Talk:Distribution (mathematics)/Archive 1

Archive 1

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. 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 Charles have an opinion on this? - 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.) 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. 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 algebras, 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. 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... --Kim Nevelsteen 21:48, 21 August 2005 (UTC)

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

extend the concept of derivative to integrable functions

I partially agree with the comment of 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". — 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. (Cj67 22:40, 9 June 2006 (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? -- 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. 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. -- Spireguy 02:56, 11 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) Mungbean 15:35, 21 March 2007 (UTC)

You're welcome to change that. :) Oleg Alexandrov (talk) 02:44, 22 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. (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. -- Spireguy 16:50, 23 March 2007 (UTC)

Angle bracket notation

Should the notation $\left\langle u,\phi \right\rangle$ be introduced in the page? --Md2perpe 23:10, 30 July 2006 (UTC)

And perhaps also the connection with inner products. The identity $\langle f,g\rangle =\int fg$ 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? --CompuChip 13:40, 25 March 2007 (UTC)

defining the topology of D(U)

"a sequence (or net) (φ_k) converges to 0 if and only if there exists a compact subset K of U such that all φ_k are identically zero outside K, and for every ε > 0 and natural number d ≥ 0 there exists a natural number k₀ such that for all k ≥ k₀ the absolute value of all d-th derivatives of φ_k 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_k be identically zero outside K, instead of almost all? --Acepectif 02:21, 26 June 2007 (UTC)

well if K contains only the support of almost all $\phi _{k}$, then you can enlarge K suitably so that it contains the support of all $\phi _{k}$. - Saibod 16:04, 8 July 2007 (UTC)

How? Let $\phi _{k}$ = 1 (if k = 1), 0 (otherwise), where the domain of $\phi _{k}$'s is Rⁿ. Then the support of $\phi _{1}$ is Rⁿ, 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. --Acepectif 16:48, 8 July 2007 (UTC)

Remember that each $\phi _{k}$ 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. -- 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 Rⁿ (meaning that all derivatives of ψ grow at most as fast as polynomials), then Sψ is again a tempered distribution and

$F(S\psi )=FS*F\psi .\,$

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). --Zvika (talk) 13:33, 19 December 2007 (UTC)

Stieltjes integral??

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

$\left\langle f,\varphi \right\rangle =\int _{\mathbf {R} }f\varphi \,dx=\int _{\mathbf {R} }\varphi \,df$

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) 89.183.10.169 (talk) 23:33, 25 March 2008 (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? Dcharaf (talk) 19:05, 10 January 2008 (UTC)dcharaf

in this case since H(x-t) is nonzero only for x>t then $T*H=\int _{x}^{\infty }dtT(t)$ —Preceding unsigned comment added by 85.85.100.144 (talk • contribs) 10:46, 26 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. siℓℓy rabbit (talk) 05:09, 28 June 2008 (UTC)

Another example of the impossibility of multiplication is given by convolution theory since $(2\pi )D^{m}\delta (u)=\int _{-\infty }^{\infty }dxe^{iux}(ix)^{m}$ then we would have $(2\pi )^{2}i^{m+n}D^{m}\delta (u)D^{n}\delta (u)=\int _{-\infty }^{\infty }dx\int _{-\infty }^{\infty }dt(x-t)^{m}t^{n}e^{-iux}$ , 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"? DavidLHarden (talk) 18:38, 30 August 2008 (UTC)DavidLHarden

Fixed! Thanks for catching the ommission. siℓℓy rabbit (talk) 19:04, 30 August 2008 (UTC)

Derivative of dirac delta

The definition of the derivative of the dirac delta distribution currently reads $\delta '(\varphi )=-\varphi '(0)$.

Shouldn't that be $\langle \delta ',\varphi \rangle =-\varphi '(0)$ instead? 134.58.253.57 (talk) 12:49, 5 November 2008 (UTC)

Typography.

Someone came through and changed all φ's to Φ's because (I assume) the html φ's doesn't look like latex $\phi$'s. Does any one object to my changing the Φ's back to φ's and also changing the $\phi$'s to $\varphi$'s? Thenub314 (talk) 12:38, 2 February 2009 (UTC)

I certainly don't object! --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.

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 $\varphi$. This makes it pretty confusing since the text uses φ, but the formulas use $\varphi$. Anyone know how to fix this? 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_k) in D'(U) converges to a distribution S if and only if

$\langle S_{k},\varphi \rangle \to \langle S,\varphi \rangle$

for all test functions φ. This is the case if and only if S_k 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_n such that every φ in E has its support in K and has its n-th derivatives bounded by ::d_n.)

