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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. 1 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. 1).

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. 1 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 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?.

References

  • 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)