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, nonsmooth, or nonintegrable 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 welldefined foundation of Quantum Mechanics (http://planetmath.org/QuantumGroupsAndVonNeumannAlgebras) based on actions of selfadjoint 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 $\mathbb{R}$ is a linear mapping that takes a smooth function^{} (with compact support) on $\mathbb{R}$ into a real number. For example, the delta distribution is the map,
$$f\mapsto f(0)$$ 
while any smooth function $g$ on $\mathbb{R}$ induces a distribution
$$f\mapsto {\int}_{\mathbb{R}}fg.$$ 
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 $U\subseteq {\mathbb{R}}^{n}$ is traditionally denoted by $\mathcal{D}(U)$. Let us also denote by ${\mathcal{D}}_{K}(U)$ the subset of $\mathcal{D}(U)$ 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 $\mathcal{D}(U)$. 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 ${\mathcal{D}}_{K}(U)$, such that for any multiindex $\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 $\u2102$.

3.
For any compact set $K$ in $U$, there are constants $C>0$ and $k\in \{1,2,\mathrm{\dots}\}$ such that for all $u\in {\mathcal{D}}_{K}(U)$, we have
$T(u)$ $\le $ $C{\displaystyle \sum _{\alpha \le k}}{{D}^{\alpha}u}_{\mathrm{\infty}},$ (1) where $\alpha $ is a multiindex, 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 ${\mathcal{D}}^{\prime}(U)$
The standard topology for ${\mathcal{D}}^{\prime}(U)$ is the weak${}^{\ast}$ topology^{}. In this topology, a sequence ${\{{T}_{i}\}}_{i=1}^{\mathrm{\infty}}$ of distributions (in ${\mathcal{D}}^{\prime}(U)$) converges^{} to a distribution $T\in {\mathcal{D}}^{\prime}(U)$ if and only if
$${T}_{i}(u)\to T(u)\text{(in}\u2102\text{) as}i\to \mathrm{\infty}$$ 
for every $u\in \mathcal{D}(U)$.
Notes
A common notation for the action of a distribution $T$ onto a test function $u\in \mathcal{D}(U)$ (i.e., $T(u)$ with above notation) is $\u27e8T,u\u27e9$. The motivation for this comes from this example (http://planetmath.org/EveryLocallyIntegrableFunctionIsADistribution).
References
 1 W. Rudin, Functional Analysis^{}, McGrawHill Book Company, 1973.
 2 L. H$\mathrm{\xf6}$rmander, The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis), 2nd ed, SpringerVerlag, 1990.
Title  distribution 
Canonical name  Distribution 
Date of creation  20130322 13:44:08 
Last modified on  20130322 13:44:08 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  23 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 4600 
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 