nLab non-singular distribution

Idea

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

Definition

Definition

(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.

Definition

(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. ).

Properties

Non-singular distributions are dense in all distributions

Proposition

(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. 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

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

(Hörmander 90, theorem 4.1.5)

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)

Remark

Proposition 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?.

References

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