Distribution (mathematics)

Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative.

Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions (weak solutions) than classical solutions, or where appropriate classical solutions may not exist. 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.

A function ${\displaystyle f}$ is normally thought of as acting on the points in the function domain by "sending" a point ${\displaystyle x}$ in the domain to the point ${\displaystyle f(x).}$ Instead of acting on points, distribution theory reinterprets functions such as ${\displaystyle f}$ as acting on test functions in a certain way. In applications to physics and engineering, test functions are usually infinitely differentiable complex-valued (or real-valued) functions with compact support that are defined on some given non-empty open subset ${\displaystyle U\subseteq \mathbb {R} ^{n}}$. (Bump functions are examples of test functions.) The set of all such test functions forms a vector space that is denoted by ${\displaystyle C_{c}^{\infty }(U)}$ or ${\displaystyle {\mathcal {D}}(U).}$

Most commonly encountered functions, including all continuous maps ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ if using ${\displaystyle U:=\mathbb {R} ,}$ can be canonically reinterpreted as acting via "integration against a test function." Explicitly, this means that such a function ${\displaystyle f}$ "acts on" a test function ${\displaystyle \psi \in {\mathcal {D}}(\mathbb {R} )}$ by "sending" it to the number ${\textstyle \int _{\mathbb {R} }f\,\psi \,dx,}$ which is often denoted by ${\displaystyle D_{f}(\psi ).}$ This new action ${\textstyle \psi \mapsto D_{f}(\psi )}$ of ${\displaystyle f}$ defines a scalar-valued map ${\displaystyle D_{f}:{\mathcal {D}}(\mathbb {R} )\to \mathbb {C} ,}$ whose domain is the space of test functions ${\displaystyle {\mathcal {D}}(\mathbb {R} ).}$ This functional ${\displaystyle D_{f}}$ turns out to have the two defining properties of what is known as a distribution on ${\displaystyle U=\mathbb {R} }$: it is linear, and it is also continuous when ${\displaystyle {\mathcal {D}}(\mathbb {R} )}$ is given a certain topology called the canonical LF topology. The action (the integration ${\textstyle \psi \mapsto \int _{\mathbb {R} }f\,\psi \,dx}$) of this distribution ${\displaystyle D_{f}}$ on a test function ${\displaystyle \psi }$ can be interpreted as a weighted average of the distribution on the support of the test function, even if the values of the distribution at a single point are not well-defined. Distributions like ${\displaystyle D_{f}}$ that arise from functions in this way are prototypical examples of distributions, but there exist many distributions that cannot be defined by integration against any function. Examples of the latter include the Dirac delta function and distributions defined to act by integration of test functions ${\textstyle \psi \mapsto \int _{U}\psi d\mu }$ against certain measures ${\displaystyle \mu }$ on ${\displaystyle U.}$ Nonetheless, it is still always possible to reduce any arbitrary distribution down to a simpler family of related distributions that do arise via such actions of integration.

More generally, a distribution on ${\displaystyle U}$ is by definition a linear functional on ${\displaystyle C_{c}^{\infty }(U)}$ that is continuous when ${\displaystyle C_{c}^{\infty }(U)}$ is given a topology called the canonical LF topology. This leads to the space of (all) distributions on ${\displaystyle U}$, usually denoted by ${\displaystyle {\mathcal {D}}'(U)}$ (note the prime), which by definition is the space of all distributions on ${\displaystyle U}$ (that is, it is the continuous dual space of ${\displaystyle C_{c}^{\infty }(U)}$); it is these distributions that are the main focus of this article.

Definitions of the appropriate topologies on spaces of test functions and distributions are given in the article on spaces of test functions and distributions. This article is primarily concerned with the definition of distributions, together with their properties and some important examples.