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== The article is far too technical ==
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The series of edits made by the same user since May 2020 has made this article overly technical, and no longer in the spirit of a Wikipedia article on the subject. To see thepoint, compare the [https://en.wikipedia.org/w/index.php?title=Distribution_(mathematics)&oldid=979774015 Sep 22, 2020 version] with the last version prior to the series of edits by the same editor, the [https://en.wikipedia.org/w/index.php?title=Distribution_(mathematics)&oldid=952024459 April 20, 2020 version].
== Sobolev ==
''Saaska, 27 Nov 2003''
I thought it would be fair to include Sobolev here.


The April 2020 version opens with a Basic Idea section that illustrates the essence of the subject. The Sep 2020 version introduces a lot of notation and ancillary concepts, much of which does not seem necessary to introduce the definition of a test function.
== composition of a distribution with a differentiable injective function ==


I would strongly urge that the article be edited back to resemble the April 2020 version, and any advanced material being added after the the simplified introduction to the subject.
Is it possible to define the composition of a distribution with a differentiable injective function? Formally, it should be like


[[User:Undsoweiter|Undsoweiter]] ([[User talk:Undsoweiter|talk]]) 19:08, 23 September 2020 (UTC)
:<math>\left\langle T\circ f,\ \varphi\right\rangle
:The article that you linked to contained false information such as the last sentence of this claim: " The elements of D(''U'') are the infinitely differentiable functions ''<math>\varphi</math>'' : ''U'' → '''R''' with compact support – also known as [[bump function]]s. This is a real [[vector space]]. It can be given a [[topology]] by defining the [[limit of a sequence]] of elements of D(''U'')."
=\left\langle T,\ \frac{\varphi\circ f^{-1}}{f'\circ f^{-1}}\right\rangle
:You can ''not'' define the topology using sequences and this is a non-trivial fact that has been proven. The above false statement should ''not'' appear in this article. The correct definition of the canonical LF-topology on the space of test functions is unfortunately technical.
</math>
:Also, the old article was missing important information such as how to extend differential operators to distributions, which is arguably is one of the more important uses of distributions.
:Distributions are unfortunately an innately technical topic. However, I am fine with simplifying the article but not at the expense of adding false-but-simple information or removing important information. [[User:Mgkrupa|<span style="text-shadow:grey 0.3em 0.3em 0.1em; class=texhtml">Mgkrupa</span>]] 22:26, 23 September 2020 (UTC)
::My comment about technicality and style of the article does not mean I endorse the accuracy of every claim in the April version I refer to. If there are incorrect statements, they can be corrected without resorting to the drastic changes you have made to the article. By all means, fix errors. But this does not justify the excessively technical approach you have followed in your rewrite.
[[User:Undsoweiter|Undsoweiter]] ([[User talk:Undsoweiter|talk]]) 20:31, 30 September 2020 (UTC)
:::Over time, I will try to make this article less technical. However, we first need to establish what assumptions can and can not be made about this article and its "typical" reader. According to [[Wikipedia:Manual of Style/Mathematics]], this article should follow the following guidelines (as well as others not listed here). It should be be written [[WP:ONEDOWN|"one level down"]], which means:
:::* "consider the typical level where the topic is studied (for example, secondary, undergraduate, or postgraduate) and write the article for readers who are at the previous level." Also,
:::* "articles on undergraduate topics can be aimed at a reader with a secondary school background, and articles on postgraduate topics can be aimed at readers with some undergraduate background."
:::* "Articles should be as accessible as possible to [[MOS:MATH#TONE|readers not already familiar with the subject]] matter."
:::* "When in doubt, articles should define the notation they use."</li>
:::* "If an article requires extensive notation, consider introducing the [[Wikipedia:Manual of Style/Mathematics#Article body|notation as a bulleted list]] or separating it into a "Notation" section."</li>
:::* "An article about a mathematical object should '''<em>provide an exact definition of the object</em>''', perhaps in a "Definition" section after section(s) of motivation."
:::* "Writing one level down also supports our goal to provide a tertiary source on the topic, which readers can use before they begin to read other sources about it."
:::I think that it is safe to assume that the reader has knowledge of calculus. But before we start editing this article to make it less technical, it's important to know what else we can assume about a "typical" reader of this article. This is important because, for example, whenever it is reasonable and possible to do so, then terminology that a reader is unlikely to be familiar with should be briefly defined/described within this article, instead of just having a link to the article about the term (this is because ideally, a "typical" reader should not have to go down a rabbit hole of Wikipedia links and search through various articles in order to understand something stated in <em>this</em> article about distributions). So we need to agree on the following (non-exhaustive) list of assumptions before we can start rewriting this article:
:::#Is it safe to assume that the reader is likely an advanced undergraduate or higher? (I personally think so).
:::#Is it safe to assume that the reader is likely a graduate student or higher?
:::#Is it safe to assume that the reader is likely a mathematics, physics, or engineering student?
:::#Is it safe to assume that the reader has studied metric spaces? (I personally think that it is).
:::#Is it safe to assume that the reader has knowledge of general topology (in particular, of non-metrizable topological spaces)?
:::#Is it safe to assume that the reader has studied Banach spaces? (If not, then the Fréchet spaces and related notions that are used in this article will need more detailed explanations).
:::#Is it safe to assume that the reader has studied Fréchet spaces? My guess is probably not and so the reader should not be assumed to know about Fréchet space. But if they are familiar with the basics of Banach spaces then the required knowledge for Fréchet spaces can be described using Banach space terminology.
:::#Is it safe to assume that the reader has studied non-metrizable [[topological vector space]]s? I think that this can not be assumed. However, unfortunately, neither the canonical LF topology nor the topology on the space of distributions is a [[sequential space]] so this topology can <em>not</em> be described using sequences (let along a metric). Suggestions about how to define and describe these topologies to readers who are not used to dealing with non-sequential (and also non-metrizable) spaces would be welcome. The current description of these topologies is (unfortunately) technical and I'd like for it to be less technical but I'm not sure how to make it less technical.
::: [[User:Mgkrupa|<span style="text-shadow:grey 0.3em 0.3em 0.1em; class=texhtml">Mgkrupa</span>]] 18:51, 26 October 2020 (UTC)
::::I believe distributions are common in physics (please correct me if I'm wrong), and I am not sure that physics students will know about Frechet spaces (I am not a physics student myself, so I don't even know if physics undergrads learn any topology) – so perhaps the ultimate goal is to make this article accessible to physicists? It would be helpful if others could give their thoughts on this.
::::Also, I agree that ''ideally'' "a 'typical' reader should not have to go down a rabbit hole of wikipedia links" to understand an article, but I would say that this article is (at least somewhat) an exception to this rule, because it requires a much higher than average amount of prerequisite knowledge (again, please correct me if I am wrong) [[User:Joel Brennan|Joel Brennan]] ([[User talk:Joel Brennan|talk]]) 00:15, 18 November 2020 (UTC)


