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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 than classical solutions, or 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 distributions, such as the Dirac delta function.

A function ${\displaystyle f}$ is normally thought of as acting on the points in its domain by "sending" a point x in its 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. Test functions are usually infinitely differentiable complex-valued (or sometimes real-valued) functions with compact support (bump functions are examples of test functions). Many "standard functions" (meaning for example a function that is typically encountered in a Calculus course), say for instance a continuous map ${\displaystyle f:\mathbb {R} \to \mathbb {R} ,}$ can be canonically reinterpreted as acting on test functions (instead of their usual interpretation as acting on points of their domain) via the action known as "integration against a test function"; explicitly, this means that ${\displaystyle f}$ "acts on" a test function g by "sending" g to the number ${\displaystyle \textstyle \int _{\mathbb {R} }fg\,dx.}$A function ${\displaystyle f}$ is normally thought of as acting on the points in its domain by "sending" a point x in its 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. Test functions are usually infinitely differentiable complex-valued (or sometimes real-valued) functions with compact support (bump functions are examples of test functions). Many "standard functions" (meaning for example a function that is typically encountered in a Calculus course), say for instance a continuous map ${\displaystyle f:\mathbb {R} \to \mathbb {R} ,}$ can be canonically reinterpreted as acting on test functions (instead of their usual interpretation as acting on points of their domain) via the action known as "integration against a test function"; explicitly, this means that ${\displaystyle f}$ "acts on" a test function g by "sending" g to the number ${\displaystyle \textstyle \int _{\mathbb {R} }fg\,dx.}$ This new action of ${\displaystyle f}$ is thus a complex (or real)-valued map, denoted by ${\displaystyle D_{f},}$ whose domain is the space of test functions; this map turns out to have two additional properties[note 1] that make it into what is known as a distribution on ${\displaystyle \mathbb {R} .}$ Distributions that arise from "standard functions" in this way are the prototypical examples of a distributions. But there are many distributions that do not arise in this way and these distributions are known as "generalized functions." Examples include the Dirac delta function or some distributions that arise via the action of "integration of test functions against measures." However, by using various methods it is nevertheless still possible to reduce any arbitrary distribution down to a simpler family of related distributions that do arise via such actions of integration.

In applications to physics and engineering, the space of test functions usually consists of smooth functions with compact support that are defined on some given non-empty open subset ${\displaystyle U\subset \mathbb {R} ^{n}.}$ This space of test functions is denoted by ${\displaystyle C_{c}^{\infty }(U)}$ or ${\displaystyle {\mathcal {D}}(U)}$ and a distribution on U is by definition a linear functional on ${\displaystyle C_{c}^{\infty }(U)}$ that is continuous when ${\displaystyle C_{c}^{\infty }(U)}$ given a topology called the canonical LF topology. This leads to the space of (all) distributions on 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.

There are other possible choices for the space of test functions, which lead to other different spaces of distributions. If ${\displaystyle U=\mathbb {R} ^{n}}$ then the use of Schwartz functions[note 2] as test functions gives rise to a certain subspace of ${\displaystyle {\mathcal {D}}'(U)}$ whose elements are called tempered distributions. These are important because they allow the Fourier transform to be extended from "standard functions" to tempered distributions. The set of tempered distributions forms a vector subspace of the space of distributions ${\displaystyle {\mathcal {D}}'(U)}$ and is thus one example of a space of distributions; there are many other spaces of distributions.

There also exist other major classes of test functions that are not subsets of ${\displaystyle C_{c}^{\infty }(U),}$ such as spaces of analytic test functions, which produce very different classes of distributions. The theory of such distributions has a different character from the previous one because there are no analytic functions with non-empty compact support.[note 3] Use of analytic test functions lead to Sato's theory of hyperfunctions.

The practical use of distributions can be traced back to the use of Green 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. According to his autobiography, Schwartz introduced the term "distribution" by analogy with a distribution of electrical charge, possibly including not only point charges but also dipoles and so on. Gårding (1997) comments that although the ideas in the transformative book by Schwartz (1951) were not entirely new, it was Schwartz's broad attack and conviction that distributions would be useful almost everywhere in analysis that made the difference.

## Notation

• ${\displaystyle n}$ is a fixed positive integer and ${\displaystyle U}$ is a fixed non-empty open subset of Euclidean space ${\displaystyle \mathbb {R} ^{n}.}$
• ${\displaystyle \mathbb {N} =\{0,1,2,\ldots \}}$ denotes the natural numbers.
• ${\displaystyle k}$ will denote a non-negative integer or ${\displaystyle \infty .}$
• If ${\displaystyle f}$ is a function then ${\displaystyle \operatorname {Dom} (f)}$ will denote its domain and the support of ${\displaystyle f,}$ denoted by ${\displaystyle \operatorname {supp} (f),}$ is defined to be the closure of the set