Talk:Fourier series/Archive 3

Archive 1

Archive 2

Archive 3

Lead and images

The images in the lead are used inappropriately (in "the below animation") per MOS:SEEIMAGE. More importantly, the lead as currently written gets a bit into the weeds trying to give the reader an intuitive sense of how Fourier series "work"; per WP:LEAD it should really just be summarizing the article. The use of certain vague phrases like "mathematically represent," "basic waves," "the process of summing," etc. seem to be part of that pedagogical effort, but I think they should be replaced with more rigorous language. One of these somewhat vague phrases, "Fourier synthesis," is especially problematic since it would appear to refer to something technical yet is not explained at all in the body of the article (which runs counter to MOS:INTRO). The same can be said of "basic waves." Wuffuwwuf (talk) 19:41, 5 April 2022 (UTC)

A clarification: when I say "that pedagogical effort," I mean the effort "to give the reader an intuitive sense of how Fourier series 'work'". I'm saying that effort should be replaced by the effort to just summarize the article, as is proper. Wuffuwwuf (talk) 19:44, 5 April 2022 (UTC)

"Fourier synthesis" is explained in the Definition section, starting with the first sentence. And I think the lede is very good. From what you have written above, I can't envision an improvement. It might help if you show us what your rewrite would actually look like.

--Bob K (talk) 21:36, 5 April 2022 (UTC)

I see that the phrase gets referred to again, but I don't see why the word "synthesis" is actually needed to explain anything here. We can disagree about that, though, and still fix the more pressing issue that the lead doesn't adhere to MOS:SEEIMAGE. The previous version here does not have this problem, and I think it lacks the other issues I pointed out as well. I don't understand why it was ever changed, honestly. Wuffuwwuf (talk) 22:07, 5 April 2022 (UTC)

Sorry I was the one that made those images inline. I think for regular folk to get a basic introduction to fourier series, they really need to see animations up front. Previously without inlining the images, they would appear way off in the bottom-right corner outside of the introduction. Since they are now inlined I think it is okay to introduce the images with "below" and a colon. The MOS:SEEIMAGE only seems to be talking about non-inlined images. Em3rgent0rdr (talk) 22:30, 5 April 2022 (UTC)

I'm not sure why you say it's only talking about images that aren't inlined. Wuffuwwuf (talk) 22:45, 5 April 2022 (UTC)

More precisely than inline, I should say I set those images "align" to "none", which forces them to appear exactly after the line of text it follows and exactly before the subsequent line of text.

But that MOS:SEEIMAGE section you referenced says "Image placement varies with platform and screen size, especially mobile platforms, and is meaningless to screen readers", which from it's wording sounds like it only applies to the other alignments and doesn't apply to to "align"=="none".

Regardless, I just made an edit to the introduction to remove the use of the directional word "Below" and direct references to the image so it can now adhere to MOS:SEEIMAGE while still keeping those images inside the introduction between the paragraphs where I strongly feel they belong. Em3rgent0rdr (talk) 04:08, 6 April 2022 (UTC)

I am noticing that § Convergence utilizes the "gallery" template, which nicely indents the image(s), which I'm now thinking looks a little nicer than the simple "align==none" that I used, while still making the image(s) appear inbetween paragraphs of text rather that lost off to a corner of the side of the screen. Em3rgent0rdr (talk) 04:49, 6 April 2022 (UTC)

Yeah, that way of doing it does look good. Thanks for switching the wording in the lead, by the way. I'd support using the formatting in the Convergence section; I have no idea how to work such things myself though. Wuffuwwuf (talk) 05:51, 6 April 2022 (UTC)

I've changed the image formatting to use GALLERY view so is like Convergence section: https://wiki.riteme.site/w/index.php?title=Fourier_series&oldid=1081343202 Em3rgent0rdr (talk) 21:51, 6 April 2022 (UTC)

On second thought, it's also really a stretch to call what's in the first sentence of the Definition section an explanation. Wuffuwwuf (talk) 22:36, 5 April 2022 (UTC)

"Hesham's identity"

Removed the subsection on "Hesham's identity", which was promptly reverted. Before starting an edit war, I explain here why the section does not belong to WP:

• Google search "Hesham identity Fourier" only turns up this page and the cited source. As far as I know, the identity does not carry a name in the general mathematical community.
• The cited source is a self-published document on ResearchGate, which is not a peer-reviewed journal. A secondary source that names the identity would be necessary.
• The revertion comment claims that the article is in publication process of a journal. This is not enough; the article should be accepted. Moreover, the edit comment suggests that the editor is the same person as the writer of the preprint.
• The identity (or a variant thereof; I did not check the details) is a direct consequence of Parseval's theorem and convolution theorems, which are both listed in the same section.

