Distribution (mathematics)

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 (which is a distribution, not a function as its historical name might suggest). 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 (which is a distribution, not a function as its historical name might suggest). The practical use of distributions can be traced back to the use of Green functions in the 1830s to solveordinary 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. Distribution theory reinterprets functions as linear functionals acting on a space of test functions. Standard functions act by integration against a test function, but many other linear functionals do not arise in this way, and these are the 'generalized functions'. There are different possible choices for the space of test functions, leading to different spaces of distributions. The basic space of test function consists of smooth functions with compact support, leading to standard distributions. Use of the space of smooth, rapidly (faster than any polynomial increases) decreasing test functions (these functions are called Schwartz functions) gives instead the tempered distributions, which are important because they have a well-defined distributional Fourier transform. Every tempered distribution is a distribution in the normal sense, but the converse is not true: in general the larger the space of test functions, the more restrictive the notion of distribution. On the other hand, the use of spaces of analytic test functions leads to Sato's theory of hyperfunctions; this theory has a different character from the previous ones because there are no analytic functions with non-empty compact support. Distributions are a class of linear functionals that map a set of test functions (conventional and well-behaved functions) into the set of real numbers. In the simplest case, the set of test functions considered is D(R), which is the set of functions φ : R → R having two properties: A distribution T is a linear mapping T : D(R) → R. Instead of writing T( φ {displaystyle varphi } ), it is conventional to write ⟨ T , φ ⟩ {displaystyle langle T,varphi angle } for the value of T acting on a test function φ {displaystyle varphi } . A simple example of a distribution is the Dirac delta δ, defined by meaning that δ evaluates a test function at 0. Its physical interpretation is as the density of a point source. As described next, there are straightforward mappings from both locally integrable functions and Radon measures to corresponding distributions, but not all distributions can be formed in this manner. Suppose that f : R → R is a locally integrable function. Then a corresponding distribution Tf may be defined by This integral is a real number which depends linearly and continuously on φ {displaystyle varphi } . Conversely, the values of the distribution Tf on test functions in D(R) determine the pointwise almost everywhere values of the function f on R. In a conventional abuse of notation, f is often used to represent both the original function f and the corresponding distribution Tf. This example suggests the definition of a distribution as a linear and, in an appropriate sense, continuous functional on the space of test functions D(R).