Distribution (mathematics)

Distributions (also known as generalized functions) are mathematical objects that are important in physics and engineering where many non-continuous problems are formulated in terms of distributions.

An important example of a distribution is the Dirac delta. Dirac's intuitive ideas were placed on firm mathematical footing by S. L. Sobolev in 1936, who studied the uniqueness of solutions of the Cauchy problem for linear hyperbolic equations. In 1950 Laurent Schwartz published his Théorie des Distributions. In this book he systematizes the theory of generalized functions unifying all earlier approaches and extending the results.

The physicist's definition of the Dirac delta function

$\int _{-\infty }^{\infty }\,\delta (x-x_{0})\phi (x)\,dx=\phi (x_{0}),$

is recognized by the mathematician as a linear functional acting on a set of "well-behaved" test functions φ(x).

In order to generalize this, the concept of "test functions" is needed. Let the set K consist of all real functions φ(x) with continuous derivatives of all orders and bounded support. This means that the function φ(x) vanishes outside some bounded region, which may differ for different functions. The set K is the space of test functions. It can be shown that K is a linear space.

Secondly, the concept of linear functional is needed. We call f a continuous linear functional on K, if f maps all elements of K onto a real number $\scriptstyle (f,\phi )\,$ such that

(i). For any two real numbers $\scriptstyle \alpha _{1}\,$ and $\scriptstyle \alpha _{2}\,$ and any two functions in K, $\scriptstyle \phi _{1}(x)\,$ and $\scriptstyle \phi _{2}(x)\,$, we have linearity

$\,(f,\alpha _{1}\phi _{1}+\alpha _{2}\phi _{2})=\alpha _{1}(f,\phi _{1})+\alpha _{2}(f,\phi _{2})$

(ii). If the sequence $\,\scriptstyle \phi _{1},\phi _{2},\ldots ,\phi _{\nu },\ldots$ converges to zero in K (i.e. functions $\phi _{i}$ have support contained in a common compact set A and the functions and their derivatives converge to 0 in the supremum norm) then

$(f,\phi _{1}),(f,\phi _{2}),\ldots ,(f,\phi _{\nu }),\ldots$

converges to zero (this is continuity of f). Equivalently, the functional f on K is continuous if for any compact set $\scriptstyle A\subset \mathbb {R} ^{n}$ there exist $\scriptstyle c\in \mathbb {R}$ and $\scriptstyle k\in \mathbb {N}$ such that for any test function $\scriptstyle \phi$ with support in A

$(f,\phi )\leq c\sup _{|\alpha |\leq k,\;x\in A}|D^{\alpha }\phi (x)|.$

Here $\scriptstyle \alpha$ denotes a multi-index and $\scriptstyle D^{\alpha }$ is usual partial derivative described by $\scriptstyle \alpha .$

A distribution (generalized function) is defined as any linear continuous functional on K.