Science Fair Project Encyclopedia
Saaska, 27 Nov 2003 I thought it would be fair to include Sobolev here.
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 f - 1)
David 18 Dec 2004
I think that:
If u is a distribution in Dí(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).
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)
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