Distribution (mathematics)

In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. They extend the concept of derivative to all integrable functions and beyond, and are used to formulate generalized solutions of partial differential equations. They are important in physics and engineering where many non-continuous problems naturally lead to differential equations whose solutions are distributions, such as the Dirac delta distribution.

"Generalized functions" were introduced by Sergei Sobolev in 1935. They were independently introduced in the late 1940s by Laurent Schwartz, who developed a comprehensive theory of distributions.

Basic idea

The basic idea is to identify functions with abstract linear functionals on a space of unproblematic test functions (conventional and well-behaved functions). Operators on distributions can be understood by moving them to the test function.

For example, let

f : R → R

be a locally integrable function, and let

φ : R → R

be a smooth (that is, infinitely differentiable) function with compact support (i.e., identically zero outside of some bounded set). The function φ is the "test function." We then set

$\left\langle f,\varphi \right\rangle =\int _{\mathbf {R} }f\varphi \,dx$.

This is a real number which linearly and continuously depends on φ. One can therefore think of the function f as a continuous linear functional on the space which consists of all the "test functions" φ.

Similarly, if P is a probability distribution on the reals and φ is a test function, then

$\left\langle P,\varphi \right\rangle =\int _{\mathbf {R} }\varphi \,dP$

is a real number that continuously and linearly depends on φ: probability distributions can thus also be viewed as continuous linear functionals on the space of test functions. This notion of "continuous linear functional on the space of test functions" is therefore used as the definition of a distribution.

Such distributions may be multiplied with real numbers and can be added together, so they form a real vector space. In general it is not possible to define a multiplication for distributions, but distributions may be multiplied with integrable functions.

To define the derivative of a distribution, we first consider the case of a differentiable and integrable function f : R → R. If φ is a test function, then we have

$\int _{\mathbf {R} }{}{f'\varphi \,dx}=-\int _{\mathbf {R} }{}{f\varphi '\,dx}$

using integration by parts (note that φ is zero outside of a bounded set and that therefore no boundary values have to be taken into account). This suggests that if S is a distribution, we should define its derivative S' by

$\left\langle S',\varphi \right\rangle =-\left\langle S,\varphi '\right\rangle$.

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: The Dirac delta (so-called Dirac delta function) is the distribution defined by

$\left\langle \delta ,\varphi \right\rangle =\varphi (0)$

It is the derivative of the Heaviside step function: For any test function $\varphi$,

$\left\langle H',\varphi \right\rangle =-\left\langle H,\varphi '\right\rangle =-\int _{-\infty }^{\infty }H(x)\varphi '(x)dx=-\int _{0}^{\infty }\varphi '(x)dx=\varphi (0)-\varphi (\infty )=\varphi (0)=\left\langle \delta ,\varphi \right\rangle ,$

so $H'=\delta$. $\varphi (\infty )=0$ because of compact support. Similarly, the derivative of the Dirac delta is the distribution

$\delta '(\varphi )=-\varphi '(0).$

This latter distribution is our first example of a distribution which is neither a function nor a probability distribution.

Formal definition

In the sequel, 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 smooth manifold.)

First, the space D(U) of test functions on U needs to be explained. A function φ : U → R is said to have compact support if there exists a compact subset K of U such that φ(x) = 0 for all x in U \ K. The elements of D(U) are the infinitely differentiable functions φ : U → R with compact support (also known as bump functions). This is a real vector space. We turn it into a topological vector space by stipulating that a sequence (or net) (φ_k) converges to 0 if and only if there exists a compact subset K of U such that all φ_k are identically zero outside K, and for every ε > 0 and natural number d ≥ 0 there exists a natural number k₀ such that for all k ≥ k₀ the absolute value of all d-th derivatives of φ_k is smaller than ε. With this definition, D(U) becomes a complete topological vector space (in fact, a so-called LF-space).

The dual space of the topological vector space D(U), consisting of all continuous linear functionals S : D(U) → R, is the space of all distributions on U; it is a vector space and is denoted by D'(U). The dual pairing between a distribution S in D′(U) and a test function φ in D(U) is denoted using angle brackets thus:

$\mathrm {D} '(U)\times \mathrm {D} (U)\ni (S,\varphi )\mapsto \langle S,\varphi \rangle \in \mathbf {R} .$

The function f : U → R is called locally integrable if it is Lebesgue integrable over every compact subset K of U. This is a large class of functions which includes all continuous functions. The topology on D(U) is defined in such a fashion that any locally integrable function f yields a continuous linear functional on D(U) whose value on the test function φ is given by the Lebesgue integral ∫_U fφ dx. Two locally integrable functions f and g yield the same element of D'(U) if and only if they are equal almost everywhere. Similarly, every Radon measure μ on U (which includes the probability distributions) defines an element of D'(U) whose value on the test function φ is ∫φ dμ.

