nLab point-supported distribution

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 $u$.

(e.g. Hörmander 90, theorem 2.3.4)

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

Related concepts

• extension of distributions

References

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