Talk:Dirichlet–Jordan test

Edit Request - Dirichlet's Theorem for 1-Dimensional Fourier Series

The section: Dirichlet's Theorem for 1-Dimensional Fourier Series, the following sentence: "The analogous statement holds..." appears right in the middle of the explanation of what the Theorem is. I am left wondering which "statement" it is referring to?? Apparently for a periodic function of any period, (even an infinite period?? doesn't this contradict the condition of boundedness??) The more I read this section the worse it appears. "For all x..." - are we supposed to know that "x" is a Real number? The Fourier Series is generally a FINITE integral, which contradicts (seemingly) the -∞ to +∞ integration in this section. "We state..." is about as pompous as you can get, but a trivial point. How does a function "oscillate" at a point?? BY DEFINITION, a function must be single valued at any point (in its domain). Why does the definition assume the given period? It either IS necessary or is NOT necessary. (Apparently it is not). Can someone either delete or move the offending sentence and possibly rewrite the entire section in a more clear fashion? e.g. "Dirichlet's theorem: If (a function [f:x ∈ R → C (?)] satisfies Dirichlet conditions, then for all x, the Fourier series, given by [insert formula here] is convergent. Where etc. etc."173.189.75.163 (talk) 05:17, 7 July 2014 (UTC)

Ooops

"These three conditions are satisfied if f is a function of bounded variation over a period."

This seems to be a false statement. Here are some examples.

(1) f=0 has bounded variation, yet it does not satisfy the three conditions. Indeed, it has an infinite number of maxima and minima in each finite interval. So the thrid condition is violated.

(2) FIx a decreasing sequence of positive numbers x_i -->0. Let g be a function with g(0)=0, g(x_i) = (-1/2)^i for i=0,1,2,..., and monotone between successive x_i. Then g is BV, in fact even continuous, so the Fourier series actually converges uniformly, but g has an infinite number of maxima and minima, so again the third condition is violated.

(3) A BV function can easily have a countably infinite number of jumps, so the second condition is easy to violate. 73.231.247.16 (talk) 03:32, 30 March 2016 (UTC)

The above mentioned statement regarding BV is valid only for the 2nd Condition "finite number of extrema" out of the 3 conditions. I think, this condition is not violated for the 2nd condition as you have stated in (3) because a BV function can not have infinite number of maxima or minima unless it is a constant function.

So, I suggest, we modify the condition 2

from

2. f must have a finite number of extrema in any given bounded interval, i.e. there must be a finite number of maxima and minima in the interval.

to

2. f must be of bounded variation in any given bounded interval, i.e. there must be a finite number of maxima and minima in the interval.

Please refer to this book p.198 available at [1]

—Pardhu Madipalli ^talk 20:49, 23 December 2017 (UTC)

Updated the article with respect to the above mentioned points.

—Pardhu Madipalli ^talk 11:11, 25 December 2017 (UTC)

References

Bounded Variation = Finite Extrema?

I believe the use of "i.e." in the second condition is incorrect. There exist BV functions with infinitely many extrema on a closed interval. For example, consider the function $x^{2}\sin {\frac {1}{x}}$ on any closed interval of positive length containing zero (after filling a removable singularity at zero). This function is BV with infinitely many critical points in the interval. While this example is not periodic, it may be easily modified to serve as a counterexample to the implied statement in condition (2). I feel that the use of "e.g." would be more appropriate here. Samuel Li (talk) 20:45, 30 March 2018 (UTC)

Notification

the first condition is not necessary because it is implied by the other conditions , but Dirichlet conditions are different from your conditions. — Preceding unsigned comment added by Boutarfa Nafia (talk • contribs) 20:00, 15 February 2022 (UTC)

Requested move 4 November 2022

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: moved. Much of the discussion here was about the article scope, rather than the proposed move per se, but based on the current state of the discussion, it seems like the article scope is appropriate as is and there is no need for a split. As for the article title itself: the main titling-related argument in this discussion was that the proposed title is preferable, in order to avoid confusion with Dirichlet boundary conditions. This argument was raised by most participants in the discussion, and was not contested, leading me to find a consensus to move. WP:COMMONNAME was also raised once on each side of the discussion, with neither side providing evidence for the claim, so the COMMONNAME argument does not affect the consensus. (non-admin closure) ModernDayTrilobite (talk • contribs) 21:56, 7 December 2022 (UTC)

---

Dirichlet conditions → Dirichlet–Jordan test – Should be moved there, since there is no danger of confusion with Dirichlet boundary conditions, and this is a more usual name for the test. 164.52.242.130 (talk) 18:04, 4 November 2022 (UTC) — Relisting. Favonian (talk) 19:26, 11 November 2022 (UTC)

• Comment. It seems you've essentially rewritten the article and soft-deleted the one on Dirichlet conditions to replace it with one on the Dirichlet-Jordan test. Perhaps the article title should be at whatever is the broader topic, the deleted content restored, and both of them should be discussed? SnowFire (talk) 04:17, 10 November 2022 (UTC)

