Editing Distribution (mathematics) (section)

===Distributions of finite order===

Let <math>k \in \N.</math> The inclusion map <math>\operatorname{In}: C_c^\infty(U) \to C_c^k(U)</math> is a continuous injection whose image is dense in its codomain, so the [[transpose]] <math>{}^{t}\operatorname{In}: (C_c^k(U))'_b \to \mathcal{D}'(U) = (C_c^\infty(U))'_b</math> is also a continuous injection. Consequently, the image of <math>{}^{t}\operatorname{In},</math> denoted by <math>\mathcal{D}'^{k}(U),</math> forms a space of distributions. The elements of <math>\mathcal{D}'^k(U)</math> are '''{{em|the distributions of order <math>\,\leq k.</math>}}'''{{sfn|Trèves|2006|pp=258-264}} The distributions of order <math>\,\leq 0,</math> which are also called '''{{em|distributions of order {{math|0}}}}''' are exactly the distributions that are Radon measures (described above).

For <math>0 \neq k \in \N,</math> a '''{{em|distribution of order {{mvar|k}}}}''' is a distribution of order <math>\,\leq k</math> that is not a distribution of order <math>\,\leq k - 1</math>.{{sfn|Trèves|2006|pp=258-264}}

A distribution is said to be of '''{{em|finite order}}''' if there is some integer <math>k</math> such that it is a distribution of order <math>\,\leq k,</math> and the set of distributions of finite order is denoted by <math>\mathcal{D}'^{F}(U).</math> Note that if <math>k \leq l</math> then <math>\mathcal{D}'^k(U) \subseteq \mathcal{D}'^l(U)</math> so that <math>\mathcal{D}'^{F}(U) := \bigcup_{n=0}^\infty \mathcal{D}'^n(U)</math> is a vector subspace of <math>\mathcal{D}'(U)</math>, and furthermore, if and only if <math>\mathcal{D}'^{F}(U) = \mathcal{D}'(U).</math>{{sfn|Trèves|2006|pp=258-264}}

====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}}