{{hr|1}}
even if ''f'' is not injective, but the support of ''T'' does not include any critical point of ''f'' it should work (summing up for all the values of <math>f^{-1}</math>)
:I am in favor of creating a separate article for the canonical LF topology (this topology has been studied enough to warrant its own article) and in this way we can simplify the article on distributions by placing some technical details into this new article. [[User:Mgkrupa|<span style="text-shadow:grey 0.3em 0.3em 0.1em; class=texhtml">Mgkrupa</span>]] 23:27, 23 September 2020 (UTC)
::For instance, we can make the presentation less general by replacing ''k'' with ∞. [[User:Mgkrupa|<span style="text-shadow:grey 0.3em 0.3em 0.1em; class=texhtml">Mgkrupa</span>]] 23:51, 23 September 2020 (UTC)


:I recently learned distributions and I found it frustrating the first time I glanced at this article, hoping for an overview of the subject. This issue was not even technicality: I understand Fréchet spaces and topological linear spaces. The article was just not reader-friendly and felt more like a giant list of notation. Compare this to the article on [[Sobolev space]], which starts off with motivation and looks much less cluttered. [[Special:Contributions/74.101.253.193|74.101.253.193]] ([[User talk:74.101.253.193|talk]]) 22:11, 16 August 2021 (UTC)
---


== Completely agree about moving discussion of TVS topologies to another article ==
David 18 Dec 2004


No one looking for a straightforward explanation of distributions will be able to get through the current article.
I think that:


As far as the continuity of distributions is concerned, all one needs to say in this article is that a linear functional <math>T : \mathcal{D}(U) \to \mathbf{R}</math> (or <math>\mathbf{C}</math>) is continuous if and only if it is sequentially continuous i.e., if <math>\varphi_n \to \varphi</math> in <math>\mathcal{D}(U)</math> (supports of all <math>\varphi_n</math> are contained in a compact subset <math>K</math> of <math>U</math> and the sequence and all of its derivatives converge uniformly on <math>K</math>), then <math>T(\varphi_n) \to T(\varphi)</math> in <math>\mathbf{R}</math>. Explanations of sequential continuity vs. continuity in TVS can be given elsewhere.
If u is a distribution in D?(A) and T is a C^00(A) invertible function:


Alternatively, you could say that <math>T</math> is continuous if and only if for every compact subset <math>K</math> of <math>U</math> there exist <math>C > 0</math>, <math>n\in \mathbf{N}</math> such that <math>|T(\varphi)| \le C\sum_{|\alpha|\le n} \|\partial^\alpha \varphi\|_\infty</math> for every <math>\varphi</math> supported in <math>K</math>. But I think the sequentially definition is more accessible.
<nowiki><u o T, g> =<u, g o T^(-1) |det J|></nowiki>


However, these changes would require a lot of rewriting of the first part of this article.[[User:Reader634|Reader634]] ([[User talk:Reader634|talk]]) 07:45, 12 December 2021 (UTC)
where g is a test function and J is the Jacobian matrix of T^(-1).


== Comment ==
:yes, this is the same formula as above, but it may not be general enough


I think the recent changes are absolutely in the right direction. I made a few minor edits, mostly to remove "scare" quotes about the difficulties of defining the topology on the space of distributions (it's enough to refer to the relevant article). Also, I think more of the discussion on topologies can be moved elsewhere, and all (or nearly all?) of the material on <math>C^k</math> spaces can be removed from this article, since <math>C^\infty</math> test functions are the fundamental ones for distributions. As a minor point, I'd denote the regular distribution associated with multiplication by a locally integrable function <math>f</math> by <math>T_f</math> rather than <math>D_f</math> (which could be confused with differentiation) but I don't think there's a standard notation here. [[User:Reader634|Reader634]] ([[User talk:Reader634|talk]]) 09:08, 31 December 2021 (UTC)
== Typo? ==
''This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense: ''


== Some notations not explained ==
:''d/dx (S * T) = (d/dx S) * T + S * (d/dx T). ''


Same as in [https://en.wikipedia.org/wiki/Spaces_of_test_functions_and_distributions| Spaces of test functions and distributions]: notation <math>L^p_c</math> first appearing in 6 ([https://en.wikipedia.org/wiki/Distribution_(mathematics)#Spaces_of_distributions| Spaces of distributions]) seems not to be explained anywhere on this page. [[User:Mamuka Jibladze|Mamuka Jibladze]] ([[User talk:Mamuka Jibladze|talk]]) 17:23, 31 January 2022 (UTC)\]
Is this a typo? Seems to me it should be


<big><big><big>'''KINDLY DO NOT ERASE OTHER PEOPLE'S POSTS WHEN YOU POST YOUR OWN'''</big></big></big>. [[Special:Contributions/2601:200:C000:1A0:7097:9C72:4BFB:717D|2601:200:C000:1A0:7097:9C72:4BFB:717D]] ([[User talk:2601:200:C000:1A0:7097:9C72:4BFB:717D|talk]]) 00:11, 18 May 2022 (UTC)
:d/dx (S * T) = (d/dx S) * T = S * (d/dx T).


== Tempered distributions ==
[[User:Josh Cherry|Josh Cherry]] 14:47, 18 Apr 2004 (UTC)


"Tempered distributions" is used but not defined in this article. I've heard they are "distributions that are upper bounded by polynomials", which I'm not quite sure how to make precise. I've also heard the more [https://math.stackexchange.com/questions/491512/definition-of-distribution-pseudofunction-and-tempered-distribution#:~:text=A%20tempered%20distribution%20is%20a,of%20smooth%20compactly%20supported%20functions.|technical and difficult definition] in terms of (the extension of?) linear functions on Schwartz function. Needs to be made clear in this article. [[User:Jess Riedel|Jess_Riedel]] ([[User talk:Jess Riedel|talk]]) [[User:Jess Riedel|Jess_Riedel]] ([[User talk:Jess Riedel|talk]]) 19:07, 16 March 2024 (UTC)
Don't think so. [[User:Charles Matthews|Charles Matthews]] 15:33, 18 Apr 2004 (UTC)

OK, help me out here. My reasoning is a follows:
*Differentiation corresponds to convolution with the derivative of the delta function. From this and the commutativity and associativity of convolution, my version seems to follow.
*Differentiation corresponds to multiplication by i&omega; in the frequency domain. From this and the convolution theorem, the same result seems easily derived.
*For concreteness, let T be the &delta; function. Clearly d/dx(S * T) = d/dx S. Clearly (d/dx S) * T = d/dx S. And S * (d/dx T), the convolution of S with the derivative of the &delta; function, is also d/dx S.

So where have I gone wrong? [[User:Josh Cherry|Josh Cherry]] 16:10, 18 Apr 2004 (UTC)

I now think you have a point ... [[User:Charles Matthews|Charles Matthews]] 16:50, 18 Apr 2004 (UTC)

So, this was changed by an anonymous user on 31 January; should be changed back.

