Editing Distribution (mathematics) (section)

===Global structure of distributions===

The formal definition of distributions exhibits them as a subspace of a very large space, namely the topological dual of <math>\mathcal{D}(U)</math> (or the [[Schwartz space]] <math>\mathcal{S}(\R^n)</math> for tempered distributions). It is not immediately clear from the definition how exotic a distribution might be. To answer this question, it is instructive to see distributions built up from a smaller space, namely the space of continuous functions. Roughly, any distribution is locally a (multiple) derivative of a continuous function. A precise version of this result, given below, holds for distributions of compact support, tempered distributions, and general distributions. Generally speaking, no proper subset of the space of distributions contains all continuous functions and is closed under differentiation. This says that distributions are not particularly exotic objects; they are only as complicated as necessary.

====Distributions as [[Sheaf (mathematics)|sheaves]]====

{{Math theorem|name=Theorem{{sfn|Trèves|2006|pp=258-264}}|math_statement=
Let {{mvar|T}} be a distribution on {{mvar|U}}.
There exists a sequence <math>(T_i)_{i=1}^\infty</math> in <math>\mathcal{D}'(U)</math> such that each {{mvar|T<sub>i</sub>}} has compact support and every compact subset <math>K \subseteq U</math> intersects the support of only finitely many <math>T_i,</math> and the sequence of partial sums <math>(S_j)_{j=1}^\infty,</math> defined by <math>S_j := T_1 + \cdots + T_j,</math> converges in <math>\mathcal{D}'(U)</math> to {{mvar|T}}; in other words we have:
<math display=block>T = \sum_{i=1}^\infty T_i.</math>
Recall that a sequence converges in <math>\mathcal{D}'(U)</math> (with its strong dual topology) if and only if it converges pointwise.
}}

====Decomposition of distributions as sums of derivatives of continuous functions====

By combining the above results, one may express any distribution on {{mvar|U}} as the sum of a series of distributions with compact support, where each of these distributions can in turn be written as a finite sum of distributional derivatives of continuous functions on {{mvar|U}}. In other words, for arbitrary <math>T \in \mathcal{D}'(U)</math> we can write:
<math display=block>T = \sum_{i=1}^\infty \sum_{p \in P_i} \partial^p f_{ip},</math>
where <math>P_1, P_2, \ldots</math> are finite sets of multi-indices and the functions <math>f_{ip}</math> are continuous.

{{Math theorem|name=Theorem{{sfn|Rudin|1991|pp=169-170}}|math_statement=
Let {{mvar|T}} be a distribution on {{mvar|U}}. For every multi-index {{mvar|p}} there exists a continuous function <math>g_p</math> on {{mvar|U}} such that

# any compact subset {{mvar|K}} of {{mvar|U}} intersects the support of only finitely many <math>g_p,</math> and
# <math>T = \sum\nolimits_p \partial^p g_p.</math>

Moreover, if {{mvar|T}} has finite order, then one can choose <math>g_p</math> in such a way that only finitely many of them are non-zero.
}}

Note that the infinite sum above is well-defined as a distribution. The value of {{mvar|T}} for a given <math>f \in \mathcal{D}(U)</math> can be computed using the finitely many <math>g_\alpha</math> that intersect the support of <math>f.</math>