With this background, it should be obvious that the identity is not novel (it is used in quite a lot of research literature, also for infinite series), and certainly not one that has a name.

Petrilaarne (talk) 15:29, 14 February 2023 (UTC)

1- The identity is new and getting known and used slowly by AI researchers. Don't expect to get a lot of results right now in google search, and remember everything has a start. Wait for a year or so and try to google search again.

2- I told you it is in the process of publication and I didn't claim this is enough as you lied and wrote. What I want to say is I will add the published version as a source once the paper is published.

3- yes, I am the same person who published the preprint until it is peer-reviewed and published, what is wrong about that?

4- The identity is not a direct consequence of Parseval's theorem and convolution theorems. I challenge you to derive it from both as you claim, and you will fail.

5- The identity is not included in any literature, however, bounds are given but not the exact value as I proposed. The bounds are applicable for infinite series, but it is still a bound not closed form. So the identity is novel.

6- You are not the first to contact me about the results, many reported it saves time in simulation so stop trying to act as you are saving the world by removing this useful result.

Regards,

Hesham 156.205.201.77 (talk) 17:57, 14 February 2023 (UTC)

Agreed. This doesn't seem notable to me. Thenub314 (talk) 16:15, 14 February 2023 (UTC)

Looking at how fast you commented on the above (Only 46 minutes after), I can say you are either the same person, or a colleague of the above. 156.205.201.77 (talk) 17:59, 14 February 2023 (UTC)

I understand your frustration, let me assure you in this case, it is simply a fact of the random arrival times. I am personally just returning form a rather long wikibreak. I do invite you to convince us, maybe we are wrong. It will also help to open an account, it makes things a lot easier, access to tools such as watch lists, a persistent talk page where people can message you, etc. Thenub314 (talk) 21:34, 14 February 2023 (UTC)

Hi Thenub314,

I am not frustrated, and no one can frustrate me, OK?! So let's say it was a random arrival, and welcome back after your wikibreak. Thanks for your invitation, I do accept it with pleasure. This is my account, please message me with your thoughts and let me explain my thoughts. It helps also if I know your background. I am a research Engineer in telecommunication engineering, with a strong foundation of mathematics and physics.

regards,

Hesham Hesham.Sharkas (talk) 06:08, 15 February 2023 (UTC)

Before coming here and finding this discussion, I wrote a little Octave script to test the identity, for fun and practice. It passed with flying colors, but apparently that's not the issue here. Oh well, here is the script in case anyone has a use for it.
--Bob K (talk) 00:10, 15 February 2023 (UTC)

Hello Bob,

Thanks for your acknowledge of correctness of the identity. It is always nice when seniors like you show appreciation to work of younger. In other countries, senior professors who are also reviewers in some journals simply reject such papers of lone researchers to republish the same idea under their names. One from Sweden -this is very common there- tried to steal this identity but he didn't noticed I registred a preprint, he is now in troubles hahaha.

Thanks for your code, it is efficient btw. It also show clearly the benefit of using the identity. When calculating LHS, the higher the sampling frequency the closer the resulted value to the exact value, but higher computation complexity. Using RHS, one get the exact value with much lower computation complexity, which makes it useful. It can be emplemented easily on a DSP, also AI engineers reported it reduce simulation time significantly.

Kind regards,

Hesham Hesham.Sharkas (talk) 06:32, 15 February 2023 (UTC)

pkg load statistics

function SUM = SUM2(S,M,k,L)
  if k >= L
    m   = (k-L:M-1) +1;                % +1 is for 1-based indexing
    SUM = sum(conj(S(m)).*S(m-(k-L)));
  else
    m   = (L-k:M-1) +1;
    SUM = sum(conj(S(m-(L-k))).*S(m));
  endif
endfunction

% generate random Fourier series coefficients
M = 10;
S= unifrnd(0,1,1,M) +i*unifrnd(0,1,1,M);