[[User:Charles Matthews|Charles Matthews]] 18:06, 18 Apr 2004 (UTC)

OK, I've made the change. [[User:Josh Cherry|Josh Cherry]] 20:19, 18 Apr 2004 (UTC)

==Proposal to move this page==

Common sense would suggest that a page titled [[distribution]] should be a disambiguation page. Within mathematics, the meaning of ''distribution'' in probability theory is quite different from the meaning contemplated in this article, and what of all the business operations pages that link here? I propose moving this to [[Schwartz distribution]] (which already has at least one link to it) or something else (mathematicians please suggest titles). This will be a lot of work because many links to this page need to get altered, some of them pointing to [[distribution (business)]] and some here, and perhaps some elsewhere. [[User:Michael Hardy|Michael Hardy]] 20:27, 3 May 2004 (UTC)

This is a good idea. I like the title [[Schwartz distribution]] for the current content. The current article should be a disambiguation page. Does [[User:Charles Matthews|Charles]] have an opinion on this? - [[User:Gauge|Gauge]] 01:17, 6 Jan 2005 (UTC)

=== title not adequate to contents ===

I think the definition should be make more readable and more correct by simplifying it: it currently applies to '' D' '' only and gives lengthy description of '' D '', while there are other spaces of distributions.
Why not give a concise definition (continuous linear forms), and then explain in more detail the different examples ?

Secondly, I think it would be good to make several pages on the different issues like Fourier transform,....
In putting "everything" on this page, much will be duplicated in many other places, which is a loss of 'energy' and of quality (because things are done superficially in many places, instead of thoroughly in one place.)
[[User:MFH|MFH]] 23:53, 21 Mar 2005 (UTC)

Firstly, I think we should have a [[generalized function]] page that discusses the various theories and some history. I am happy enough to have [[Schwartz distribution]] hanging off that; but are we going to have [[tempered distribution]] called that, or ''Schwartz tempered'' or ''tempered Schwartz'' or what? Well, that could wait. It is probably now overdue to have this page title as the disambiguation, and a splitting-up of topics. [[User:Charles Matthews|Charles Matthews]] 09:30, 22 Mar 2005 (UTC)

: We *do* have a page "[[generalized function]]" with some links and a history "stub". It's very incomplete, please feel free to complete it even partially! It's maybe a bit biased via what should be rather [[generalized function algebra]]s, if you dislike 'that'', I understand and I'll try to fix this: Say, let's put a 1-phrase description of Schwartz distributions (D's in the sequel) there, and move the "worked" example of Colombeau type algebra to a 'GF algebra' page. (I don't like too much the [[Colombeau algebra]] page which is too... "specialized", say....)
:Maybe some sheaf theoretic (supp, supp sing,...) aspects can remain on the "GF" page as far as they concern "ALL" theories of GF's, also the embedding stuff is to some extend "universal".
:I suppose your
:: ''Schwartz tempered'' or ''tempered Schwartz''
:is a joke...(unless you tacitly understood "distribution" added). Notions like "tempered" etc. are special cases of Schwartz D's and should go there, or better, "Schwartz D'" should contain only what applies to ALL Schwartz D's, and links to such special cases.
:On the other hand, I think it is justified that "distributions" concerns mainly Schwartz D's, with the "disambig stub" (regarding probability or other D's) at the top, I suppose if it's not precised, > 95% of all visitors will indeed look here for Schwartz D's.
:Finally, [[IMHO]], Fourier transformation of D's should be discussed or referred to on the FT page and only referenced, but not worked out, on the "D'" page. [[User:MFH|MFH]] 14:35, 24 Mar 2005 (UTC)

---
Jun 10, 2005
"Tempered distribution" article required. [[Fourier_transformation]] has a link to
"Tempered distribution" which redirects to "Distribution" - that's useless in the context of the
article "Fourier transformation". (The context was Lebesgue-integrable functoions and
the Delta function)

== Distribution ==

General "software distribution" is missing! Downloading, etc... --[[User:K.Nevelsteen|Kim Nevelsteen]] 21:48, 21 August 2005 (UTC)

* I guess you need to write that one in the [[distribution]] disambiguation page. [[User:Oleg Alexandrov|Oleg Alexandrov]] 23:24, 21 August 2005 (UTC)

== extend the concept of derivative to integrable functions ==

I partially agree with the comment of [http://en.wikipedia.org/w/index.php?title=Distribution_%28mathematics%29&curid=51955&diff=56550571&oldid=56502063 Cj67's edit].
But I think "integrable" is too restrictive - in some sense more restrictive than "continuous" (concerning decrease at infinity).
In the sense of "extend the concept of derivative of (resp. to ...) functions", I still believe the correct term is "locally integrable".

Of course not any distribution is a locally integrable function. In what this is concerned, I would even
advocate to put something before the first section starting with "The basic idea is..." - since for me, this may be the basic idea for "generalized function", but the basic idea in "distribution" is the idea of continuous linear forms; the fact that L¹loc and other spaces can be (densely) embedded then comes as a "surprise". &mdash; [[User:MFH|MFH]]:[[User talk:MFH|Talk]] 20:22, 2 June 2006 (UTC)

: It isn't really restrictive to say integrable, since there is also the phrase "and beyond". I think it is better in the introduction to be as un-technical as possible, so my preference is against the "locally", since probably many people don't know what that means. I won't fight too much against "locally integrable", but still there needs to be the phrase "and beyond", so I'm not sure how important it is be so exact regarding which functions are distributions. ([[User:Cj67|Cj67]] 22:40, 9 June 2006 (UTC))

== Angle bracket notation ==

Should the notation <math>\left\langle u, \phi \right\rangle</math> be introduced in the page? --[[User:Md2perpe|Md2perpe]] 23:10, 30 July 2006 (UTC)

:And perhaps also the connection with inner products. The identity <math>\langle f, g \rangle = \int f g</math> has some (read: a lot of) similarity with the inner product in function spaces. Is this actually the reason for this notation, or is this just a co-incidence? --[[User:CompuChip|CompuChip]] 13:40, 25 March 2007 (UTC)

== Added section; LaTeX ==

I added a short section about distributions as derivatives of continuous functions. I think it's an important result for understanding the idea. Also, I changed a display in the preceding section to LaTeX, so that it's clear that these are partial derivatives. Any objections to redoing that whole section in LaTeX? -- [[User:Spireguy|Spireguy]] 22:55, 9 January 2007 (UTC)

== Path integrals ==

The discussion of hyperfunctions includes the sentence,
"This extends the range of symbolic methods that can be made into rigorous mathematics, for example Feynman integrals."
I don't understand what that means -- the vast majority of Feynman integrals are still outside the
realm of rigorous mathematics, are they not? I would prefer to remove this sentence, or else to indicate more clearly
just which Feynman integrals can be understood better using the notion of hyperfunction. [[User:66.180.184.38|66.180.184.38]] 03:26, 10 March 2007 (UTC)

:I'll second that...I have never heard of hyperfunctions being used for path integrals. That claim needs explanation and a reference. -- [[User:Spireguy|Spireguy]] 02:56, 11 March 2007 (UTC)

==Cleaning Up==

I think the page is in bad shape. Two quick improvements would be:

