Editing Distribution (mathematics) (section)

====Structure of distributions of finite order====

Every distribution with compact support in {{mvar|U}} is a distribution of finite order.{{sfn|Trèves|2006|pp=258-264}} Indeed, every distribution in {{mvar|U}} is {{em|locally}} a distribution of finite order, in the following sense:{{sfn|Trèves|2006|pp=258-264}} If {{mvar|V}} is an open and relatively compact subset of {{mvar|U}} and if <math>\rho_{VU}</math> is the restriction mapping from {{mvar|U}} to {{mvar|V}}, then the image of <math>\mathcal{D}'(U)</math> under <math>\rho_{VU}</math> is contained in <math>\mathcal{D}'^{F}(V).</math>

The following is the theorem of the structure of distributions of finite order, which shows that every distribution of finite order can be written as a sum of derivatives of [[Radon measure]]s:

{{math theorem|name=Theorem{{sfn|Trèves|2006|pp=258-264}}|math_statement=Suppose <math>T \in \mathcal{D}'(U)</math> has finite order and <math>I =\{p \in \N^n : |p| \leq k\}.</math> Given any open subset {{mvar|V}} of {{mvar|U}} containing the support of <math>T,</math> there is a family of Radon measures in {{mvar|U}}, <math>(\mu_p)_{p \in I},</math> such that for very <math>p \in I, \operatorname{supp}(\mu_p) \subseteq V</math> and
<math display=block>T = \sum_{|p| \leq k} \partial^p \mu_p.</math>}}

'''Example.''' (Distributions of infinite order) Let <math>U := (0, \infty)</math> and for every test function <math>f,</math> let <math display=block>S f := \sum_{m=1}^\infty (\partial^m f)\left(\frac{1}{m}\right).</math>

Then <math>S</math> is a distribution of infinite order on {{mvar|U}}. Moreover, <math>S</math> can not be extended to a distribution on <math>\R</math>; that is, there exists no distribution <math>T</math> on <math>\R</math> such that the restriction of <math>T</math> to {{mvar|U}} is equal to <math>S.</math>{{sfn|Rudin|1991|pp=177-181}}