As mentioned above, integration by parts suggests that the derivative ∂S/∂x_k of the distribution S in the direction x_k should be defined using the formula

$\left\langle {\frac {\partial S}{\partial x_{k}}},\varphi \right\rangle =-\left\langle S,{\frac {\partial \varphi }{\partial x_{k}}}\right\rangle$

for all test functions φ. In this way, every distribution is infinitely differentiable, and the derivative in the direction x_k is a linear operator on D′(U). In general, if α = (α₁, ..., α_n) is an arbitrary multi-index and ∂^α denotes the associated mixed partial derivative operator, the mixed partial derivative ∂^αS of the distribution S ∈ D′(U) is defined by

$\left\langle \partial ^{\alpha }S,\varphi \right\rangle =(-1)^{|\alpha |}\left\langle S,\partial ^{\alpha }\varphi \right\rangle {\mbox{ for all }}\varphi \in \mathrm {D} '(U).$

The space D'(U) is turned into a locally convex topological vector space by defining that the sequence (S_k) converges towards 0 if and only if S_k(φ) → 0 for all test functions φ; this topology is called the weak-* topology. This is the case if and only if S_k converges uniformly to 0 on all bounded subsets of D(U). (A subset of E of D(U) is bounded if there exists a compact subset K of U and numbers d_n such that every φ in E has its support in K and has its n-th derivatives bounded by d_n.) With respect to this topology, differentiation of distributions is a continuous operator; this is an important and desirable property that is not shared by most other notions of differentiation. Furthermore, the test functions (which can themselves be viewed as distributions) are dense in D'(U) with respect to this topology.

If ψ : U → R is an infinitely often differentiable function and S is a distribution on U, we define the product Sψ by (Sψ)(φ) = S(ψφ) for all test functions φ. The ordinary product rule of calculus remains valid.

Distributions as derivatives of continuous functions

The formal definition of distributions exhibits them as a subspace of a very large space, namely the algebraic dual of D(U). It is not immediately clear from the definition how exotic a distribution might be. To answer this question, it is instructive to see distributions built up from a smaller space, namely the space of continuous functions. Roughly, any distribution is locally a (multiple) derivative of a continuous function. (The precise theorem is below.) In other words, no proper subset of the space of distributions contains all continuous functions and is closed under differentiation. This says that distributions are not particularly exotic objects; they are only as complicated as necessary.

One precise version of the theorem is the following. Let S be a distribution on U. Then for every multi-index α, there exists a continuous function g_α such that any compact subset K of U intersects the supports of only finitely many g_α, and such that

$\displaystyle S=\sum _{\alpha }D^{\alpha }g_{\alpha }.$

Compact support and convolution

We say that a distribution S has compact support if there is a compact subset K of U such that for every test function φ whose support is completely outside of K, we have S(φ) = 0. Alternatively, one may define distributions with compact support as continuous linear functionals on the space C^∞(U); the topology on C^∞(U) is defined such that φ_k converges to 0 if and only if all derivatives of φ_k converge uniformly to 0 on every compact subset of U.

If both S and T are distributions on Rⁿ and one of them has compact support, then one can define a new distribution, the convolution S ∗ T of S and T, as follows: if φ is a test function in D(Rⁿ) and x, y elements of Rⁿ, write φ_x(y) = φ (x + y), ψ(x) = T(φ_x) and (S ∗ T) (φ) = S(ψ). This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense:

${\dfrac {d}{dx}}(S*T)={\dfrac {dS}{dx}}*T=S*{\dfrac {dT}{dx}}.$

This definition of convolution remains valid under less restrictive assumptions about S and T.

Tempered distributions and Fourier transform

By using a larger space of test functions, one can define the tempered distributions, a subspace of D'(Rⁿ). These distributions are useful if one studies the Fourier transform in generality: all tempered distributions have a Fourier transform, but not all distributions have one.

The space of test functions employed here, the so-called Schwartz space, is the space of all infinitely differentiable rapidly decreasing functions, where φ : Rⁿ → R is called rapidly decreasing if any derivative of φ, multiplied with any power of |x|, converges towards 0 for |x| → ∞. These functions form a complete topological vector space with a suitably defined family of seminorms. More precisely, let

$p_{\alpha ,\beta }(\phi )=\sup _{x\in \mathbf {R} ^{n}}|x^{\alpha }D^{\beta }\phi (x)|$

for α, β multi-indices of size n. Then φ is rapidly-decreasing if all the values

$p_{\alpha ,\beta }(\phi )<\infty .$

The family of seminorms p_{α, β} defines a locally convex topology on the Schwartz-space. It is metrizable and complete.

