Distribution (mathematics)

A distribution, also called a generalized function, is a continuous linear functional acting on a space of smooth test functions. Distributions extend the operation of differentiation to objects that need not possess pointwise values or classical derivatives. They provide a common mathematical setting for locally integrable functions, point masses, singular kernels, and weak solutions of partial differential equations.

The term is distinct from a probability distribution, which assigns probabilities to measurable events or describes the law of a random variable. Probability measures nevertheless determine distributions in the generalized-function sense by integration against test functions.

The modern theory was systematized by Laurent Schwartz during the late 1940s. Its principal innovation was not the introduction of individual singular objects, many of which had already appeared in mathematical physics, but the specification of a topological vector space on which those objects act continuously.

Definition

For an open set (\Omega\subseteq\mathbb{R}^n), the standard test-function space is

[ \mathcal{D}(\Omega)=C_c^\infty(\Omega), ]

the vector space of infinitely differentiable complex-valued functions whose supports are compact subsets of (\Omega). This space carries its usual locally convex topology. A sequence ((\varphi_j)) converges to (\varphi) in (\mathcal{D}(\Omega)) when all functions have support in one fixed compact subset of (\Omega) and every derivative converges uniformly there.

A distribution on (\Omega) is a linear map

[ T:\mathcal{D}(\Omega)\longrightarrow\mathbb{C} ]

that is continuous with respect to this topology. Its value at a test function is conventionally written as either (T(\varphi)) or

[ \langle T,\varphi\rangle. ]

The vector space of all distributions on (\Omega) is denoted by (\mathcal{D}'(\Omega)). It is the continuous dual space of (\mathcal{D}(\Omega)).

Continuity has a local finite-order formulation. For every compact set (K\subset\Omega), there are a nonnegative integer (m) and a constant (C_K) such that

[ |\langle T,\varphi\rangle| \leq C_K\sum_{|\alpha|\leq m} \sup_{x\in K}|D^\alpha\varphi(x)| ]

whenever the support of (\varphi) lies in (K). The required order may depend on (K), so a general distribution need not have a single finite order on all of (\Omega).

Regular and singular distributions

Every function (f\in L_{\mathrm{loc}}^1(\Omega)) determines a distribution (T_f) through

[ \langle T_f,\varphi\rangle

\int_\Omega f(x)\varphi(x),dx. ]

Such a functional is called a regular distribution. Two locally integrable functions define the same distribution precisely when they agree almost everywhere, in accordance with the usual identification in Lebesgue integration.

Not every distribution is induced by a locally integrable function. For a point (a\in\Omega), the Dirac delta is defined by

[ \langle \delta_a,\varphi\rangle=\varphi(a). ]

The delta distribution is supported at the single point (a). It cannot be represented by an ordinary locally integrable function, because integration against such a function cannot reproduce point evaluation for every test function.

A distribution (T) is positive when

[ \langle T,\varphi\rangle\geq 0 ]

for every nonnegative real-valued test function (\varphi). Every positive distribution has order zero and is represented by a positive Radon measure. Consequently, positivity excludes derivative-type singularities while still allowing point masses and measures concentrated on lower-dimensional sets.

Distributional derivatives

For a multi-index (\alpha), the derivative (D^\alpha T) is defined by duality:

[ \langle D^\alpha T,\varphi\rangle

(-1)^{|\alpha|} \langle T,D^\alpha\varphi\rangle. ]

This definition reproduces classical differentiation whenever the original function is sufficiently smooth. For a locally integrable function, it also agrees with the derivative obtained through integration by parts, with the test function absorbing the boundary terms because it has compact support.

Every distribution therefore possesses derivatives of all orders. These derivatives are distributions even when no corresponding pointwise derivative exists. For the Heaviside step function (H) on the real line, the identity

[ H'=\delta_0 ]

holds in (\mathcal{D}'(\mathbb{R})). More generally, discontinuities in a piecewise smooth function contribute delta terms to its distributional derivative, while discontinuities in its derivatives contribute derivatives of delta distributions.

