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non-singular distribution

Contents

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

n()bdvol n:C cp ( n) \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 bdvolb dvol.

This construction defines a linear inclusion

C ( n)𝒟( n) 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 nn \in \mathbb{N} let ϕ𝒟( n)\phi \in \mathcal{D}'(\mathbb{R}^n) be a distribution. Then the singular support supp sing(ϕ)Xsupp_{sing}(\phi) \subset X is the subset of points such that for every open neighbourhood U xXU_x \subset X the restriction ϕ| U x\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 ( n)𝒟( n) C^\infty(\mathbb{R}^n) \hookrightarrow \mathcal{D}'(\mathbb{R}^n)

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

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

u(ϕ)=limn nϕ(x)f n(x)dvol(x). 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 cp ( n)𝒮( n) 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

Last revised on November 2, 2017 at 14:11:27. See the history of this page for a list of all contributions to it.