A generalized function, also called a "distribution" or "ideal function," is the class of all regular sequences of particularly well-behaved functions equivalent to a given regular sequence. 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.
Generalized functions are defined as continuous linear functionals over a space of infinitely differentiable functions such
that all continuous functions have derivatives which 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 by Gel'fand and Shilov (1964abcde)
is a classic and rigorous treatment of the field. A result of Schwarz shows that
distributions can't be consistently defined over the complex numbers 
.
While it is possible to add distributions, it is not possible to multiply distributions when they have coinciding singular support. 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.
For example, given any locally integrable function 
it makes sense to ask for solutions 
of Poisson's equation
 |
(1) |
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 
. In contrast, a real function 
on 
pulls back to a function on 
, namely 
.
Distributions are, by definition, the 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 function
is
 |
(4) |
and the distribution corresponding to a measure 
is
 |
(5) |
The pushforward map of a distribution 
along 
is defined by
 |
(6) |
and the derivative of 
is defined by 
where 
is the formal adjoint of 
. For example, the first derivative of the delta
function is given by
![d/(dx)[delta(f)]=-(df)/(dx)|_(x=0).](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FGeneralizedFunction%2FNumberedEquation5.svg){kind=small} |
(7) |
As is the case for any function space, the topology determines which linear functionals are continuous, that is, are in the dual vector space. The topology is defined by the family of seminorms,
 |
(8) |
where sup denotes the supremum. It agrees with the C-infty topology on compact subsets. In this
topology, a sequence converges, 
, iff there is a compact set
such that all 
are supported in 
and every derivative 
converges uniformly to 
in 
. Therefore, the constant function 1 is a distribution, because
if 
then
 |
(9) |
See also
Convolution, Delta Function, Delta Sequence, Fourier Series, Functional, Linear Functional
Related Wolfram sites
http://functions.wolfram.com/GeneralizedFunctions/
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. and Shilov, G. E. Generalized Functions, Vol. 1: Properties and Operations. New York: Academic Press, 1964a.Gel'fand, I. M. and Shilov, G. E. Generalized Functions, Vol. 2: Spaces of Fundamental and Generalized Functions. New York: Academic Press, 1964b.Gel'fand, I. M. and Shilov, G. E. Generalized Functions, Vol. 3: Theory of Differential Equations. New York: Academic Press, 1964c.Gel'fand, I. M. and Shilov, G. E. Generalized Functions, Vol. 4: Applications of Harmonic Analysis. New York: Academic Press, 1964d.Gel'fand, I. M. and Shilov, G. E. Generalized Functions, Vol. 5: Integral Geometry and Representation Theory. New York: Academic Press, 1964e.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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Cite this as:
Rowland, Todd. "Generalized Function." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/GeneralizedFunction.html