2
$\begingroup$

Is this true?

Any compactly supported distribution $T\in \cal D'$ can be represented as finite sum of partial derivatives of functions.

  • 0
    If $T$ is compactly supported, then $T$ can be extended to the space of infinitely differentiable function, and hence have a finite order.2012-05-02
  • 0
    What do you mean exactly by sum of partial derivatives of functions?2012-05-02
  • 0
    I found the answer here. http://people.oregonstate.edu/~peterseb/mth627/docs/627w2004_compact_support.pdf, It is a theorem of Schwarz's.2012-05-02
  • 1
    The link doesn't work.2012-05-02
  • 0
    Eh...maybe you need to edit the link, just to retype the ".pdf", it is strange because the link _is_ correct.http://people.oregonstate.edu/~peterseb/mth627/docs/627w2004_compact_support.pdf2012-05-03
  • 0
    Yes, the link works now. But in fact it says that a distribution with support contained in a singleton is a linear sum of derivatives of Dirac. What do you mean by finite sum of partial derivatives of functions?2012-05-03
  • 0
    Sorry, I googled it, it was just talking about the case that the compact support is only a point... However, this is a theorem of Schwarz. I found a paper related to this. It is long. You may find the theorem in the forth one. http://uu.diva-portal.org/smash/get/diva2:170706/FULLTEXT012012-05-04
  • 0
    And I found the theorem in Schwarz's book "Théorie des distributions", Hermann, Paris, 1966.2012-05-04
  • 0
    A full treatment of this subject is Chpater 24 in the book "Topological Vector Spaces, Distributions and Kernels" by F. Treves, available online in major parts. @Yimin: Be careful with your first comment, you will need a constant there, a distribution may not have seminorm less or equal 1.2012-05-04

0 Answers 0