Distribution (mathematics)

Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense.

Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are singular, such as the Dirac delta function.

This article is primarily concerned with the definition of distributions, together with their properties and some important examples.

The practical use of distributions can be traced back to the use of Green's functions in the 1830s to solve ordinary differential equations, but was not formalized until much later.

According to Kolmogorov & Fomin (1957), generalized functions originated in the work of Sergei Sobolev (1936) on second-order hyperbolic partial differential equations, and the ideas were developed in somewhat extended form by Laurent Schwartz in the late 1940s.

The following notation will be used throughout this article: In this section, some basic notions and definitions needed to define real-valued distributions on U are introduced.

is continuous, and therefore a distribution, if and only if any of the following equivalent conditions is satisfied: We now introduce the seminorms that will define the topology on

The canonical LF-topology is not metrizable and importantly, it is strictly finer than the subspace topology that

The formal definition of distributions exhibits them as a subspace of a very large space, namely the topological dual of

Generally speaking, no proper subset of the space of distributions contains all continuous functions and is closed under differentiation.

is a linear map that is continuous with respect to the weak topology, then it is not always possible to extend

This is because the transpose allows for a unified presentation of the many definitions in the theory of distributions and also because its properties are well-known in functional analysis.

[21] enables us to arrive at the correct definition: the formal transpose induces the (continuous) canonical linear operator

With this definition of multiplication by a smooth function, the ordinary product rule of calculus remains valid.

[24] With more effort, it is possible to define a well-behaved product of several distributions provided their wave front sets at each point are compatible.

Here Henri Epstein and Vladimir Glaser developed the mathematically rigorous (but extremely technical) causal perturbation theory.

Inspired by Lyons' rough path theory,[25] Martin Hairer proposed a consistent way of multiplying distributions with certain structures (regularity structures[26]), available in many examples from stochastic analysis, notably stochastic partial differential equations.

Existence, however, requires using the change of variables formula, the inverse function theorem (locally), and a partition of unity argument.

This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense: for every multi-index

The convolution of a finite number of distributions, all of which (except possibly one) have compact support, is associative.

[34] The convolution of distributions with compact support induces a continuous bilinear map

be an ordinary (slowly growing, smooth) function within the space of tempered distributions and let

is the sinc function and both equations yield the Classical Sampling Theorem for suitable

every one of the following canonical injections is continuous and has an image (also called the range) that is a dense subset of its codomain:

can be identified as the space of Radon measures, where there is a one-to-one correspondence between the continuous linear functionals

is the space of all smooth functions that are rapidly decreasing at infinity along with all partial derivatives.

These functions form a complete TVS with a suitably defined family of seminorms.

[52] The Schwartz space is nuclear, and the tensor product of two maps induces a canonical surjective TVS-isomorphisms

To study the Fourier transform, it is best to consider complex-valued test functions and complex-linear distributions.

This extends the range of symbolic methods that can be made into rigorous mathematics, for example, Feynman integrals.