In mathematical analysis, distributions are a type of generalized function that makes 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.
The practical use of distributions can be traced back to the use of Green 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.
Distribution theory reinterprets functions as linear functionals acting on a space of . 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.
Basic idea
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 : R → R having two properties: is smooth (infinitely differentiable); has compact support (is identically zero outside some bounded interval). A distribution T is a linear mapping T : D(R) → R. Instead of writing T(), it is conventional to write for the value of T acting on a test function . 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.
Functions and measures as distributions 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 . 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).
Similarly, if μ is a Radon measure on R, then a corresponding distribution Rμ may be defined by
This integral also depends linearly and continuously on , so that Rμ is a distribution. If μ is absolutely continuous with respect to Lebesgue measure with density f and dμ = f dx, then this definition for Rμ is the same as the previous one for Tf, but if μ is not absolutely continuous, then Rμ is a distribution that is not associated with a function. For example, if P is the point-mass measure on R that assigns measure one to the singleton set {0} and measure zero to sets that do not contain zero, then
so that RP = δ is the Dirac delta.
Adding and multiplying distributions Distributions may be multiplied by real numbers and added together, so they form a real vector space. Distributions may also be multiplied by infinitely differentiable functions, but it is not possible to define a product of general distributions that extends the usual pointwise product of functions and has the same algebraic properties. This result was shown by , and is usually referred to as the Schwartz' Impossibility Theorem.
Derivatives of distributions It is desirable to choose a definition for the derivative of a distribution which, at least for distributions derived from smooth functions, has the property that . If is a test function, we can use integration by parts to see that
where the last equality follows from the fact that has compact support, so is zero outside of a bounded set. This suggests that if is a distribution, we should define its derivative by
It turns out that this is the proper definition; it extends the ordinary definition of derivative, every distribution becomes infinitely differentiable and the usual properties of derivatives hold.
Example: Recall that the Dirac delta (so-called Dirac delta function) is the distribution defined by the equation
It is the derivative of the distribution corresponding to the Heaviside step function H: For any test function ,
so H′ = δ. Note, (∞) = 0 because has compact support by our definition of a test function. Similarly, the derivative of the Dirac delta is the distribution defined by the equation
This latter distribution is an example of a distribution that is not derived from a function or a measure. Its physical interpretation is the density of a dipole source. Just as the Dirac impulse can be realized in the weak limit as a sequence of various kinds of constant norm bump functions of ever increasing amplitude and narrowing support, its derivative can by definition be realized as the weak limit of the negative derivatives of said functions, which are now antisymmetric about the eventual distribution's point of singular support.
Test functions
In the following, real-valued distributions on an open subset U of Rn will be formally defined. With minor modifications, one can also define complex-valued distributions, and one can replace Rn by any (paracompact) smooth manifold.
The first object to define is the space D(U) of test functions on U. Once this is defined, it is then necessary to equip it with a topology by defining the limit of a sequence of elements of D(U). The space of distributions will then be given as the space of continuous linear functionals on D(U).
Test function space
The space D(U) of test functions on U is defined as follows. A function : U → R is said to have compact support if there exists a compact subset K of U such that (x) = 0 for all x in U \ K. The elements of D(U) are the infinitely differentiable functions : U → R with compact support – also known as bump functions. This is a real vector space. It can be given a topology by defining the limit of a sequence of elements of D(U). A sequence (k) in D(U) is said to converge to ∈ D(U) if the following two conditions hold:According to
There is a compact set K ⊂ U containing the supports of all k:
For each multi-index α, the sequence of partial derivatives tends uniformly to .
With this definition, D(U) becomes a complete locally convex topological vector space satisfying the Heine–Borel property.See for example .
This topology can be placed in the context of the following general construction: let
be a countable increasing union of locally convex topological vector spaces and ιi : Xi → X be the inclusion maps. In this context, the inductive limit topology, or final topology, τ on X is the finest locally convex vector space topology making all the inclusion maps continuous. The topology τ can be explicitly described as follows: let β be the collection of convex balanced subsets W of X such that W ∩ Xi is open for all i. A base for the inductive limit topology τ then consists of the sets of the form x + W, where x in X and W in β.
The proof that τ is a vector space topology makes use of the assumption that each Xi is locally convex. By construction, β is a local base for τ. That any locally convex vector space topology on X must necessarily contain τ means it is the weakest one. One can also show that, for each i, the subspace topology Xi inherits from τ coincides with its original topology. When each Xi is a Fréchet space, (X, τ) is called an LF space.
Now let U be the union of Ui where {Ui} is a countable nested family of open subsets of U with compact closures Ki = i. Then we have the countable increasing union
where DKi is the set of all smooth functions on U with support lying in Ki. On each DKi, consider the topology given by the seminorms
i.e. the topology of uniform convergence of derivatives of arbitrary order. This makes each DKi a Fréchet space. The resulting LF space structure on D(U) is the topology described in the beginning of the section.
On D(U), one can also consider the topology given by the seminorms
However, this topology has the disadvantage of not being complete. On the other hand, because of the particular features of DKi's, a set this bounded with respect to τ if and only if it lies in some DKi's. The completeness of (D(U), τ) then follow from that of DKi's.
The topology τ is not metrizable by the Baire category theorem, since D(U) is the union of subspaces of the first category in D(U).See for example
Distributions
A distribution on U is a continuous linear functional T : D(U) → R (or T : D(U) → C). That is, a distribution T assigns to each test function a real (or complex) scalar T() such that
for all test functions 1, 2 and scalars c1, c2. Moreover, T is continuous if and only if
for every convergent sequence k in D(U). (Even though the topology of D(U) is not metrizable, a linear functional on D(U) is continuous if and only if it is sequentially continuous.) Equivalently, T is continuous if and only if for every compact subset K of U there exists a positive constant CK and a non-negative integer NK such that
for all test functions with support contained in K.See for example .
The space of distributions on U is denoted by D′(U) and it is the continuous dual space of D(U). No matter what dual topology is placed on D′(U), a sequence of distributions converges in this topology if and only if it converges pointwise (although this need not be true of a net), which is why the topology is sometimes defined to be the weak-* topology. But often the topology of bounded convergence, which in this case is the same as the topology of uniform convergence on compact sets, is placed on D′(U) since it is with this topology that D′(U) becomes a nuclear Montel space and it is with this topology that the kernels theorem of Schwartz holds.See for example . No matter which topology is chosen, D′(U) will be a non-metrizable, locally convex topological vector space.