(1) Getting rid of the "Probability Dist." part of "Basic Idea" as it has its own page.
(2) The section on Formal definition needs major shake up. For example we have pages for "compact support" or "locally integrable" so there is no need to redefine it here.
([[User:Hesam7|Hesam7]] 12:14, 20 March 2007 (UTC))

:Can you explain further how the page is "in bad shape"? It's frustrating to see a vague comment like that on a talk page without much support. Comments on the particular points:
:(1) The mention of probability distributions, as currently in the article, is appropriate: the intent is not to define what a probability distribution is, but to show that it provides an example of a distribution (in the sense of a generalized function). So I don't see a need to change this. It could perhaps be changed to mention arbitrary signed measures instead, but then one would lose the link between the terms.
:(2) It may be better to remove the explicit definitions of "compact support" and "locally integrable", but I'm not sure. One has to strike a balance between brevity and clarity; if you make the reader click on links for every associated definition, it gets quite tiresome. As it stands, I don't think that the section is overburdened with associated definitions, so I wouldn't feel a great need to change that. -- [[User:Spireguy|Spireguy]] 16:50, 23 March 2007 (UTC)

== compact support ==

I don't like the definition of compact support
(so, identically zero except on some closed, bounded set)

as it suggests that it must be non zero on the closed bounded set. This is not the case

suggest replacing with
(so, identically zero *outside of* some closed, bounded set) or
(so, identically zero *in the complement of* some closed, bounded set)
[[User:Mungbean|Mungbean]] 15:35, 21 March 2007 (UTC)
: You're welcome to change that. :) [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 02:44, 22 March 2007 (UTC)

== defining the topology of D(U) ==

"a [[sequence]] (or [[net (mathematics)|net]]) (φ<sub>''k''</sub>) converges to 0 if and only if there exists a compact subset ''K'' of ''U'' such that all φ<sub>''k''</sub> are identically zero outside ''K'', and for every ε > 0 and [[natural number]] ''d'' ≥ 0 there exists a [[natural number]] ''k''<sub>0</sub> such that for all ''k'' ≥ ''k''<sub>0</sub> the [[absolute value]] of all ''d''-th derivatives of φ<sub>''k''</sub> is smaller than ε."

I find the above definition of convergence in the formal definition section absurd. Normally, convergence does not depend on the behavior of the first few terms of a sequence. Why should ''all'' φ<sub>''k''</sub> be identically zero outside ''K'', instead of ''almost all''? --[[User:Acepectif|Acepectif]] 02:21, 26 June 2007 (UTC)

: well if K contains only the support of almost all <math>\phi_k</math>, then you can enlarge K suitably so that it contains the support of all <math>\phi_k</math>. - [[User:Saibod|Saibod]] 16:04, 8 July 2007 (UTC)

:: How? Let <math>\phi_k</math> = 1 (if k = 1), 0 (otherwise), where the domain of <math>\phi_k</math>'s is '''R'''<sup>n</sup>. Then the support of <math>\phi_1</math> is '''R'''<sup>n</sup>, so there is no way to enlarge K (which should remain compact) so that it contains the support. However, this sequence obviously converges to 0. --[[User:Acepectif|Acepectif]] 16:48, 8 July 2007 (UTC)

:::Remember that each <math>\phi_k</math> must have compact support to be in D(U) at all. Hence your proposed counterexample is a non-starter, and Saibod's response is entirely correct--you can enlarge K by unioning in a finite collection of compact sets and it will still be compact. -- [[User:Spireguy|Spireguy]] 17:00, 9 July 2007 (UTC)

== Convolution and distributions ==

From the article:
:if ''S'' is a tempered distribution and ψ is a ''slowly increasing'' infinitely differentiable function on '''R'''<sup>''n''</sup> (meaning that all derivatives of ψ grow at most as fast as [[polynomial]]s), then ''S''ψ is again a tempered distribution and
::<math>F(S\psi)=FS*F\psi.\,</math>
Is this indeed true? Can anyone provide a reference for this? I looked through the relevant chapter in Rudin (''Functional Analysis'', 1973) and I can only find a statement of this theorem for rapidly decreasing ψ. (The reference I found was Theorem 7.19(c), p.179 in Rudin). --[[User:Zvika|Zvika]] ([[User talk:Zvika|talk]]) 13:33, 19 December 2007 (UTC)

== convolution of distribution ==

if ''T'' is a distribution, how can we write T*H where H is the heavide function i.e H(x)=0 of x<0 and H(x)=1 if x≥1?
[[User:Dcharaf|Dcharaf]] ([[User talk:Dcharaf|talk]]) 19:05, 10 January 2008 (UTC)dcharaf


in this case since H(x-t) is nonzero only for x>t then <math> T*H= \int_{x}^{\infty}dtT(t) </math> <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:85.85.100.144|85.85.100.144]] ([[User talk:85.85.100.144|talk]] • [[Special:Contributions/85.85.100.144|contribs]]) 10:46, 26 May 2008 (UTC)</small><!-- Template:Unsigned -->

== Stieltjes integral?? ==

In the "Basic idea" section the angle bracket is defined as

: <math>\left\langle f, \varphi \right\rangle = \int_\mathbf{R} f \varphi \,dx = \int_\mathbf{R} \varphi \,df </math>

The second integral looks like a Stieltjes integral to me, but that would be pretty wrong here. So I would opt for removing it. It doesn't add any information that isn't contained in the first integral. If it should be retained, its meaning should at least be clarified. (ezander) [[Special:Contributions/89.183.10.169|89.183.10.169]] ([[User talk:89.183.10.169|talk]]) 23:33, 25 March 2008 (UTC)

== Problem of multiplication ==

There is currently the phrase "if 1/x is the distribution obtained by extending the corresponding function to a homogeneous distribution". This seems a bit confusing to me. For example we do not use the term homogeneous anywhere else in this article. Further, if I do the naive calculation of taking <math>\varphi(x)</math> to be a test function satisfying
<math>\varphi(x)\geq 0</math> and <math>\varphi(x)= 1</math> for |''x''|&lt;1, then the pairing

:<math>\int \frac{1}{x}\varphi(x)\,dx</math>

doesn't converge, so 1/''x'' doesn't define a distribution in the obvious way. I think what might be meant is the distribution

<math>T(\varphi)=\text{P.V.} \int \frac{1}{x}\varphi(x)\,dx </math>

(where we invoke the [[Cauchy principal value]]) is well defined. But I wanted a sanity check before I went polluting the article with my strange ideas. [[User:Thenub314|Thenub314]] ([[User talk:Thenub314|talk]]) 13:40, 1 May 2008 (UTC)

== Nonsense ==

I have removed the following text, since it makes no sense as stated. For one thing, of course the integral is divergent. The delta function itself would also produce a divergent integral if you were to perform such a trick. For another thing, it is incorrect to talk about the "value" of a distribution at a point. These aren't pointwise objects, and so don't usually have values. If anyone wants to improve it and restore the text, then feel free to do so. [[User:Silly rabbit|<font color="#c00000">siℓℓy rabbit</font>]] ([[User talk:Silly rabbit|<span style="color:#FF823D;font-family:Monotype Corsiva;cursor:help"><font color="#c00000">talk</font></span>]]) 05:09, 28 June 2008 (UTC)

{{quote|
Another example of the impossibility of multiplication is given by [[convolution theory ]] since <math>(2 \pi)D^{m} \delta(u) = \int_{-\infty}^{\infty}dx e^{iux}(ix)^{m} </math> then we would have <math>(2 \pi)^{2}i^{m+n}D^{m} \delta(u)D^{n}\delta(u)= \int_{-\infty}^{\infty}dx \int_{-\infty}^{\infty}dt (x-t)^{m} t^{n}e^{-iux} </math> , however the last integral is divergent for every value of 'u'.}}

== Omitted word? ==

Currently, the phrase "which is with respect to the weak-* topology" is in the article in the section titled "Operations on distrbutions".