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. 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". Sławomir Biały (talk) 20:29, 25 July 2012 (UTC)

Unclear equation

In first section,

$\left\langle T_{f},\varphi \right\rangle =\int _{\mathbf {R} }f\varphi \,dx$

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. — Preceding unsigned comment added by 78.128.190.8 (talk) 17:06, 9 October 2012 (UTC)

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) 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. Sławomir Biały (talk) 16:39, 4 March 2014 (UTC)

Deriving Dirac delta distribution from Heaviside step function

The article gives an example:

$\left\langle H',\varphi \right\rangle =-\left\langle H,\varphi '\right\rangle =-\int _{-\infty }^{\infty }H(x)\varphi '(x)dx=-\int _{0}^{\infty }\varphi '(x)dx=\varphi (0)-\varphi (\infty )=\varphi (0)=\left\langle \delta ,\varphi \right\rangle ,$

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?

Monsterman222 (talk) 20:51, 9 March 2014 (UTC)

It's the definition of the derivative $H'$ of the distribution H that

$\left\langle H',\varphi \right\rangle =-\left\langle H,\varphi '\right\rangle \ .$

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. 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.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". 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. — Preceding unsigned comment added by 50.181.78.116 (talk) 13:06, 21 March 2014 (UTC)

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 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. 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. — Preceding unsigned comment added by 50.181.78.116 (talk) 13:59, 21 March 2014 (UTC)

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. 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. — Preceding unsigned comment added by 50.181.78.116 (talk) 17:46, 29 March 2014 (UTC)

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. ᛭ 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 hatnote of that article tells us "This article is about probability distribution. For generalized functions in mathematical analysis, see Distribution (mathematics)." So the conflict in meanings already is "addressed symmetrically between these two articles", without the need to go outside the guidelines to accommodate the issue. Sławomir Biały (talk) 11:42, 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."

So you're saying that because conclusion seems absurd the argument must be invalid? All I'm asking you to imagine is this current situation.

"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."

The practice isn't supported by the guidelines, but it's not forbidden either. All the guidelines state is that one hatnote is preferred to multiple hatnotes. Repeating that it violates WP:ONESHORTHAT without explaining why it does like I asked, nor resolving the unclarities that I presented to the existence of a WP:ONESHORTHAT violation, is not contributing to the discussion. Just because you haven't seen something before doesn't mean it isn't the correct solution. Argon also uses these hatnotes in conjunction, just not to link to the same article, and that doesn't make it any more or less of a WP:ONESHORTHAT violation than this, because WP:ONESHORTHAT doesn't say anything about a link appearing in two of the hatnotes when more than one hatnote is needed.

"Why you think a special case should be made to accommodate two links to exactly the same article is rather mystifying."

Upon elimination of other solutions. I've supported two solutions, and rejected the current state of the article as I don't consider it a solution. Another solution that just occurred to me which would handle both use cases is to eliminate a mention in the Other uses template, use a Distinguish template, and modify the Distinguish template to add in "which is [often; in probability and statistics] simply called a distribution". But now that I think about it, a consequence of using Distinguish here is that if I'm reading the article and I see Probability distribution appear in Distinguish, as opposed to Other uses, then I'm going to infer that one is not a specialization of the other because when one term is a specialization of another but goes by the same name, the practice is to construe references to the general object to mean references to the special case (of ones we're interested in using, e.g. seeing as how we're studying probabilities not areas) and in that respect is always a deliberate confusion of the specific with the general for prosaity.

I wonder if the source of the original disagreement is that you feel the uses of the two template are the same here because you don't think "Not to be confused with Probability distribution" insinuates this, and only says the term "probability distribution" may not have the same definition as this article's concept called distribution, something Other uses states equally well. My argument is that Other uses would not suggest equally well that the two concepts of distribution have the same definition, but the solution I've just proposed would make that more obvious: The purpose of Distinguish here is to say not to confuse the concepts, whereas Other uses only says not to confuse the terms. "This hatnote is generally used when there is an ambiguity that is not necessarily literal ...". A reader isn't guaranteed to interpret Distinguish that way, but the template is associated with that purpose, for example from common names given to multiple animals with clearly distinct article titles. This association also gives it a salience that means it is less likely to be skimmed over than Other uses, as I described in my first post. Other uses states that there are two concepts going by the name "distribution," but it doesn't indicate that the concept of distribution in the term "probability distribution" is not the same as this article's concept of distribution, only that one of the concepts out of all concepts of distribution is the same as "probability distribution". The omitted information, that the "distribution" in "probability distribution" does not have a formal meaning of its own and in particular does not mean a distribution in the sense of this article, is much more important to convey than merely that there are multiple meanings of the term "distribution."