SUM = 0;
for k = 0:M-1
  for L = 0:M-1
    SUM = SUM + S(1+k)*conj(S(1+L))*SUM2(S,M,k,L);
  endfor
endfor
SUM

P = 10^5;
s = P*ifft(S,P);
integral = sum(abs(s).^4)/P

Look I don't mean to play lower but here is how I see it. According to the edit summaries this of 156.205.201.77 this is a paper going through the peer review process. Which means it doesn't yet meet WP:V, because we have know way to know if the peer reviewer will accept or reject the publication. Also, 156.205.201.77 is aware that the where in the process the paper is meaning they are an author, editor, or reviewer. In all of thse cases it seems to be a WP:COI. I really do think it violates wikipedia policy. Thenub314 (talk) 01:55, 15 February 2023 (UTC)

Hi Thenub314,

I got some feedback from the journal. I will add a small section about computation complexity to show clearly the efficiency of using the identity, and don't worry, the peer reviewer is willing to accept it. You need to know also that I had a lot of chances to publish in a non-open journals, but I didn't because something simple and useful like this should be availalbe to everybody. Finding an open-acces journal and going through the process was not easy and needed time, I didn't mind that, just to make it public to everybody, so kindly let DSP and AI researchers and students find it and use it. Once published, I will replace the preprint link reference with the peer-reviewed journal link.

Yes, I am the author. If the name is the problem, and I agree it is not very innocent :) kindly suggest me a short descriptive name and I will use it here and in the paper and give you a thanks in the acknowledgement. The point is, it must have a name.

Best regards,

Hesham Hesham.Sharkas (talk) 06:46, 15 February 2023 (UTC)

The main problem here is that the addition violates Wikipedia policies: Wikipedia:SELFCITE and Wikipedia:SELFPUBLISH. A secondary source (a paper that cites your article) would be necessary, see Wikipedia:SCHOLARSHIP. Otherwise you are promoting yourself (Wikipedia:SPIP).

Even if this form of the identity is more computationally efficient, it is not notable enough for Wikipedia yet (Wikipedia:Notability). Wikipedia cannot contain every possible mathematical statement; you can see how the page contains only the most essential 200-year-old properties of Fourier series. If the identity proves very useful in the coming years, somebody else can surely add it back. In the meanwhile, people can find the result on academic venues (I know that it is harder, but that's how it goes).

On a more general mathematical level: Naming things after yourself is frowned upon; it has to be someone else who calls it Hesham's identity. And still: this is a combination of Parseval's identity and the convolution identity (L^4 norm is the L^2 norm of function squared). This is used in a lot of mathematical research (see e.g. Jean Bourgain's work on cubic nonlinear Schrödinger equation, which is the field I am working in).

Petrilaarne (talk) 08:10, 15 February 2023 (UTC)

Hello Petrilaarne,

Promoting myself is if I make a page talking about me as a person, I added a useful mathematical finding and not talking about myself.I understand your concern it is not notable yet, keeping it here will make it notable very soon -which started to happen-. The published version will be availalbe in a matter of few monthes, and also citing by others, so all requirements you suggested will be fulfilled.

Mathematically, I have checked your suggested research -thanks for your time and efforts- and as I said above they used bounds (inequality) instead of equality. for example this research

https://academic.oup.com/imrn/article/1994/2/79/681161?login=false

in equation (13) the sign (<=) is used not (=) which makes is an upper bound (inequality), not an identity (equality).

The same happens in this research

https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-166/issue-1/Periodic-nonlinear-Schr%c3%b6dinger-equation-and-invariant-measures/cmp/1104271501.full

in Theorem 1, equation 1.4, the same story (<=) is there, but not (=).

I am sure you know the difference between bounds and equalities.

Finally, the fact that the identity is for finite length makes it useful in signal processing, but not in a research like what you shared. However, I may think about publishing an identity for infinite length, which may help you in the field you are working in.