Should this say "which is continuous with respect to the weak-* topology"?
[[User:DavidLHarden|DavidLHarden]] ([[User talk:DavidLHarden|talk]]) 18:38, 30 August 2008 (UTC)DavidLHarden

:Fixed! Thanks for catching the ommission. [[User:Silly rabbit|<font color="#c00000">siℓℓy rabbit</font>]] ([[User talk:Silly rabbit|<span style="color:#FF823D;font-family:Monotype Corsiva;cursor:help"><font color="#c00000">talk</font></span>]]) 19:04, 30 August 2008 (UTC)

== Derivative of dirac delta ==

The definition of the derivative of the dirac delta distribution currently reads
<math>\delta'(\varphi) = -\varphi'(0)</math>.

Shouldn't that be
<math>\langle \delta', \varphi \rangle = -\varphi'(0)</math>
instead? [[Special:Contributions/134.58.253.57|134.58.253.57]] ([[User talk:134.58.253.57|talk]]) 12:49, 5 November 2008 (UTC)

== Typography. ==

Someone came through and changed all &phi;'s to &Phi;'s because (I assume) the html &phi;'s doesn't look like latex <math>\phi</math>'s. Does any one object to my changing the &Phi;'s back to &phi;'s and also changing the <math>\phi</math>'s to <math>\varphi</math>'s? [[User:Thenub314|Thenub314]] ([[User talk:Thenub314|talk]]) 12:38, 2 February 2009 (UTC)

:I certainly don't object! --[[User:Bdmy|Bdmy]] ([[User talk:Bdmy|talk]]) 13:21, 2 February 2009 (UTC)

== Merge with generalized function article? ==

Disclaimer -- I don't know anything about distributions or generalized functions. I came across the terms in a textbook. But, I got the impression that the terms were equivalent. Is this the case? If so, why are there separate articles? If not, the difference should be mentioned, IMHO. The distribution article starts with a statement that it discusses generalized functions, so you can see there is occasion for confusion.

I'll let the knowledgeable editors sort it out. It just struck me as an opportunity for improvement.

[[User:Paul D. Anderson|Paul D. Anderson]] ([[User talk:Paul D. Anderson|talk]]) 06:05, 30 December 2009 (UTC)

== Typography 2 ==

Maybe it's a problem with my browser (Firefox 3.6.6) or computer (Mac OS 10.4.11), but for me on this page the character φ does not give the same greek letter as <math>\varphi</math>. This makes it pretty confusing since the text uses φ, but the formulas use <math>\varphi</math>. Anyone know how to fix this? [[User:Holmansf|Holmansf]] ([[User talk:Holmansf|talk]]) 16:41, 13 July 2010 (UTC)

== Topology on D(U) ==

Questions about the characterization of topology on D(U) in the article:

::In particular, a sequence (''S''<sub>''k''</sub>) in D'(''U'') converges to a distribution ''S'' if and only if
:::<math>\langle S_k, \varphi\rangle \to \langle S, \varphi\rangle</math>
::for all test functions ''&phi;''. This is the case if and only if ''S''<sub>''k''</sub> [[uniform convergence|converges uniformly]] to ::''S'' on all bounded subsets of D(''U''). (A subset ''E'' of D(''U'') is bounded if there exists a compact subset ''K'' of ''U'' and ::numbers ''d''<sub>''n''</sub> such that every ''&phi;'' in ''E'' has its support in ''K'' and has its ''n''-th derivatives bounded by ::''d''<sub>''n''</sub>.)

1. Sequences are used; is it true that the topology is metrizable?

2. Do we have a reference for that statement about uniform convergence on bounded sets?

Given that this is a weak-* topology (where the predual does not be seem to be separable), both would be very surprising. To me, anyway. [[User:Mct mht|Mct mht]] ([[User talk:Mct mht|talk]]) 19:07, 25 July 2012 (UTC)

:(1) The topology is not metrizable. Although the statement is true as stated, it's also true under more generality with "sequence" instead replaced by a filter (or net). (2) This follows easily from the [[Banach-Steinhaus theorem]], but at any rate is Theorem XIII on p. 74 of Laurent Schwartz's book "Theorie des distributions". [[User:Sławomir Biały|<span style="text-shadow:grey 0.3em 0.3em 0.1em; class=texhtml">Sławomir Biały</span>]] ([[User talk:Sławomir Biały|talk]]) 20:29, 25 July 2012 (UTC)

== Unclear equation ==
In first section,
: <math>\left\langle T_f, \varphi \right\rangle = \int_\mathbf{R} f \varphi \,dx </math>
It is no entirely clear if it's function composition or multiplication, so I've changed it by including arguments in it. I hope it's correct. <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/78.128.190.8|78.128.190.8]] ([[User talk:78.128.190.8|talk]]) 17:06, 9 October 2012 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->

== Kleinert's Work on Multiplication of Distributions ==

The quoted results of Kleinert et al. on the Multiplication of Distributions in Quantum Mechanical Calculations do not seem relevant to the general theory of distributions for me. Therefore I would opt to remove them.
(As already mentioned they concern properties of and methods to handle distributions which arise in QM, which are rather special distributions and not general Schwartz- or tempered distributions.)
(Further emphasized as they are not quoted in mathematical encyclopedias. e.g. http://www.encyclopediaofmath.org/index.php/Multiplication_of_distributions
has an entire article including references on the multiplication problem)
[[Special:Contributions/87.165.196.119|87.165.196.119]] ([[User talk:87.165.196.119|talk]]) 05:53, 8 August 2013 (UTC)

:I agree that this seems like undue weight. It really should be a separate article anyway, and could be modeled on the EoM article, provided there is an interest in writing it. [[User:Sławomir Biały|<span style="text-shadow:grey 0.3em 0.3em 0.1em; class=texhtml">Sławomir Biały</span>]] ([[User talk:Sławomir Biały|talk]]) 16:39, 4 March 2014 (UTC)

== Deriving Dirac delta distribution from Heaviside step function ==

The article gives an example:

: <math>\left\langle H', \varphi \right\rangle = - \left\langle H, \varphi' \right\rangle = - \int_{-\infty}^{\infty} H(x) \varphi'(x) dx = - \int_{0}^{\infty} \varphi'(x) dx = \varphi(0) - \varphi(\infty) = \varphi(0) = \left\langle \delta, \varphi \right\rangle,</math>

But how was the first step done? I spent a while thinking about it, and then thought it must be an application of integration by parts. I edited the line before this one to say, "For any test function φ, using [[integration by parts]] we have", but that edit got reverted. Am I incorrect or is there another reason not to explicitly state this?

