Editing Distribution (mathematics) (section)

====Distributions of finite order with support in an open subset====

{{Math theorem|name=Theorem{{sfn|Rudin|1991|pp=149-181}}|math_statement=
Suppose {{mvar|T}} is a distribution on {{mvar|U}} with compact support {{mvar|K}} and let {{mvar|V}} be an open subset of {{mvar|U}} containing {{mvar|K}}. Since every distribution with compact support has finite order, take {{mvar|N}} to be the order of {{mvar|T}} and define <math>P:=\{0,1,\ldots, N+2\}^n.</math> There exists a family of continuous functions <math>(f_p)_{p\in P}</math> defined on {{mvar|U}} '''with support in {{mvar|V}}''' such that
<math display=block>T = \sum_{p \in P} \partial^p f_p,</math>
where the derivatives are understood in the sense of distributions. That is, for all test functions <math>\phi</math> on {{mvar|U}},
<math display=block>T \phi = \sum_{p \in P} (-1)^{|p|} \int_U f_p(x) (\partial^p \phi)(x) \, dx.</math>
}}