distribution

Motivation

The main motivation behind distribution theory is toextend the common linear operators on functions,such as the derivative, convolution, and the Fourier transform,so that they also apply to the singular, non-smooth, or non-integrablefunctions that regularly appear in both theoretical and appliedanalysis.

Distribution theory also seeks to define suitable structureson the spaces of functions involvedto ensure the convergence of suitable approximating functions,and the continuity of certain operators.For example, the limit of derivatives should be equalto the derivative of the limit, with some definition of the limitingoperation.

When this program is carried out,inevitably we find that we have to enlarge the space of objects that wewould consider as “functions”. For example, the derivative of a stepfunction is the Dirac delta function with a spike at the discontinuous step;the Fourier transform of a constant function is also a Dirac deltafunction, with the spike representing infinite spectral magnitudeat one single frequency. (These facts, of course, had long beenused in engineering mathematics.)

Remark:Dirac’s notion of delta distributions was introduced to facilitate computations in Quantum Mechanics,however without having at the beginning a proper mathematical definition. In part asa (negative) reaction to such a state of affairs, von Neumann produced a mathematicallywell-defined foundation of Quantum Mechanics (http://planetmath.org/QuantumGroupsAndVonNeumannAlgebras) based on actions ofself-adjoint operators on Hilbert spaces which is still currently in use, with several significantadditions such as Frechét nuclear spaces and quantum groups.

There are several theories of such ‘generalized functions’.In this entry, we describe Schwartz’ theory of distributions,which is probably the most widely used.

Essentially, a distribution on $ℝ$ is a linear mapping that takes asmooth function (with compact support) on $ℝ$ into a real number.For example, the delta distribution is the map,

$$f\mapsto f(0)$$

while any smooth function $g$ on $ℝ$ induces a distribution

$$f\mapsto {\int }_{ℝ}fg.$$

Distributions are also well behaved under coordinate changes, andcan be defined onto a manifold. Differential forms withdistribution valued coefficients are called currents.However, it is not possible to define a product of twodistributions generalizing the product of usual functions.

Formal definition

A note on notation. In distribution theory, thetopological vector space of smooth functions with compact support onan open set $U\subseteq {ℝ}^n$is traditionally denoted by $𝒟(U)$. Let us also denote by${𝒟}_K(U)$ the subset of $𝒟(U)$ of functions with support in acompact set $K\subset U$.

Definition 1 (Distribution).

A distribution is a linear continuous functional on $\mathrm{D}\mathit{}\mathrm{(}U\mathrm{)}$,i.e., a linear continuous mapping $\mathrm{D}\mathit{}\mathrm{(}U\mathrm{)}\mathrm{\to }\mathrm{C}$.The set of all distributions on $U$ is denoted by ${\mathrm{D}}^{\mathrm{\prime }}\mathit{}\mathrm{(}U\mathrm{)}$.

Suppose $T$ is a linear functional on $𝒟(U)$.Then $T$ is continuous if and only if $T$ is continuousin the origin (see this page (http://planetmath.org/ContinuousLinearMapping)).This condition can be rewritten in various ways, andthe below theorem gives two convenient conditions that can be used to provethat a linear mapping is a distribution.

Theorem 1.

Let $U$ be an open set in ${\mathrm{R}}^n$,and let $T$ be a linear functional on $\mathrm{D}\mathit{}\mathrm{(}U\mathrm{)}$. Then thefollowing are equivalent:

1. $T$ is a distribution.

2. If $K$ is a compact set in $U$, and${\{u_i\}}_{i=1}^{\mathrm{\infty }}$ be a sequence in ${𝒟}_K(U)$, such thatfor any multi-index $\alpha$, we have

   $$D^{\alpha }{u}_i\to 0$$

   in the supremum norm as $i\to \mathrm{\infty }$,then $T(u_i)\to 0$ in $ℂ$.

3. For any compact set $K$ in $U$, there are constants $C>0$ and$k\in \{1,2,\mathrm{\dots }\}$ such that for all $u\in {𝒟}_K(U)$, we have

   $|T(u)|$$\le$$C{\displaystyle \sum _{|\alpha |\le k}}{||D^{\alpha }u||}_{\mathrm{\infty }},$(1)

   where $\alpha$ is a multi-index, and$||\cdot |{|}_{\mathrm{\infty }}$ is the supremum norm.

Proof The equivalence of (2) and (3) can befound on this page (http://planetmath.org/EquivalenceOfConditions2And3),and the equivalence of (1) and (3) is shown in[1].

Distributions of order $k$

If $T$ is a distribution on an open set $U$,and the same $k$ can be used for any $K$in the above inequality, then $T$ is adistribution of order $k$.The set of all such distributions is denoted by $D^{\prime k}(U)$.

Both usual functions and the delta distribution are of order $0$.One can also show that by differentiating a distribution its order increasesby at most one. Thus, in some sense, the order is a measure of how”smooth” a distribution is.

Topology for ${𝒟}^{\prime }(U)$

The standard topology for ${𝒟}^{\prime }(U)$ is the weak${}^{\ast }$ topology.In this topology, a sequence ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$ of distributions(in ${𝒟}^{\prime }(U)$) converges to a distribution $T\in {𝒟}^{\prime }(U)$ if and only if

$$T_i(u)\to T(u)\text{(in }ℂ\text{) as }i\to \mathrm{\infty }$$

for every $u\in 𝒟(U)$.

Notes

A common notation for the action of a distribution $T$ onto a test function $u\in 𝒟(U)$(i.e., $T(u)$ with above notation) is $⟨T,u⟩$.The motivation for thiscomes from this example (http://planetmath.org/EveryLocallyIntegrableFunctionIsADistribution).