non-singular distribution

A distribution, being a generalized function, is called *non-singular* if it is (defined by) an actual smooth function (def. 1 below). The subspace of non-singular distributions is a dense subspace of that of all distributions.

**(non-singular distributions)**

For $n \in \mathbb{N}$, a smooth function $b \in C^\infty(\mathbb{R}^n)$ induces a distribution

$\int_{\mathbb{R}^n} (-) b dvol_{\mathbb{R}^n}
\;\colon\;
C_{cp}^\infty(\mathbb{R}^n) \longrightarrow \mathbb{R}$

by integration of smooth functions against $b dvol$.

This construction defines a linear inclusion

$C^\infty(\mathbb{R}^n) \hookrightarrow \mathcal{D}'(\mathbb{R}^n)$

of the real vector space of smooth functions into that of distributions.

The distributions arising this way are called the *non-singular distributions*.

**(singular support of a distribution)**

For $n \in \mathbb{N}$ let $\phi \in \mathcal{D}'(\mathbb{R}^n)$ be a distribution. Then the *singular support* $supp_{sing}(\phi) \subset X$ is the subset of points such that for every open neighbourhood $U_x \subset X$ the restriction $\phi\vert_{U_x}$ is singular, hence not a non-singular distribution (def. 1).

**(non-singular distributions are dense in all distributions)**

The inclusion of non-singular distributions into all distributions

$C^\infty(\mathbb{R}^n) \hookrightarrow \mathcal{D}'(\mathbb{R}^n)$

from def. 1 exhibits a dense subspace of $\mathcal{D}'(\mathbb{R}^n)$ (equipped with the dual space topology, this def. or Hörmander 90, p. 38):

Every distribution $u \in \mathcal{D}'(\mathbb{R}^n)$ is the limit of a sequence $\{f_n \in C^\infty(\mathbb{R}^n)\}_{n \in \mathbb{N}}$ of smooth functions in that for every compactly supported smooth function $\phi \in C^\infty_{cp}(\mathbb{R}^n)$ we have that the value $u(\phi)$ is the limit of integrals

$u(\phi)
\;=\;
\underset{n \to \infty}{\lim} \int_{\mathbb{R}^n} \phi(x) f_n(x) dvol(x)
\,.$

Analogous statements hold for smaller classes of distributions: Also the inclusion

$C^\infty_{cp}(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)$

of compactly supported smooth functions (bump functions) into the space of tempered distributions is dense.

(e.g. Hörmander 90, lemma 7.1.8)

Proposition 1 justifies to think of distributions as “generalized functions”, and to seek generalizations of standard operations on functions, such as concepts of *product of distributions* and *composition of distributions?*.

- Lars Hörmander, section 2.3 of
*The analysis of linear partial differential operators*, vol. I, Springer 1983, 1990

Revised on November 2, 2017 14:11:27
by Urs Schreiber
(195.37.234.82)