nLab
point-supported distribution

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

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

for some pXp \in X.

Properties

Proposition

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

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

where {c α} α\{c^\alpha \in \mathbb{R}\}_\alpha, and for kk \in \mathbb{N} the order of uu.

(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. See the history of this page for a list of all contributions to it.