Note that in the taxonomy example, "Not to be confused with (scientific name)" would suffice to indicate that the second animal may be called the same thing as the first based on the very different article titles, thus covering a case where Other uses would be needed. Almost the same use case for Distinguish to this one occurs when Ketone uses Distinguish and not Other uses to link to ketone bodies which appear to be called ketones in that article. To contrast, the two terms have related meanings, and the term formation pattern "x → x body" does not match some existing pattern wherein the juxtaposition of terms has the conjunction of meaning ("x → probability x" is an instance of such a pattern when applied to measure, but not when applied to distribution; "x → x measure" is to my knowledge always an instance of such a pattern, and you would expect "distribution" to behave like "measure" in this regard). The way Hammurapi relates to Ammurapi is similar - yes, you could save a hatnote and mention Ammurapi in the other uses in addition to the disambiguation page for Hammurabi, but still, they're both people and thus there's a reason their similar names create a source of confusion which has to do with the similarity of the subject matter, though the case is weaker here because referring to Ammurapi as a different person in the Other uses template would fully explain why they might be confused and what's not to be confused between them. So I can see why the editors concerned would have either preference (the earliest hatnote set in the article was just to link to the computer game Hamurabi, to which the Distinguish note was added before the Hamurabi dismbiguation page replaced the reference to the computer game, so there was at least one point when simply adding an Other uses style mention for Ammurapi would have been done if the editors felt it would be adequate to). The point is, the two hatnotes are in practice juxtaposed judiciously based on the nature of the potential confusion. Luminance linking to Luma (video) for its abusive "luminance" synonym is another example where the Other uses hatnote is not deemed to be enough.

Maybe saying "Not to be confused with the concept in probability and statistics by the same name" would also make it clear that the concepts themselves are what's distinct (you could technically talk about general name / special case situations the same way, but you wouldn't), even if the term "probability distribution" not appearing isn't preferred, without mentioning that a probability distribution is simply called a distribution, which may or may not confuse the point about why this and that are not to be confused.

"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.'"

For each thing I've said that you think doesn't show the current hatnote is unclear, please explain that it doesn't or ask for detailed elaboration as to how it does. Otherwise you're just saying you failed to comprehend the argument, an effect you can induce in yourself when you're not willing to examine someone else's position. Also when you call two paragraphs dedicated to opening a discussion about a collaborative dispute between 3 people while taking into account policy questions posed to date a wall of text it makes you look like an infant. And by the way, the length of my responses to disputes tends to be inversely proportional to the specificity of the complaint.

"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."

And I've explained why that isn't specific enough information to prevent confusion.

The Other uses template often doesn't tell the reader enough when Distinguish is needed unless the other article has the same name, say with a (different) subject distinguisher in parentheses. If we created Distribution (probability and statistics) as a redirect and used that link in the Other uses template instead, that would also solve the problem (not saying it wouldn't create another). ᛭ LokiClock (talk) 15:19, 26 July 2015 (UTC)

"So you're saying that because conclusion seems absurd the argument must be invalid? " Well, yes. If the result is an absurd situation, then there isn't any point in having an argument in the first place. This is an encyclopedia, with guidelines, not an amateur debate club. I don't see further discussion as constructive, so this will be my last post in this thread. Sławomir Biały (talk) 15:39, 26 July 2015 (UTC)

So you disagree with me. We can agree that the problem has been addressed, even if we disagree that it is a satisfying solution. If that's all we can agree on, that's enough. Hugs, ᛭ LokiClock (talk) 18:07, 26 July 2015 (UTC)

Fourier transform changes x^a to x^a ?

"Because the Fourier transform changes differentiation by x^α into multiplication by x^α and vice versa... " I know little about this topic, but perhaps "differentiation by D^a into multiplication by (ix)^a" is meant, or some such? — Preceding unsigned comment added by 66.188.89.180 (talk) 19:12, 31 July 2015 (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 $\varphi (x)$ to be a test function satisfying $\varphi (x)\geq 0$ and $\varphi (x)=1$ for |x|<1, then the pairing

$\int {\frac {1}{x}}\varphi (x)\,dx$

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

$T(\varphi )={\text{P.V.}}\int {\frac {1}{x}}\varphi (x)\,dx$

(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. Thenub314 (talk) 13:40, 1 May 2008 (UTC)

You are right. See, e.g., Hunter: "Applied Analysis". Just correct it. --TN (talk) 17:07, 6 December 2015 (UTC)

Ultra-Distributions

Would be nice if someone with experience could add some words on ultra-distributions. --TN (talk) 17:33, 6 December 2015 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Distribution (mathematics)/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Article getting close to B+. Should add motivation for needing distributions as well as discussion on applications. Further references and further reading would be good too. Stca74 18:15, 15 May 2007 (UTC)

Last edited at 18:15, 15 May 2007 (UTC). Substituted at 02:01, 5 May 2016 (UTC)

tempered distribution and feynman path integral

Abdelmalek Abdesselam writes on mathoverflow:[1] "It is not fair to say there is no rigorous definition of a Feynman path integral. It is a probability measure on the space of tempered distributions. The difficulty is in constructing nontrivial examples. This is what constructive field theory is about." (Feb 26 '11 at 18:59).

