# Talk:Distribution (mathematics)

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Mathematics rating:
 B Class
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Field:  Analysis

## 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

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

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 ${\displaystyle f^{-1}}$)

---

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)

## 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)