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|WikiProject Mathematics||(Rated B-class, High-importance)|
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)
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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)|
Last edited at 18:15, 15 May 2007 (UTC). Substituted at 02:01, 5 May 2016 (UTC)