distribution

Motivation

The main motivation behind distribution theory is to extend 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-integrable functions that regularly appear in both theoretical and applied analysis.

Distribution theory also seeks to define suitable structures on the spaces of functions involved to ensure the convergence of suitable approximating functions, and the continuity of certain operators. For example, the limit of derivatives should be equal to the derivative of the limit, with some definition of the limiting operation.

When this program is carried out, inevitably we find that we have to enlarge the space of objects that we would consider as “functions”. For example, the derivative of a step function is the Dirac delta function with a spike at the discontinuous step; the Fourier transform of a constant function is also a Dirac delta function, with the spike representing infinite spectral magnitude at one single frequency. (These facts, of course, had long been used 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 as a (negative) reaction to such a state of affairs, von Neumann produced a mathematically well-defined foundation of Quantum Mechanics (http://planetmath.org/QuantumGroupsAndVonNeumannAlgebras) based on actions of self-adjoint operators on Hilbert spaces which is still currently in use, with several significant additions 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 a smooth 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, and can be defined onto a manifold. Differential forms with distribution valued coefficients are called currents. However, it is not possible to define a product of two distributions generalizing the product of usual functions.

Formal definition

A note on notation. In distribution theory, the topological vector space of smooth functions with compact support on an 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 a compact 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 continuous in the origin (see this page (http://planetmath.org/ContinuousLinearMapping)). This condition can be rewritten in various ways, and the below theorem gives two convenient conditions that can be used to prove that 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 the following 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 that for 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 be found 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 a distribution 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 increases by 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 this comes from this example (http://planetmath.org/EveryLocallyIntegrableFunctionIsADistribution).

Title	distribution
Canonical name	Distribution
Date of creation	2013-03-22 13:44:08
Last modified on	2013-03-22 13:44:08
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	23
Author	matte (1858)
Entry type	Definition
Classification	msc 46-00
Classification	msc 46F05
Synonym	‘generalized function’
Related topic	ExampleOfDiracSequence
Related topic	DiracDeltaFunction
Related topic	DiscreteTimeFourierTransformInRelationWithItsContinousTimeFourierTransfrom
Related topic	QuantumGroups
Related topic	FourierStieltjesAlgebraOfAGroupoid
Related topic	QuantumOperatorAlgebrasInQuantumFieldTheories
Related topic	QFTOrQuantumFieldTheories
Related topic	QuantumGroup
Defines	distribution of finite order