[[User:Monsterman222|Monsterman222]] ([[User talk:Monsterman222|talk]]) 20:51, 9 March 2014 (UTC)

:It's the ''definition'' of the derivative <math>H'</math> of the distribution ''H'' that
::<math>\left\langle H', \varphi \right\rangle = - \left\langle H, \varphi' \right\rangle \ .</math>
:This section is about showing that this definition of derivative for ''H'' yields the Dirac delta distribution. The definition is chosen so that for distributions defined by functions, the derivative of the distribution agrees with the distribution obtained from the derivative of the function, and in that case the definition is formally integration by parts on the associated functions. [[User:Deltahedron|Deltahedron]] ([[User talk:Deltahedron|talk]]) 20:59, 9 March 2014 (UTC)

== Distributions are functions ==

There is an *extremely* basic mistake that is repeated several places in this article, which says that distributions are not functions. The article even links to the Wikipedia article on functions, which contradicts this article. What the article means to say is that distributions are not functions from a real vector space back to the reals. But it defines distributions as linear functionals. So they are not only functions, but they are linear functions. This needs to be corrected. Wikipedia should not have mathematical mistakes.[[Special:Contributions/50.181.78.116|50.181.78.116]] ([[User talk:50.181.78.116|talk]]) 00:21, 11 March 2014 (UTC)

:(Note: originally I did not see the above post, because it was posted at the top of this discussion page contrary to the instructions.) Recently an IP editor has been trying to push some very technical material into the first paragraph of the article, apparently with the goal of emphasizing that distributions ''are'' functions of a kind (continuous linear functionals on a topological vector space). Although it is technically true that distributions are modeled as continuous linear functionals, I don't think that is a helpful perspective to start the article with. The reason one models distributions in this fashion is precisely because one wants a nice generalization of functions. This is Laurent Schwartz's own perspective in the "Théorie des distributions". It is not a "mathematical error" to consider distributions as a generalization of functions, as the IP editor insists. When we say "distribution on U", we do not mean "continuous linear functional on U". [[User:Sławomir Biały|<span style="text-shadow:grey 0.3em 0.3em 0.1em; class=texhtml">Sławomir Biały</span>]] ([[User talk:Sławomir Biały|talk]]) 01:12, 11 March 2014 (UTC)

:: Sorry for posting at the top, that was an accident. A function is an object that assigns a unique element of some set to an element of another set. In modern language, it's a morphism in the category of Set. This has been the prevailing definition for at least the last 50 or 60 years. This is the definition in Wikipedia, and it accurately describes distributions. There is a historical usage of "function" which makes it different from "mapping" or "transformation" but which is no longer used and is not mentioned by Wikipedia. You are confusing this old definition of "function" with the current one. For example, on the old definition, a map between two spheres would not be a function, but a map from the sphere to the real numbers, or from the sphere to the plane would be. This usage is *sometimes* still used in analysis and differential geometry books, but not very commonly and only if they define precisely what they mean by function. Instead of doing this, you link to Wikipedia's definition of function, which is the universally accepted mathematical definition, but which contradicts the usage of your article. So either (1) stop linking to the Wikipedia article on functions and instead define your own notion of function, or (2) accurately explain the relationship between distributions and functions. Most pure math students that have taken up to Sophomore or Junior courses would recognize functions as being mappings between any two sets. If they are confused by your insistence that distributions are not functions, and try to resolve their confusion by clicking on the link to the function article, they are going to be even more bewildered. So for the sake of having a truthful, precise, and encyclopedic article the issue must be addressed. You are free to make it as non-confusing as you'd like. But you can't both say distributions generalize functions and also claim to be using the standard definition of function.
::And of course, you don't mean that "f is a function on U" when you say "f is a distribution on U". But there is a large difference between saying "f is a function on a set associated to U, which for convenience we treat as being somewhat like a function on U" and "f is not a function." For example, the former is true but the latter is false. <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/50.181.78.116|50.181.78.116]] ([[User talk:50.181.78.116|talk]]) 13:06, 21 March 2014 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->
{{od}} The article does explain that distributions are continuous linear functionals on a topological vector space in the ''relevant'' section. This doesn't obviate the fact that distributions generalize functions. In fact, a large part of the world calls distributions "generalized functions". As I pointed out, a distribution on ''U'' is not a function on ''U''. Rather the idea of distributions generalizes the idea of a function on ''U''. The point is that functions on ''U'' are distributions, but the converse is not true. Now it is true that at some level, distributions are ultimately described as functions. But it is very misleading to say that they "are" functions, even if it is actually true at some level. An easier analogy might be sections of a [[line bundle]]. These are also functions (in fact, they are very special kinds of functions), but it is grossly misleading to describe them informally as functions. They also generalize functions on a manifold by allowing the values of functions to be globally twisted by the topology of the manifold. A student would be more likely to come away with the wrong idea if we said that "sections of line bundles are just special kinds of functions" just as if we were to say "distributions on a subset of Euclidean space are just special kinds of functions". Arguably, in fact, the latter statement would not be true. And although it isn't exactly what [https://en.wikipedia.org/w/index.php?title=Distribution_%28mathematics%29&oldid=599065154 this revision] states, it's hard to see how that is not going to be the takeaway message for the hypothetical Sophomore or Junior that you want me to consider. In fact, that revision would likely be utterly bewildering to anyone, regardless of their mathematical background, since the first paragraph asserts at least two contradictory things without any clear explanation.

In light of a disagreement between editors, I propose that we just go by what high quality reliable sources say, sources like [[Laurent Schwartz]] and [[Israel Gel'fand]], as well as both cited sourced by Vladimirov, that distributions generalize functions. This conveys the correct intuition at an informal level, and the formal details are contained in the body of the article. [[User:Sławomir Biały|<span style="text-shadow:grey 0.3em 0.3em 0.1em; class=texhtml">Sławomir Biały</span>]] ([[User talk:Sławomir Biały|talk]]) 13:20, 21 March 2014 (UTC)

