Contents

# Contents

## 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

###### Definition

A distribution $u \in \mathcal{D}'(X)$ is point-supported if its support of a distribution is a singleton set:

$supp(u) = \{p\}$

for some $p \in X$.

## 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 $u$.

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

• Lars Hörmander, The analysis of linear partial differential operators, vol. I, Springer 1983, 1990