout definitely needs some work, but I'm not sure it's such a good idea to add yet another section to mention a trivial property... No offense meant - just my 2 pence. -- Deimos 28 07:08, 28 April 2006 (UTC)
replies: text: Ah, didn't see it. Yes, that's probably quite enough, you're right; I'll take it out. Thanks --Entangledphotons 12:56, 28 April 2006 (UTC)
thread_title: sine and cosine series

message: The canonical definition section reads: "many mathematicians prefer this form for its simplicity and elegance, whereas scientists and engineers..." While I understand that mathematicians can be very annoying at times, I still think they qualify as scientists, don't you?
--
Deimos 28 07:04, 28 March 2006 (UTC)
replies: text: -D ! At least some of them... The author certainly was thinking of somehow more "applied" scientists. Maybe you can rephrase it in the sense of "for practical use / concrete calculations / ..." or so. — MFH:Talk 16:10, 29 March 2006 (UTC)
replies: text: OK, so I just made an issue of the whole thing by creating a new subsection... Please feel free to comment: I wrote it fairly quicly so I'm sure it's pretty lame... -- Deimos 28 17:25, 29 March 2006 (UTC)
thread_title:  Are mathematicians scientists? 

message: I created an archive, and removed some of the stuff that was older than 6 months or so. I would support archiving things that are older than 2 months, but don't want to make that decision myself. The link is at the top of this talk page. Comments? Fresheneesz 02:19, 24 May 2006 (UTC)
thread_title: Archive

message: I changed the limits of integration for the coefficients of the fourier series to be "t_1" and "t_2", where t2-t1 = T (the period). This is more general, and is very often used - especialy when the function is continuous through a region that isn't -T/2 to T/2. For example, a sawtooth wave that is 0 at t=0 and 1 when t=2π. Fresheneesz 02:19, 24 May 2006 (UTC)
replies: text: In what way is this more general? Aren't we dealing with T-periodic functions? How is [t1,t2] better than [0,T] or even [-T/2,T/2]? -- Deimos 28 14:48, 29 May 2006 (UTC)
thread_title: My edits in the definition