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 is the Dirac delta function. 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 "well-behaved" test function 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 normed linear space. Notice that this norm defines a metric on K. For simplicity we give the expression for the norm only when the underlying space is the real line

$||f||=\sup {x\in \mathbb {R} ,\;n\in \mathbb {N} }|{\frac {d^{n}f}{dx^{n}}}(x)|.$

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). Continuity. If the sequence $\,\scriptstyle \phi _{1},\phi _{2},\ldots ,\phi _{\nu },\ldots$ converges uniformly to zero in K, then

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

converges to zero (continuity of f).

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