::If I'm reading your response correctly, I think I see the kernel of the issue. You say that sections generalize functions on a manifold by allowing them to be twisted by the topology. That suggests to me that you think of a function on a manifold to be a map from the manifold to Euclidean space. Is my reading correct? If so, this is the older more restricted notion of function that I've been talking about. It's used primarily in physics, analysis, and some differential geometry, but it conflicts with the modern notion of function. Most importantly, it conflicts with the definition of function used on Wikipedia. And for that reason, the article needs to make it clear that they're using a non-standard definition of function (even if that definition may be standard among people that use distributions the most). This article needs to be readable by people who are not in fields that use that restricted definition of function. If I am incorrect in my assessment that you think of functions on a manifold as mappings into Euclidean space, please let me know. I personally would find it rather odd if somebody told me that sections generalize functions. And certainly sections are considered ordinary everyday functions on Wikipedia: "In the mathematical field of topology, a section (or cross section)[1] of a fiber bundle π is a continuous right inverse of the function π." There *are* objects that generalize the notion of a function; for example, the functor assigning to each based topological space its fundamental group. Since the collection of topological spaces and the collection of groups are both proper classes rather than sets, the functor is not a function.
::I get what you're saying about distributions not being a function "on U". But you have the ability to say precisely what you mean. You can say, for example, "a distribution on U generalizes the idea of a function on U". But you can't say "distributions generalize the idea of functions." It's the logical difference between saying "f is not an A with property B" and "f is not an A". As it stands, the article imports the definition of functions from Wikipedia by linking to the article. It claims that distributions are not functions, but then defines them as functions. That is self-contradictory. You are free to use older sources such as Schwartz, but make sure that the definition of function in the article is explicitly the same as the one used by Schwartz. I haven't read any French analysis papers from the 1950s, but there's a chance he's using the older non-Wikipedia definition. You are free to use that definition, but define it explicitly and be clear that it differs from the definition given in the function article. <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/50.181.78.116|50.181.78.116]] ([[User talk:50.181.78.116|talk]]) 13:59, 21 March 2014 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->
::: I think you might have the wrong idea about what the lead of the article is supposed to do. It is supposed to serve as an informal introduction to the topic. Formal details appear in the body of the article. Basically all authors I have ever seen (not just "older sources") describe distributions as generalizing functions. Perhaps this is not true in a sense that is immediately made mathematically precise, but that's okay for the purposes of the lead. Precise details are quick to follow, and the sense in which distributions do generalize functions, as well as the kind of functions they generalize, is already made quite clear in the text of the article. In any event, I have removed the link to "function", since that seems to be your main bone of contention with the article. I consider this issue settled. [[User:Sławomir Biały|<span style="text-shadow:grey 0.3em 0.3em 0.1em; class=texhtml">Sławomir Biały</span>]] ([[User talk:Sławomir Biały|talk]]) 14:37, 21 March 2014 (UTC)

:::: Hi, sorry to have disappeared for so long. I saw the change in the first line, and I am happy with how it reads now. I agree that the lead should be a less formal discussion. The issue wasn't so much the informality of the discussion, it was that the language used before was incorrect. I think it's possible (but challenging) to make precise statements without being technical. I think the article now is a good compromise, so thank you. <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/50.181.78.116|50.181.78.116]] ([[User talk:50.181.78.116|talk]]) 17:46, 29 March 2014 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->

== Distinguish hatnote for Probability distribution ==

Regarding [[Special:Diff/672841592/672844760]], I'm not sure why this counts as a [[WP:ONESHORTHAT]] violation and the appropriate remedy is to remove the Distinguish template. The sentences I'm reading that seem relevant are addressed as follows:

: "Direct links to other articles should be limited to circumstances immediately following a page move or redirect change or if the other article could be reasonably expected by a significant number of readers to be at the title in question"

A significant number of readers would expect probability distributions to be distributions in the sense of this article, not just for users to think that everything in math referred to only as a distribution will be covered by the Wikpedia article called "Distribution (mathematics)". This is a consequence of the fact that, since probability distributions are not always just called distributions, the naming conflict is indirect, and it's not apparent to everyone who meets the term distribution in the term probability distribution that in that case it has a definition that conflicts with the sense meant here. Consider also that [[probability distribution]] has so many alternative definitions that the fact that the definitions of distribution conflict is not apparent to the reader. It could be stated in that article that in the context of probability and statistics probability distributions are often just called distributions, and that this is not a special case of the linear functionals (in many fields there are conventions where a special case is so pervasive and central that the general term is used to refer to it), but the confusion can arise in someone on either side of the pursuit of information, so the conflicting definitions have to be addressed symmetrically between these two articles.

: "One single hatnote is greatly preferred to two (or worse, more). There are many different templates that can easily accommodate several links; use one of them rather than two hatnote templates."

Yes this is preferred, but that doesn't imply this constraint must be imposed on all articles. There is a very clear reason why the one hatnote used is insufficient. Is David Eppstein saying a single custom hatnote should be used to inform the reader about both the risk of confusion and that the term distribution can be used to refer to a probability distribution? I am concerned that a reader scanning the hatnotes will classify the hatnote as a simple other uses template based on how the hatnote starts and ends and will dismiss the importance of the appearance of the highlighted term [[probability distribution]] occuring in the template before reading the hatnote in full. So mentioning the risk of confusion in the lede would be preferable to putting it in a hatnote that readers don't associate with risk of confusion. ᛭ [[User:LokiClock|LokiClock]] ([[User talk:LokiClock|talk]]) 07:56, 24 July 2015 (UTC)

:This is a lot of words to justify the ridiculous situation of having ''two'' hatnotes that contain the same link. It's hard to imagine any situation where that would be acceptable. That practice is certainly not supported by the guidelines, nor is it something that I've ever seen on any article. In addition, it does actually violate [[WP:ONESHORTHAT]]. Why you think a special case should be made to accommodate two links to exactly the same article is rather mystifying.
:In all of this wall of text, I cannot see any reason why the current hatnote is not already clear enough: "This article is about generalized functions in mathematical analysis. For the probability meaning, see [[Probability distribution]]." So, this tells us that there is another meaning of the term in probability theory, and for that meaning we should go see that article. The conflict in meanings already is "addressed systematically between these two articles", without the need to go outside the guidelines to accommodate the issue. [[User:Sławomir Biały|<span style="text-shadow:grey 0.3em 0.3em 0.1em; class=texhtml">Sławomir Biały</span>]] ([[User talk:Slawekb|talk]]) 11:42, 24 July 2015 (UTC)

Latest revision as of 07:20, 17 March 2024

The article is far too technical

[edit]

The series of edits made by the same user since May 2020 has made this article overly technical, and no longer in the spirit of a Wikipedia article on the subject. To see thepoint, compare the Sep 22, 2020 version with the last version prior to the series of edits by the same editor, the April 20, 2020 version.

The April 2020 version opens with a Basic Idea section that illustrates the essence of the subject. The Sep 2020 version introduces a lot of notation and ancillary concepts, much of which does not seem necessary to introduce the definition of a test function.

I would strongly urge that the article be edited back to resemble the April 2020 version, and any advanced material being added after the the simplified introduction to the subject.

Undsoweiter (talk) 19:08, 23 September 2020 (UTC)[reply]

The article that you linked to contained false information such as the last sentence of this claim: " The elements of D(U) are the infinitely differentiable functions  : U → R with compact support – also known as bump functions. This is a real vector space. It can be given a topology by defining the limit of a sequence of elements of D(U)."
You can not define the topology using sequences and this is a non-trivial fact that has been proven. The above false statement should not appear in this article. The correct definition of the canonical LF-topology on the space of test functions is unfortunately technical.
Also, the old article was missing important information such as how to extend differential operators to distributions, which is arguably is one of the more important uses of distributions.
Distributions are unfortunately an innately technical topic. However, I am fine with simplifying the article but not at the expense of adding false-but-simple information or removing important information. Mgkrupa 22:26, 23 September 2020 (UTC)[reply]
My comment about technicality and style of the article does not mean I endorse the accuracy of every claim in the April version I refer to. If there are incorrect statements, they can be corrected without resorting to the drastic changes you have made to the article. By all means, fix errors. But this does not justify the excessively technical approach you have followed in your rewrite.

