# distribution

## Motivation

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.)

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 $U\subseteq\mathbbmss{R}^{n}$ 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 $\mathcal{D}(U)$, i.e., a linear continuous mapping $\mathcal{D}(U)\to\mathbb{C}$. The set of all distributions on $U$ is denoted by $\mathcal{D}^{\prime}(U)$.

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 $\mathbb{R}^{n}$, and let $T$ be a linear functional on $\mathcal{D}(U)$. Then the following are equivalent      :

1. 1.

$T$ is a distribution.

2. 2.

If $K$ is a compact set in $U$, and $\{u_{i}\}_{i=1}^{\infty}$ be a sequence in $\mathcal{D}_{K}(U)$, such that for any multi-index $\alpha$, we have

 $D^{\alpha}u_{i}\to 0$

$i\to\infty$, then $T(u_{i})\to 0$ in $\mathbb{C}$.

3. 3.

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

 $\displaystyle|T(u)|$ $\displaystyle\leq$ $\displaystyle C\sum_{|\alpha|\leq k}||D^{\alpha}u||_{\infty},$ (1)

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

### 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 $\{T_{i}\}_{i=1}^{\infty}$ of distributions (in $\mathcal{D}^{\prime}(U)$) converges  to a distribution $T\in\mathcal{D}^{\prime}(U)$ if and only if

 $T_{i}(u)\to T(u)\,\,\,\mbox{(in \mathbb{C}) as i\to\infty}$

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 $\langle T,u\rangle$. The motivation for this comes from this example (http://planetmath.org/EveryLocallyIntegrableFunctionIsADistribution).

## References

• 1
• 2 L. H$\"{o}$rmander, The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis), 2nd ed, Springer-Verlag, 1990.
 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