Distribution (mathematics) - Distributions As Derivatives of Continuous Functions

Distributions As Derivatives of Continuous Functions

The formal definition of distributions exhibits them as a subspace of a very large space, namely the topological dual of D(U) (or S(Rd) for tempered distributions). It is not immediately clear from the definition how exotic a distribution might be. To answer this question, it is instructive to see distributions built up from a smaller space, namely the space of continuous functions. Roughly, any distribution is locally a (multiple) derivative of a continuous function. A precise version of this result, given below, holds for distributions of compact support, tempered distributions, and general distributions. Generally speaking, no proper subset of the space of distributions contains all continuous functions and is closed under differentiation. This says that distributions are not particularly exotic objects; they are only as complicated as necessary.

Tempered distributions

If ƒ ∈ S′(Rn) is a tempered distribution, then there exists a constant C > 0, and positive integers M and N such that for all Schwartz functions φ ∈S(Rn)

This estimate along with some techniques from functional analysis can be used to show that there is a continuous slowly increasing function F and a multiindex α such that

Compactly supported distributions

Let U be an open set, and K a compact subset of U. If ƒ is a distribution supported on K, then there is a continuous function F compactly supported in U (possibly on a larger set than K itself) such that

for some multi-index α. This follows from the previously quoted result on tempered distributions by means of a localization argument.

Distributions with point support

If ƒ has support at a single point {P}, then ƒ is in fact a finite linear combination of distributional derivatives of the δ function at P. That is, there exists an integer m and complex constants a_α for multi-indices |α| ≤ m such that

where τ_P is the translation operator.

General distributions

A version of the above theorem holds locally in the following sense (Rudin 1991). Let S be a distribution on U. Then one can find for every multi-index α a continuous function g_α such that

and that any compact subset K of U intersects the supports of only finitely many g_α; therefore, to evaluate the value of S for a given smooth function f compactly supported in U, we only need finitely many g_α; hence the infinite sum above is well-defined as a distribution. If the distribution S is of finite order, then one can choose g_α in such a way that only finitely many of them are nonzero.