If that is a generally accepted idea it would be great if the article could explain it a bit. Thanks. 173.228.123.166 (talk) 17:21, 16 August 2018 (UTC)

“It is abuse of language to assert that distributions are not functions”

The first paragraph says: “While a distribution, such the Dirac delta function is, indeed a function, there does not exist a function on the space of the real line that is equivalent to the Dirac delta function. Instead, distributions are constructed from alternative domains. Since the formal treatment of distributions is often considerably different from the typical functions of more elementary theories, it is a common abuse of language to assert that distributions are not functions.” This sounds highly dubious. Function is a map from numbers to numbers. Distribution is traditionally defined as a map from functions to numbers. They are linear functionals: if we view functions as vectors (eg. of a Hilbert space), distributions are covectors. So this first paragraph sounds very controversial to me. Michal Grňo (talk) 07:32, 3 December 2019 (UTC)

Distributions, a.k.a. generalised functions, are indeed not functions proper. The cited part of the introductory paragraph has been removed accordingly by me. — JivanP (talk) 06:13, 8 December 2019 (UTC)

Sobolev

Saaska, 27 Nov 2003 I thought it would be fair to include Sobolev here.

composition of a distribution with a differentiable injective function

Is it possible to define the composition of a distribution with a differentiable injective function? Formally, it should be like

$\left\langle T\circ f,\ \varphi \right\rangle =\left\langle T,\ {\frac {\varphi \circ f^{-1}}{f'\circ f^{-1}}}\right\rangle$

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 $f^{-1}$)

---

David 18 Dec 2004

I think that:

If u is a distribution in D?(A) and T is a C^00(A) invertible function:

<u o T, g> =<u, g o T^(-1) |det J|>

where g is a test function and J is the Jacobian matrix of T^(-1).

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

Typo?

This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense:

d/dx (S * T) = (d/dx S) * T + S * (d/dx T).

Is this a typo? Seems to me it should be

d/dx (S * T) = (d/dx S) * T = S * (d/dx T).

Josh Cherry 14:47, 18 Apr 2004 (UTC)

Don't think so. 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 iω in the frequency domain. From this and the convolution theorem, the same result seems easily derived.
• For concreteness, let T be the δ 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 δ function, is also d/dx S.

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

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

So, this was changed by an anonymous user on 31 January; should be changed back. Charles Matthews 18:06, 18 Apr 2004 (UTC)

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

Symbols

According to Wikipedia:Manual of Style/Mathematics, the symbols ∀ ∃ and ⇔ should not be used unnecessarily so over time, I will be removing instances of these symbols. Also, words like "we," "recall," and especially "clearly" and "obviously" should be avoided (since something that is clear to you may not be clear to the reader). Mgkrupa 17:52, 26 October 2020 (UTC)

PROPOSAL: Move content related to SPACES of tests functions and distributions (e.g. definitions of topologies) to new article

This article is extremely long and (as the talk page shows) some people are not happy with how information is presented. The primary issue being that the technical topological vector space (TVS)-related information is not friendly to people new to this subject, most of whom are unlikely to be familiar with the general theory of locally convex TVSs. So I propose that a new article

Proposed name: Spaces of test functions and distributions

be created using the following content from this article:

1. Distribution (mathematics)#Definitions of test functions and distributions
2. Distribution (mathematics)#Spaces of distributions

This will move most of the content related to TVSs, which many (probably most) readers are not familiar with, into its own new article. Those readers want information about the construction of the canonical LF-topology and the various distribution-related TVSs will be able to find it there. What remains in this article will primarily be information about individual distributions (because this article is named "Distribution (mathematics)" after all). I propose that the following information will remain in this article: varphi)

1. Distribution (mathematics)#Localization of distributions
2. Distribution (mathematics)#Operations on distributions

Some information can be repeated in both article, such as notation and possibly also some important information related to TVS topologies (e.g. the seminorms in the section Distribution (mathematics)#Topology on Ck(U)).

Of course, what content is and is not moved is entirely up for discussion. The above was just my initial suggestion. Mgkrupa 19:38, 28 May 2021 (UTC)