Talk:Distribution (mathematics)

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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)

1 Sobolev
2 composition of a distribution with a differentiable injective function
3 Typo?
4 Proposal to move this page
- 4.1 title not adequate to contents
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

[edit] Sobolev

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

[edit] 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:

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

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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))

[edit] 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 $\langle f, g \rangle = \int f g$ 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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

φ k

, then you can enlarge K suitably so that it contains the support of all

φ k

. - Saibod 16:04, 8 July 2007 (UTC)

How? Let

φ k

= 1 (if k = 1), 0 (otherwise), where the domain of

φ k

's is Rⁿ. Then the support of

φ 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

φ 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)