The deleted content had been recently written and was self-sourced. I have also added sources of a much better pedigree than the source before. (Antoni Zygmund is recognized as a world authority on Fourier series, as evidenced by his 1979 Leroy P. Steele Prize and 1986 National Medal of Science award for example. The term he uses is "Dirichlet-Jordan test".) 164.52.242.130 (talk) 12:19, 10 November 2022 (UTC)

Hey 164. I saw your comment on my talk page. You called the addition a "self-published source", but IEEE Signal Processing Magazine seems to be used over 100 times across Wikipedia. Did you just mean that the editor who added this, spushp, was probably Pushpenda Singh? That's more accurately called a Conflict of interest in Wikipedia-ese ("self-published" is used for fake / vanity journals). And while it's not encouraged, academics adding their own work - when it's published - is not unheard of on Wikipedia. Are you claiming that Singh's work is somehow "bad" or "incorrect", or merely that he's overstating his own contributions? If you're not saying that it's somehow invalid, this really seems like something that could be dealt with by simply making a new, separate article - revert to the old article on Dirichlet conditions (and the cited journal article certainly seems to be a valid source for it), and create a new article on the Dirichlet–Jordan test. If you are saying it's invalid, then why? Is this journal article the only one in the world that thinks this is a real topic, say? SnowFire (talk) 17:45, 11 November 2022 (UTC)

That source is garbage. It makes basic errors. There are much better sources for this topic, as I have provided. Making a separate article for the same topic seems like a very stupid idea. 74.111.96.48 (talk) 23:30, 11 November 2022 (UTC)

Note: WikiProject Mathematics has been notified of this discussion. Favonian (talk) 19:25, 11 November 2022 (UTC)

Relisting comment: Experts needed; math project notified. Favonian (talk) 19:26, 11 November 2022 (UTC)

• Oppose per WP:COMMONNAME.  — Amakuru (talk) 12:24, 28 November 2022 (UTC)

Sources overwhelmingly call this the Dirchlet-Jordan test (or some variant thereof). So perhaps you mean "support per WP:COMMONNAME? 74.111.96.48 (talk) 00:11, 29 November 2022 (UTC)

• Comment. Since nobody took the bait from the Wikiproject Math notification, I think that split is the only valid option. Revert this article to its old form about the conditions, create a new article called Dirichlet–Jordan test with the new content, and let the IP editor nominate Dirichlet conditions for deletion at AFD if the main source used is really so poor. This avoids a "backdoor deletion via a move". SnowFire (talk) 20:49, 30 November 2022 (UTC)

Good grief, I already told you that it was a stupid idea to have two articles on the same topic. I improved the article, with better sources, and the standard name for this topic. Do you... not want improvements to Wikipedia? 74.111.96.48 (talk) 01:05, 1 December 2022 (UTC)

Look. You may well be correct. If you are, then thank you for your service. But think about how this looks on the outside. Suppose editor A and editor B each have radically different ideas of how to take some arbitrary topic - could be math, could be the history of Afghanistan, could be anything. One editor is a crank, the other editor is a genuine expert. How are passerby supposed to know which editor is which? I know that you think you're the expert, but how am I supposed to know? On the Internet, nobody knows you're a dog, etc. The answer is to look at the sources. Unfortunately for us, the old "Proper Definitions of Dirichlet Conditions and Convergence of Fourier Representations" source only shows the abstract without a login. You've added sources but I can't actually verify them. "Fourier Series" seems to talk about "Jordan's Test" but the Google Books preview is too incomplete. Archive.org has Volume I and Volume II of Zygmund's book, but the index in Volume II is woefully lacking, mentioning only a "Dirichlet problem" on page 97 back in Volume I, yet I can't find any such mention there. Do you have the page number to look for? And even if these sources confirmed your claims about the Dirichlet-Jordan test, it's not clear that they'd verify it is the larger over-arching topic of which "Dirichlet conditions" are a part of. So do you have anything better? Hell, take cell phone pictures of the pages in your textbook if you have to, or upload a PDF to Google Drive with "share with the link" and use the "Send email" function if you want to be scrupulous about copyright. Improving the citations a tad to include page numbers would be nice as well. If you add that, I can take a look, but bear in mind that these citations need to ideally not merely set up a "Dirichlet-Jordan test" article, but also explain the relationship with the conditions used in signal processing. SnowFire (talk) 03:25, 1 December 2022 (UTC)

I have added the page numbers you requested. An explicit source linking the two versions of the test is here: https://encyclopediaofmath.org/wiki/Dirichlet_theorem . I do not plan to follow your suggestion about uploading textbook pages so that you can check up on me. That's a ridiculous suggestion. No idea why you're questioning a clear preponderance of sources in my favor, especially as you now say you were even unable to verify the older source! But I have provided yet more. No one asked *you specifically* to attempt to verify things. But if you wish to volunteer for this task, then go to a fucking library.74.111.96.48 (talk) 11:06, 1 December 2022 (UTC)

