Editing Distribution (mathematics) (section)

====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>