|This is the talk page for discussing improvements to the Distribution (mathematics) article.|
|WikiProject Mathematics||(Rated B-class, High-importance)|
- 1 Sobolev
- 2 composition of a distribution with a differentiable injective function
- 3 Typo?
- 4 Proposal to move this page
- 5 Distribution
- 6 extend the concept of derivative to integrable functions
- 7 Angle bracket notation
- 8 Added section; LaTeX
- 9 Path integrals
- 10 Cleaning Up
- 11 compact support
- 12 defining the topology of D(U)
- 13 Convolution and distributions
- 14 convolution of distribution
- 15 Stieltjes integral??
- 16 Problem of multiplication
- 17 Nonsense
- 18 Omitted word?
- 19 Derivative of dirac delta
- 20 Typography.
- 21 Merge with generalized function article?
- 22 Typography 2
- 23 Topology on D(U)
- 24 Unclear equation
- 25 Kleinert's Work on Multiplication of Distributions
- 26 Deriving Dirac delta distribution from Heaviside step function
- 27 Distributions are functions
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
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 )
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
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)
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)
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))
Angle bracket notation
Should the notation be introduced in the page? --Md2perpe 23:10, 30 July 2006 (UTC)
- And perhaps also the connection with inner products. The identity 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)
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)
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. 126.96.36.199 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)
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)
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)
defining the topology of D(U)
"a sequence (or net) (φk) converges to 0 if and only if there exists a compact subset K of U such that all φk 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 φk 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 φk 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 , then you can enlarge K suitably so that it contains the support of all . - Saibod 16:04, 8 July 2007 (UTC)
- How? Let = 1 (if k = 1), 0 (otherwise), where the domain of 's is Rn. Then the support of 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 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
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)
convolution of distribution
In the "Basic idea" section the angle bracket is defined as
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) 188.8.131.52 (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 to be a test function satisfying and for |x|<1, then the pairing
doesn't converge, so 1/x doesn't define a distribution in the obvious way. I think what might be meant is the distribution
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 then we would have , however the last integral is divergent for every value of 'u'.
Currently, the phrase "which is with respect to the weak-* topology" is in the article in the section titled "Operations on distrbutions".
Derivative of dirac delta
The definition of the derivative of the dirac delta distribution currently reads .
Someone came through and changed all φ's to Φ's because (I assume) the html φ's doesn't look like latex 's. Does any one object to my changing the Φ's back to φ's and also changing the 's to 's? Thenub314 (talk) 12:38, 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.
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 . This makes it pretty confusing since the text uses φ, but the formulas use . 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 (Sk) in D'(U) converges to a distribution S if and only if
- for all test functions φ. This is the case if and only if Sk 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 dn such that every φ in E has its support in K and has its n-th derivatives bounded by ::dn.)
- In particular, a sequence (Sk) in D'(U) converges to a distribution S if and only if
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?
- (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)
In first section,
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 184.108.40.206 (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) 220.127.116.11 (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:
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?
- It's the definition of the derivative of the distribution H that
- 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.18.104.22.168 (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 22.214.171.124 (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) 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 126.96.36.199 (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 188.8.131.52 (talk) 17:46, 29 March 2014 (UTC)