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Is this true?

Any compact supported distribution can be represented as finite sum of partial derivatives of functions.

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    $\begingroup$ This is Theorem 6.27 in Rudin's Functional Analysis $\endgroup$
    – Jochen Wengenroth
    Commented May 3, 2012 at 8:20

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This is a theorem in Laurent Schwartz' classic monograph Theorie des Distributions, Thm. XXVI, Chap. III, Sec. 7.

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http://people.oregonstate.edu/~peterseb/mth627/docs/627w2004_compact_support.pdf

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