Distribution (mathematical analysis)


Distributions, also known as Schwartz distributions are a kind of generalized function 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 where 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 singular, such as the Dirac delta function.
A function is normally thought of as on the in the function domain by "sending" a point in the domain to the point. Instead of acting on points, distribution theory reinterprets functions such as as acting on in a certain way. In applications to physics and engineering, ' are usually infinitely differentiable complex-valued functions with compact support that are defined on some given non-empty open subset. The set of all such test functions forms a vector space that is denoted by or.
Most commonly encountered functions, including all continuous maps if using can be canonically reinterpreted as acting via "integration against a test function". Explicitly, this means that such a function "acts on" a test function by "sending" it to the number which is often denoted by. This new action of defines a scalar-valued map whose domain is the space of test functions. This functional turns out to have the two defining properties of what is known as a : it is linear, and it is also continuous when is given a certain topology called. The action of this distribution on a test function can be interpreted as a weighted average of the distribution on the [|support] of the test function, even if the values of the distribution at a single point are not well-defined. Distributions like that arise from functions in this way are prototypical examples of distributions, but there exist many distributions that cannot be defined by integration against any function. Examples of the latter include the Dirac delta function and distributions defined to act by integration of test functions against certain measures on. Nonetheless, it is still always possible to [|reduce any arbitrary distribution] down to a simpler of related distributions that do arise via such actions of integration.
More generally, a is by definition a linear functional on that is continuous when is endowed with the '. The space of all distributions on is usually denoted by.
Definitions of the appropriate topologies on spaces of test functions and distributions are given in the article on spaces of test functions and distributions. This article is primarily concerned with the definition of distributions, together with their properties and some important examples.

History

The practical use of distributions can be traced back to the use of Green's functions in the 1830s to solve ordinary differential equations, but was not formalized until much later. According to, generalized functions originated in the work of 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. comments that although the ideas in the transformative book by 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. A detailed history of the theory of distributions was given by.

Notation

The following notation will be used throughout this article:
  • is a fixed positive integer and is a fixed non-empty open subset of Euclidean space.
  • denotes the natural numbers.
  • will denote a non-negative integer or.
  • If is a function then will denote its domain and the ' of denoted by is defined to be the closure of the set in.
  • For two functions the following notation defines a canonical pairing:
  • A of size is an element in . The ' of a multi-index is defined as and denoted by. Multi-indices are particularly useful when dealing with functions of several variables, in particular, we introduce the following notations for a given multi-index : We also introduce a partial order of all multi-indices by if and only if for all. When we define their multi-index binomial coefficient as:

    Definitions of test functions and distributions

In this section, some basic notions and definitions needed to define real-valued distributions on are introduced. Further discussion of the topologies on the spaces of test functions and distributions is given in the article on spaces of test functions and distributions.
File:Bump.png|thumb|350x350px|The graph of the bump function, where and. This function is a test function on and is an element of. The support of this function is the closed unit disk in. It is non-zero on the open unit disk and it is equal to everywhere outside of it.
For all and any compact subsets and of, we have:
Distributions on are continuous linear functionals on when this vector space is endowed with a particular topology called the '. The following proposition states two necessary and sufficient conditions for the continuity of a linear function on that are often straightforward to verify.
Proposition: A linear functional on is continuous, and therefore a ', if and only if any of the following equivalent conditions is satisfied:
  1. For every compact subset there exist constants and such that for all with support contained in,
  2. For every compact subset and every sequence in whose supports are contained in, if converges uniformly to zero on for every multi-index, then.

    Topology on ''C''''k''(''U'')

We now introduce the seminorms that will define the topology on. Different authors sometimes use different families of seminorms so we list the most common families below. However, the resulting topology is the same no matter which family is used.
All of the functions above are non-negative -valued seminorms on. As explained in this article, every set of seminorms on a vector space induces a locally convex vector topology.
Each of the following sets of seminorms
generate the same locally convex vector topology on .
With this topology, becomes a locally convex Fréchet space that is normable. Every element of is a continuous seminorm on.
Under this topology, a net in converges to if and only if for every multi-index with and every compact, the net of partial derivatives converges uniformly to on For any any bounded subset of is a relatively compact subset of In particular, a subset of is bounded if and only if it is bounded in for all The space is a Montel space if and only if
A subset of is open in this topology if and only if there exists such that is open when is endowed with the subspace topology induced on it by.

Topology on ''C''''k''(''K'')

As before, fix Recall that if is any compact subset of then
If is finite then is a Banach space with a topology that can be defined by the norm

Trivial extensions and independence of ''C''''k''(''K'')'s topology from ''U''

Suppose is an open subset of and is a compact subset. By definition, elements of are functions with domain , so the space and its topology depend on to make this dependence on the open set clear, temporarily denote by
Importantly, changing the set to a different open subset will change the set from to so that elements of will be functions with domain instead of
Despite depending on the open set, the standard notation for makes no mention of it.
This is justified because, as this subsection will now explain, the space is canonically identified as a subspace of .
It is enough to explain how to canonically identify with when one of and is a subset of the other. The reason is that if and are arbitrary open subsets of containing then the open set also contains so that each of and is canonically identified with and now by transitivity, is thus identified with
So assume are open subsets of containing
Given its is the function defined by:
This trivial extension belongs to and it will be denoted by . The assignment thus induces a map that sends a function in to its trivial extension on This map is a linear injection and for every compact subset ,
If is restricted to then the following induced linear map is a homeomorphism :
and thus the next map is a topological embedding:
Using the injection
the vector space is canonically identified with its image in Because through this identification, can also be considered as a subset of
Thus the topology on is independent of the open subset of that contains which justifies the practice of writing instead of

Canonical LF topology

Recall that denotes all functions in that have compact support in where note that is the union of all as ranges over all compact subsets of Moreover, for each is a dense subset of The special case when gives us the space of test functions.
The canonical LF-topology is metrizable and importantly, it is Comparison of topologies| than the subspace topology that induces on However, the canonical LF-topology does make into a complete reflexive nuclear Montel bornological barrelled Mackey space; the same is true of its strong dual space. The canonical LF-topology can be defined in various ways.

Distributions

As discussed earlier, continuous linear functionals on a are known as distributions on Other equivalent definitions are described below.
There is a canonical duality pairing between a distribution on and a test function which is denoted using angle brackets by
One interprets this notation as the distribution acting on the test function to give a scalar, or symmetrically as the test function acting on the distribution