A generalized function, also called a "distribution" or "ideal function," is a continuous linear functional on the function space of test functions, namely infinitely differentiable functions with compact support, equipped with the topology described below. Equivalent regular sequences of particularly well-behaved functions determine the same generalized function when their integrals against every Schwartz function have the same limit. This construction uses functions in the Schwartz space and describes a restricted class of distributions, rather than all distributions. As its name implies, a generalized function is a generalization of the concept of a function. For example, in physics, a baseball being hit by a bat encounters a force from the bat, as a function of time. Since the transfer of momentum from the bat is modeled as taking place at an instant, the force is not actually a function. Instead, it is a multiple of the delta function. The set of distributions contains functions (locally integrable) and Radon measures. Note that the term "distribution" is closely related to statistical distributions.

All continuous functions have derivatives in the sense of distributions, and these derivatives are themselves generalized functions. The most commonly encountered generalized function is the delta function. Vladimirov (1971) contains a nice treatment of distributions from a physicist's point of view, while the multivolume work (Gel'fand and Shilov 1964, Gel'fand and Vilenkin 1964, Gel'fand et al. 1966, Gel'fand and Shilov 1967, 1968) is a classic and rigorous treatment of the subject. Distributions can be defined over the complex numbers using complex-linear functionals on complex-valued test functions.

Distributions can be added, but a product of arbitrary distributions is not always defined. Coinciding singular supports alone do not rule out a product. For example, the Heaviside step function is locally integrable and satisfies . Despite this, it is possible to take the derivative of a distribution, to get another distribution. Consequently, they may satisfy a linear partial differential equation, in which case the distribution is called a weak solution in the sense of a weak formulation. For example, given any locally integrable function it makes sense to ask for solutions of Poisson's equation

by only requiring the equation to hold in the sense of distributions, that is, both sides are the same distribution. The definitions of the derivatives of a distribution are given by

Distributions also differ from functions because they are covariant, that is, they push forward. Given a smooth function , a distribution on pushes forward to a distribution on provided is proper on the support of . This condition ensures that pullbacks of test functions have compact support when restricted to the support of . In contrast, a real function on pulls back to a function on , namely .

Distributions are, by definition, the continuous dual to the smooth functions of compact support, with a particular topology. For example, the delta function is the linear functional . The distribution corresponding to a locally integrable function is

and the distribution corresponding to a Radon measure is

Under this support condition, the pushforward map of a distribution along is defined by

and the coordinate derivative , where , is defined by , with the formal adjoint of . For example, the first derivative of the delta function is given by

As is the case for any function space, the topology determines which linear functionals are continuous, that is, are in the continuous dual space. For each fixed compact set , the function space of smooth functions supported in has the topology defined by the seminorms

Here is a derivative multi-index and denotes the supremum. The full function space of test functions has the inductive-limit topology of these fixed-support function spaces. In this topology, a sequence converges, , iff there is a compact set such that all and are supported in and every derivative converges uniformly to in . Therefore, the constant function 1 is a distribution, because if then

See also

Convolution, Delta Function, Delta Sequence, Fourier Series, Functional, Linear Functional, Test Function

Related Wolfram sites

https://functions.wolfram.com/GeneralizedFunctions/

Portions of this entry contributed by Todd Rowland

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References

Brychkov, Yu. A. and Prudnikov, A. P. Integral Transforms of Generalized Functions. New York: Gordon and Breach, 1989.Gel'fand, I. M.; Graev, M. I.; and Vilenkin, N. Ya. Generalized Functions, Vol. 5: Integral Geometry and Representation Theory. New York: Academic Press, 1966.Gel'fand, I. M. and Shilov, G. E. Generalized Functions, Vol. 1: Properties and Operations. New York: Academic Press, 1964.Gel'fand, I. M. and Shilov, G. E. Generalized Functions, Vol. 2: Spaces of Fundamental and Generalized Functions. New York: Academic Press, 1968.Gel'fand, I. M. and Shilov, G. E. Generalized Functions, Vol. 3: Theory of Differential Equations. New York: Academic Press, 1967.Gel'fand, I. M. and Vilenkin, N. Ya. Generalized Functions, Vol. 4: Applications of Harmonic Analysis. New York: Academic Press, 1964.Kanwal, R. P. Generalized Functions: Theory and Technique, 2nd ed. Boston, MA: Birkhäuser, 1998.Vladimirov, V. S. Equations of Mathematical Physics. New York: Dekker, 1971.

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Generalized Function

Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland. "Generalized Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GeneralizedFunction.html

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