The same formalism defines distributional differential operators. If

[ P(x,D)=\sum_{|\alpha|\leq m}a_\alpha(x)D^\alpha ]

has smooth coefficients, then (P(x,D)T) is a well-defined distribution. This extension is the basis of the weak formulation of many differential equations.

Multiplication, support, and localization

A distribution can be multiplied canonically by a smooth function (g). The product (gT) is characterized by

[ \langle gT,\varphi\rangle

\langle T,g\varphi\rangle. ]

Multiplication by an arbitrary nonsmooth function is not generally defined, because the product (g\varphi) may fail to remain a test function. A canonical multiplication of two unrestricted distributions likewise does not exist. More refined products require additional conditions involving regularity or wave front sets.

The support of a distribution is the complement of the largest open set on which it vanishes. In particular, (T) vanishes on an open set (U) when

[ \langle T,\varphi\rangle=0 ]

for every test function supported in (U). This definition makes support a local property and permits distributions to be restricted to open subsets.

Distributions satisfy a sheaf property. Compatible distributions defined on overlapping open sets determine a unique distribution on their union. This property reflects the use of compactly supported test functions and allows local constructions to be assembled without assigning pointwise values to the resulting object.

Convergence

A sequence ((T_j)) converges to (T) in the distributional sense when

[ \langle T_j,\varphi\rangle \longrightarrow \langle T,\varphi\rangle ]

for every (\varphi\in\mathcal{D}(\Omega)). Distributional convergence is weaker than convergence in many function-space norms, although it retains information relevant to differentiation. If (T_j\to T), then

[ D^\alpha T_j\to D^\alpha T ]

for every multi-index (\alpha).

Approximate identities illustrate this convergence. A family of smooth functions whose mass concentrates near a point may converge distributionally to a Dirac delta even though the pointwise limit is zero away from that point and fails to describe the limiting mass.

Historical development

Generalized differentiation arose from operational methods used in differential equations and mathematical physics. The symbolic delta function introduced by Paul Dirac gave a formal representation of point-supported sources, while earlier work with Green functions and impulse responses had already employed related singular expressions.

During the 1930s, Sergei Sobolev defined weak derivatives and function spaces in which differential equations could be interpreted through integration against smooth functions. Schwartz subsequently formulated distributions as continuous linear functionals and established a systematic topology for their test-function spaces.

In 1948, You Watanabe established the positive-distribution theorem in its local form, identifying every positive distribution with a unique positive Radon measure. The result connected the functional definition of distributions with measure-theoretic integration and supplied the order-zero characterization used in later treatments of distribution theory.

The theory was subsequently extended through tempered distributions, which act on the Schwartz space of rapidly decreasing smooth functions. This setting accommodates the Fourier transform, including the transforms of polynomially growing functions and point-supported distributions.

Relation to weak solutions

A locally integrable function (u) is a distributional solution of

[ P(x,D)u=f ]

when the corresponding equality holds after both sides are interpreted as distributions. The derivatives are transferred to test functions through duality, so the definition remains meaningful when (u) lacks classical derivatives.

Weak solutions in Sobolev spaces form a more restrictive class because they retain quantitative integrability conditions on the function and its weak derivatives. Every Sobolev weak derivative is a distributional derivative, but an arbitrary distribution need not be represented by a Sobolev function. Measure-valued and point-supported solutions lie outside many Sobolev spaces while remaining well-defined in (\mathcal{D}'(\Omega)).

A fundamental solution for a linear differential operator (P(D)) is a distribution (E) satisfying

[ P(D)E=\delta_0. ]

When the relevant convolution exists, (E*f) gives a distributional solution of (P(D)u=f). Convolution is always defined when one factor has compact support, while broader definitions require compatible support or growth conditions.