The space of tempered distributions is defined as the dual of the Schwartz space. In other words, a distribution F is a tempered distribution if and only if

$\lim _{m\to \infty }\sup _{x\in \mathbf {R} ^{n}}|x^{\alpha }D^{\beta }\phi _{m}(x)|=0$

for all multi-indices α, β implies

$\lim _{n\to \infty }F(\phi _{n})=0.$

The derivative of a tempered distribution is again a tempered distribution. Tempered distributions generalize the bounded (or slow-growing) locally integrable functions; all distributions with compact support and all square-integrable functions are tempered distributions. All locally integrable functions f with at most polynomial growth, i.e. such that f(x)=O(|x|^r) for some r, are tempered distributions. This includes all functions in L^p(Rⁿ) for p≥1.

To study the Fourier transform, it is best to consider complex-valued test functions and complex-linear distributions. The ordinary continuous Fourier transform F yields then an automorphism of Schwartz-space, and we can define the Fourier transform of the tempered distribution S by (FS)(φ) = S(Fφ) for every test function φ. FS is thus again a tempered distribution. The Fourier transform is a continuous, linear, bijective operator from the space of tempered distributions to itself. This operation is compatible with differentiation in the sense that

$F{\dfrac {dS}{dx}}=ixFS$

and also with convolution: if S is a tempered distribution and ψ is a slowly increasing infinitely differentiable function on Rⁿ (meaning that all derivatives of ψ grow at most as fast as polynomials), then Sψ is again a tempered distribution and

$F(S\psi )=FS*F\psi .\,$

Using holomorphic functions as test functions

The success of the theory led to investigation of the idea of hyperfunction, in which spaces of holomorphic functions are used as test functions. A refined theory has been developed, in particular by Mikio Sato, using sheaf theory and several complex variables. This extends the range of symbolic methods that can be made into rigorous mathematics, for example Feynman integrals.

Problem of multiplication

A possible limitation of the theory of distributions (and hyperfunctions) is that it is a purely linear theory, in the sense that the product of two distributions cannot consistently be defined (in general), as has been proved by Laurent Schwartz in the 1950s. For example, if 1/x is the distribution obtained by extending the corresponding function to a homogeneous distribution, and δ is the Dirac delta distribution then

(δ × x) × 1/x = 0

but

δ × (x × 1/x) = δ

so the product of a distribution by a smooth function (which is always well defined) cannot be extended to an associative product on the space of distributions.

Thus, nonlinear problems cannot be posed in general and thus not solved within distribution theory alone. In the context of quantum field theory, however, solutions can be found. In more than two spacetime dimensions the problem is related to the regularization of "divergencies". Here Henri Epstein and Vladimir Glaser developed the mathematically rigorous (but extremely technical) causal perturbation theory. This does not solve the problem in other situations. Many other interesting theories are non linear, like for example Navier-Stokes equations of fluid dynamics.

In view of this, several not entirely satisfactory theories of algebras of generalized functions have been developed, among which Colombeau's (simplified) algebra is maybe the most popular in use today.

Another example of the imposibility of multiplication is the convolution since $(2\pi )D^{m}\delta (u)=\int _{-\infty }^{\infty }dxe^{iux}(ix)^{m}$ then we would have $(2\pi )^{2}i^{m+n}D^{m}\delta (u)D^{n}\delta (u)=\int _{-\infty }^{\infty }dx\int _{-\infty }^{\infty }dt(t-x)^{m}t^{n}e^{iut}$ , however the last integral is divergent for every value of 'u'

A simple solution of the multiplication problem is dictated by the path integral formulation of quantum mechanics. Since this is required to be equivalent to the Schrödinger theory of quantum mechanics which is invariant under coordinate transformations, this property must be shared by path integrals. This fixes all products of distributions as shown by H. Kleinert and A. Chervyakov. The result is equivalent to what can be derived from dimensional regularization.

See also

• Generalized function
• Colombeau algebra
• Weak solution

Books

• M. J. Lighthill (1959). Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press. ISBN 0-521-09128-4 (requires very little knowledge of analysis; defines distributions as limits of sequences of functions under integrals)
• R. Strichartz (1994). A Guide to Distribution Theory and Fourier Transforms. CRC Press. ISBN 0849382734 (the standard way of defining distributions, as presented in this article)
• L. Schwartz (1954), Sur l'impossibilité de la multiplications des distributions, C.R.Acad. Sci. Paris 239, pp 847-848.
• H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th edition, World Scientific (Singapore, 2006)(also available online here). See Chapter 11 for defining products of distributions from the physical requirement of coordinate invariance.