I agree it's ridiculous, but it was also ridiculous to ask me to essentially read some very dense textbooks straight through without giving page numbers. Thank you for adding more sources now, but those weren't there when I made my above comment. And yes, I am the type who is willing to "go to a fucking library" to confirm this if need be, but it's a lot easier with something more concrete to go on. SnowFire (talk) 20:18, 5 December 2022 (UTC)

• • With Silly Rabbit's comment confirming the IP's claim that the "Lecture Notes" source (that I couldn't access) is weak, I suppose a split isn't needed. Weak support to avoid the confusion with Dirichlet boundary conditions although it seems like Dirichlet-Jordan test has some issues too, and there will need to be some explanations of both the original version and the modern improvements. SnowFire (talk) 20:18, 5 December 2022 (UTC)
• Revert and split per SnowFire. I'm prepared to change my !vote if a new knowledgeable user shows up and gives their opinion on (1) the name of the article and (2) the contents of the article before and now. [Feel free to ping me in such case.] But until then, these look sufficiently different that separate articles is the best approach. I definitely oppose a backdoor deletion-via-move, though. - CRGreathouse (t | c) 21:06, 1 December 2022 (UTC)

  Withdrawn per discussion with Silly rabbit. - CRGreathouse (t | c) 15:23, 5 December 2022 (UTC)

• Oppose split. I have no strong position on whether the article should be moved, although the argument that "Dirichlet conditions" would under ordinary circumstances refer to Dirichlet boundary conditions, rather than the subject of this article, has merit. However, I strongly disagree with the proposed splitting of the article. Sources adequately show that there really is one topic here. This is a well-known test in elementary Fourier series (see the article by Jaak Peetre referenced in the article, for example). Dirichlet, in the early 19th century, formulated the test for what we would nowadays call "piecewise monotone functions". Dirichlet had various spurious continuity assumptions in his paper, because the technology of Riemann integration was not yet available. Later in that century, the result was generalized by Jordan to functions of bounded variation. This is not much of a generalization to the formulation of the Dirichlet condition for piecewise monotone functions, since every function of bounded variation is a sum of two monotone functions (a result I believe also due to Jordan). There seems to be a fair amount of confusion in the literature on this topic, mostly in poor quality sources (e.g., undergraduate engineering textbooks, or the "lecture notes" referenced in the original article Singh, Pushpendra; Singhal, Amit; Fatimah, Binish; Gupta, Anubha; Joshi, Shiv Dutt (September 2022). "Proper Definitions of Dirichlet Conditions and Convergence of Fourier Representations [Lecture Notes]". IEEE Signal Processing Magazine. 39 (5): 77–84. doi:10.1109/MSP.2022.3172620.), that attempt to follow Dirichlet's original formulation a little bit too rigidly and run into various sorts of trouble. (Remember: Dirichlet did not even have Riemann integral available, so such formulations are necessarily mathematically anachronistic.) Clearly there is room in one article for the modern formulation (using the Riemann integral), and some mention of the original formulation due to Dirichlet. @CRGreathouse: siℓℓy rabbit (talk) 00:09, 3 December 2022 (UTC)

• What do you think of the contents of the article before the rewrite vs. now? - CRGreathouse (t | c) 03:00, 3 December 2022 (UTC)

The new version is an improvement with regard to sourcing (in particular referencing the key secondary source of Antoni Zygmund, to which most tertiary sources on this topic refer back to), and in regard to mathematical and historical content. Arguably, the old version had some slightly better cosmetic wikignoming and cosmetics, but I would say is inferior in substance. siℓℓy rabbit (talk) 16:09, 3 December 2022 (UTC)

I agree with this. Two pages are overkill. Walt Pohl (talk) 12:52, 3 December 2022 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Editor keen on adding a specific reference

An editor seems quite keen to add links to the article Singh, Pushpendra; Singhal, Amit; Fatimah, Binish; Gupta, Anubha; Joshi, Shiv Dutt (September 2022). "Proper Definitions of Dirichlet Conditions and Convergence of Fourier Representations [Lecture Notes]". IEEE Signal Processing Magazine. 39 (5): 77–84. doi:10.1109/MSP.2022.3172620.. He has spammed this reference over several pages. What is the relationship of this editor to the author of that paper I wonder? WP:SELFCITE warns against self-citation, especially in cases like this where there are much better references available on the topic. From a purely mathematical point of view, it is incorrect to say that a function is of bounded variation if it has only finitely many finite discontinuities and countably infinite number of local extrema. (This condition is neither necessary nor sufficient.) I have removed the incorrect edits from this page, and the other two pages that the author spammed with this identical content. 2600:1016:B021:BB70:2536:C7FE:F927:38E1 (talk) 11:22, 30 March 2023 (UTC)