A generalized function, also called a "distribution" or "ideal function," is a continuous linear functional on the function space of infinitely differentiable functions with compact support, equipped with the test-function space 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 test 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. 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
 |  |  |
(2)
|
 |  |  |
(3)
|
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
 |
(4)
|
and the distribution corresponding to a Radon measure 
is
 |
(5)
|
Under this support condition, the pushforward map of a distribution 
along 
is defined by
 |
(6)
|
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
 |
(7)
|
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
 |
(8)
|
Here 
is a derivative multi-index
and 
denotes the supremum. The full test-function
space 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 
are supported in 
and every derivative 
converges uniformly
to 
in 
.
Therefore, the constant function
1 is a distribution, because if 
then
 |
(9)
|
