Editing Distribution (mathematics) (section)

====Support in a point set and Dirac measures====

For any <math>x \in U,</math> let <math>\delta_x \in \mathcal{D}'(U)</math> denote the distribution induced by the Dirac measure at <math>x.</math> For any <math>x_0 \in U</math> and distribution <math>T \in \mathcal{D}'(U),</math> the support of {{mvar|T}} is contained in <math>\{x_0\}</math> if and only if {{mvar|T}} is a finite linear combination of derivatives of the Dirac measure at <math>x_0.</math>{{sfn|Trèves|2006|pp=264-266}} If in addition the order of {{mvar|T}} is <math>\leq k</math> then there exist constants <math>\alpha_p</math> such that:{{sfn|Rudin|1991|p=165}}
<math display=block>T = \sum_{|p| \leq k} \alpha_p \partial^p \delta_{x_0}.</math>

Said differently, if {{mvar|T}} has support at a single point <math>\{P\},</math> then {{mvar|T}} is in fact a finite linear combination of distributional derivatives of the <math>\delta</math> function at {{mvar|P}}. That is, there exists an integer {{mvar|m}} and complex constants <math>a_\alpha</math> such that
<math display=block>T = \sum_{|\alpha|\leq m} a_\alpha \partial^\alpha(\tau_P\delta)</math>
where <math>\tau_P</math> is the translation operator.