Undsoweiter (talk) 20:31, 30 September 2020 (UTC)[reply]

Over time, I will try to make this article less technical. However, we first need to establish what assumptions can and can not be made about this article and its "typical" reader. According to Wikipedia:Manual of Style/Mathematics, this article should follow the following guidelines (as well as others not listed here). It should be be written "one level down", which means:
  • "consider the typical level where the topic is studied (for example, secondary, undergraduate, or postgraduate) and write the article for readers who are at the previous level." Also,
  • "articles on undergraduate topics can be aimed at a reader with a secondary school background, and articles on postgraduate topics can be aimed at readers with some undergraduate background."
  • "Articles should be as accessible as possible to readers not already familiar with the subject matter."
  • "When in doubt, articles should define the notation they use."
  • "If an article requires extensive notation, consider introducing the notation as a bulleted list or separating it into a "Notation" section."
  • "An article about a mathematical object should provide an exact definition of the object, perhaps in a "Definition" section after section(s) of motivation."
  • "Writing one level down also supports our goal to provide a tertiary source on the topic, which readers can use before they begin to read other sources about it."
I think that it is safe to assume that the reader has knowledge of calculus. But before we start editing this article to make it less technical, it's important to know what else we can assume about a "typical" reader of this article. This is important because, for example, whenever it is reasonable and possible to do so, then terminology that a reader is unlikely to be familiar with should be briefly defined/described within this article, instead of just having a link to the article about the term (this is because ideally, a "typical" reader should not have to go down a rabbit hole of Wikipedia links and search through various articles in order to understand something stated in this article about distributions). So we need to agree on the following (non-exhaustive) list of assumptions before we can start rewriting this article:
  1. Is it safe to assume that the reader is likely an advanced undergraduate or higher? (I personally think so).
  2. Is it safe to assume that the reader is likely a graduate student or higher?
  3. Is it safe to assume that the reader is likely a mathematics, physics, or engineering student?
  4. Is it safe to assume that the reader has studied metric spaces? (I personally think that it is).
  5. Is it safe to assume that the reader has knowledge of general topology (in particular, of non-metrizable topological spaces)?
  6. Is it safe to assume that the reader has studied Banach spaces? (If not, then the Fréchet spaces and related notions that are used in this article will need more detailed explanations).
  7. Is it safe to assume that the reader has studied Fréchet spaces? My guess is probably not and so the reader should not be assumed to know about Fréchet space. But if they are familiar with the basics of Banach spaces then the required knowledge for Fréchet spaces can be described using Banach space terminology.
  8. Is it safe to assume that the reader has studied non-metrizable topological vector spaces? I think that this can not be assumed. However, unfortunately, neither the canonical LF topology nor the topology on the space of distributions is a sequential space so this topology can not be described using sequences (let along a metric). Suggestions about how to define and describe these topologies to readers who are not used to dealing with non-sequential (and also non-metrizable) spaces would be welcome. The current description of these topologies is (unfortunately) technical and I'd like for it to be less technical but I'm not sure how to make it less technical.
Mgkrupa 18:51, 26 October 2020 (UTC)[reply]
I believe distributions are common in physics (please correct me if I'm wrong), and I am not sure that physics students will know about Frechet spaces (I am not a physics student myself, so I don't even know if physics undergrads learn any topology) – so perhaps the ultimate goal is to make this article accessible to physicists? It would be helpful if others could give their thoughts on this.
Also, I agree that ideally "a 'typical' reader should not have to go down a rabbit hole of wikipedia links" to understand an article, but I would say that this article is (at least somewhat) an exception to this rule, because it requires a much higher than average amount of prerequisite knowledge (again, please correct me if I am wrong) Joel Brennan (talk) 00:15, 18 November 2020 (UTC)[reply]
I am in favor of creating a separate article for the canonical LF topology (this topology has been studied enough to warrant its own article) and in this way we can simplify the article on distributions by placing some technical details into this new article. Mgkrupa 23:27, 23 September 2020 (UTC)[reply]
For instance, we can make the presentation less general by replacing k with ∞. Mgkrupa 23:51, 23 September 2020 (UTC)[reply]
I recently learned distributions and I found it frustrating the first time I glanced at this article, hoping for an overview of the subject. This issue was not even technicality: I understand Fréchet spaces and topological linear spaces. The article was just not reader-friendly and felt more like a giant list of notation. Compare this to the article on Sobolev space, which starts off with motivation and looks much less cluttered. 74.101.253.193 (talk) 22:11, 16 August 2021 (UTC)[reply]

Completely agree about moving discussion of TVS topologies to another article

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No one looking for a straightforward explanation of distributions will be able to get through the current article.

As far as the continuity of distributions is concerned, all one needs to say in this article is that a linear functional (or ) is continuous if and only if it is sequentially continuous i.e., if in (supports of all are contained in a compact subset of and the sequence and all of its derivatives converge uniformly on ), then in . Explanations of sequential continuity vs. continuity in TVS can be given elsewhere.

Alternatively, you could say that is continuous if and only if for every compact subset of there exist , such that for every supported in . But I think the sequentially definition is more accessible.

However, these changes would require a lot of rewriting of the first part of this article.Reader634 (talk) 07:45, 12 December 2021 (UTC)[reply]

Comment

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I think the recent changes are absolutely in the right direction. I made a few minor edits, mostly to remove "scare" quotes about the difficulties of defining the topology on the space of distributions (it's enough to refer to the relevant article). Also, I think more of the discussion on topologies can be moved elsewhere, and all (or nearly all?) of the material on spaces can be removed from this article, since test functions are the fundamental ones for distributions. As a minor point, I'd denote the regular distribution associated with multiplication by a locally integrable function by rather than (which could be confused with differentiation) but I don't think there's a standard notation here. Reader634 (talk) 09:08, 31 December 2021 (UTC)[reply]

Some notations not explained

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Same as in Spaces of test functions and distributions: notation first appearing in 6 (Spaces of distributions) seems not to be explained anywhere on this page. Mamuka Jibladze (talk) 17:23, 31 January 2022 (UTC)\][reply]

KINDLY DO NOT ERASE OTHER PEOPLE'S POSTS WHEN YOU POST YOUR OWN. 2601:200:C000:1A0:7097:9C72:4BFB:717D (talk) 00:11, 18 May 2022 (UTC)[reply]

Tempered distributions

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"Tempered distributions" is used but not defined in this article. I've heard they are "distributions that are upper bounded by polynomials", which I'm not quite sure how to make precise. I've also heard the more and difficult definition in terms of (the extension of?) linear functions on Schwartz function. Needs to be made clear in this article. Jess_Riedel (talk) Jess_Riedel (talk) 19:07, 16 March 2024 (UTC)[reply]

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