nLab
point-supported distribution
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Functional analysis
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Idea
A point-supported distribution is a distribution whose support of a distribution is a single point. These turn out to be precisely the sums of multiples of the delta distribution and its derivatives at that point (prop. below).

Definition
Properties
Proposition
Every point-supported distribution $u$ (def. ) with $supp(u) = \{p\}$ is a finite sum of multiplies of derivatives of the delta distribution at that point:

$u =
\underset{ {\alpha \in \mathbb{N}^n} \atop { {\vert \alpha\vert} \leq k } }{\sum} c^\alpha \partial_\alpha \delta(p)$

where $\{c^\alpha \in \mathbb{R}\}_\alpha$ , and for $k \in \mathbb{N}$ the order of $u$ .

(e.g. Hörmander 90, theorem 2.3.4 )

Clearly a point-supported distribution is in particular a compactly supported distribution .

References
Lars Hörmander , The analysis of linear partial differential operators , vol. I, Springer 1983, 1990
Created on August 6, 2017 at 14:31:01.
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