Best regards and appreciating your time,

Hesham Sharkas Hesham.Sharkas (talk) 09:13, 15 February 2023 (UTC)

• You still have a conflict of interest as the author of the cited source. See Wikipedia:Neutral_point_of_view. I would also argue that naming a theorem after oneself does count as self-promotion. (Since mathematicians have strong opinions on who gets things named after themselves, I feel that this is not good publicity for you or your work.)
• For the notability, Wikipedia is intended to lag behind the outside world (see Wikipedia:SUSTAINED). It is okay to add the result back once sufficiently many sources cite it. Since this page only contains an extremely small subset of all known Fourier series results, I argue that "sufficiently many" means "over a few decades".
• "Putting [X] on Wikipedia makes [X] notable in the future" is a circular argument.
• Just last year, over 80 000 mathematics research articles were published. Most of them contain useful mathematical findings, yet we are not including all of them in Wikipedia. WP is an encyclopedia, not an exhaustive reference. [1]https://zbmath.org/?q=py%3A2022+dt%3Aj

This is why the result cannot be on Wikipedia at this time. I'm more than happy to see it here once it is notable enough, but that will take years, and has to be added by someone independent of you.

And for the mathematical notability, the full infinite-length proof goes like this: We can write the L^4 norm as

$$\int _{0}^{1}u(x)^{4}\,\mathrm {d} x=\int _{0}^{1}[u\cdot u](x)^{2}=\sum _{k\in \mathbb {Z} }{\widehat {[u\cdot u]}}(k)^{2}$$

by Parseval's identity. Now by the convolution theorem this equals

$$\sum _{k\in \mathbb {Z} }\left[\sum _{m\in \mathbb {Z} }{\hat {u}}(m){\hat {u}}(k-m)\right]^{2}=\sum _{k,\ell ,m\in \mathbb {Z} }{\hat {u}}(m){\hat {u}}(\ell ){\hat {u}}(k-m){\hat {u}}(k-\ell ).$$

We rewrite $n=k-m$, in which case the sum becomes

$$\sum _{k,\ell ,n\in \mathbb {Z} }{\hat {u}}(m){\hat {u}}(\ell ){\hat {u}}(n){\hat {u}}(n+m-\ell ),$$

which is exactly the claim. All equalities. The finite-length version is a special case of this. I don't think this formula has any name since it is used implicitly in so much of the literature (that's why you usually won't find it as a separate theorem either).

Petrilaarne (talk) 10:38, 15 February 2023 (UTC)

I added a Conflict of Interest template to the section. I suggest that neither of us touch the page, and let an unrelated editor make the decision on whether to include the section or not. The tag links to this Talk page discussion. Petrilaarne (talk) 10:55, 15 February 2023 (UTC)

Agreed. We will wait for some new opinion. Though, if nobody comes by within a couple of months, then I may take it out. Thenub314 (talk) 11:21, 15 February 2023 (UTC)

Hello Petrilaarne,

The above proof is not correct. Because you know the final result in advance you forcibly derived it. I will write you where exactly you made the mistakes, but this needs time that I don't have right now. Expect a reply by the weekend.

For now, we can leave the COI template, no problem. Hesham.Sharkas (talk) 11:39, 15 February 2023 (UTC)

Hi all, agree the naming was not common practice, so I removing the subtitle and embedded subsection with Parseval's identity section. The information by user (Hesham.Sharkas) are useful and correct and reduced simulation time for me. There is issues with the proof by user (Petrilaarne) because it doesn't give the correct results, also if the integral is definite how get infinite length series please? I wait correct proof from by user (Hesham.Sharkas) in weekend as he/she said. I don't know WP policy so I don't touch COI template. Thank you! XinZou802 (talk) 06:34, 18 February 2023 (UTC)

broken links

§ Complex-valued functions is broken.

WikiLink Fourier series in article Discrete-time_Fourier_transform is broken.

Bob K (talk) 19:21, 18 February 2023 (UTC)

Error in derivation?

It seems that Eq. (4) should not have /P in exponent. 157.82.122.40 (talk) 07:42, 20 February 2023 (UTC)

There was a mistake, but it was in the definition of $c_{n}$, it is necessary to divide by P when reconstructing the function so that the complex exponential have the correct frequency. To see this consider first exponential that multiplies $c_{1}$, it should have a period of $P$ because we are reconstructing a function whose period is $P$. Checking that we have

$e^{\frac {2\pi i(x+P)}{P}}=e^{{\frac {2\pi ix}{P}}+{\frac {2\pi iP}{P}}}=e^{\frac {2\pi ix}{P}}e^{2\pi i}=e^{\frac {2\pi ix}{P}}\cdot 1$

Without dividing by $P$ the resulting sums would not be $P$ periodic. Thenub314 (talk) 15:00, 20 February 2023 (UTC)