Editing Distribution (mathematics) (section)

===Support of a distribution===

This last corollary implies that for every distribution {{mvar|T}} on {{mvar|U}}, there exists a unique largest subset {{mvar|V}} of {{mvar|U}} such that {{mvar|T}} vanishes in {{mvar|V}} (and does not vanish in any open subset of {{mvar|U}} that is not contained in {{mvar|V}}); the complement in {{mvar|U}} of this unique largest open subset is called {{em|the '''support''' of {{mvar|T}}}}.{{sfn|Trèves|2006|pp=253-255}} Thus
<math display=block> \operatorname{supp}(T) = U \setminus \bigcup \{V \mid \rho_{VU}T = 0\}.</math>

If <math>f</math> is a locally integrable function on {{mvar|U}} and if <math>D_f</math> is its associated distribution, then the support of <math>D_f</math> is the smallest closed subset of {{mvar|U}} in the complement of which <math>f</math> is [[almost everywhere]] equal to 0.{{sfn|Trèves|2006|pp=253-255}} If <math>f</math> is continuous, then the support of <math>D_f</math> is equal to the closure of the set of points in {{mvar|U}} at which <math>f</math> does not vanish.{{sfn|Trèves|2006| pp=253-255}} The support of the distribution associated with the [[Dirac measure]] at a point <math>x_0</math> is the set <math>\{x_0\}.</math>{{sfn|Trèves|2006|pp=253-255}} If the support of a test function <math>f</math> does not intersect the support of a distribution {{mvar|T}} then <math>Tf = 0.</math> A distribution {{mvar|T}} is 0 if and only if its support is empty. If <math>f \in C^\infty(U)</math> is identically 1 on some open set containing the support of a distribution {{mvar|T}} then <math>f T = T.</math> If the support of a distribution {{mvar|T}} is compact then it has finite order and there is a constant <math>C</math> and a non-negative integer <math>N</math> such that:{{sfn|Rudin|1991|pp=149-181}}
<math display=block>|T \phi| \leq C\|\phi\|_N := C \sup \left\{\left|\partial^\alpha \phi(x)\right| : x \in U, |\alpha| \leq N \right\} \quad \text{ for all } \phi \in \mathcal{D}(U).</math>

If {{mvar|T}} has compact support, then it has a unique extension to a continuous linear functional <math>\widehat{T}</math> on <math>C^\infty(U)</math>; this function can be defined by <math>\widehat{T} (f) := T(\psi f),</math> where <math>\psi \in \mathcal{D}(U)</math> is any function that is identically 1 on an open set containing the support of {{mvar|T}}.{{sfn|Rudin|1991|pp=149-181}}

If <math>S, T \in \mathcal{D}'(U)</math> and <math>\lambda \neq 0</math> then <math>\operatorname{supp}(S + T) \subseteq \operatorname{supp}(S) \cup \operatorname{supp}(T)</math> and <math>\operatorname{supp}(\lambda T) = \operatorname{supp}(T).</math> Thus, distributions with support in a given subset <math>A \subseteq U</math> form a vector subspace of <math>\mathcal{D}'(U).</math>{{sfn|Trèves|2006|pp=255-257}} Furthermore, if <math>P</math> is a differential operator in {{mvar|U}}, then for all distributions {{mvar|T}} on {{mvar|U}} and all <math>f \in C^\infty(U)</math> we have <math>\operatorname{supp} (P(x, \partial) T) \subseteq \operatorname{supp}(T)</math> and <math>\operatorname{supp}(fT) \subseteq \operatorname{supp}(f) \cap \operatorname{supp}(T).</math>{{sfn|Trèves|2006|pp=255-257}}