Distribution (mathematics)

Distributions (or generalized functions) are objects that generalize the classical notion of functions 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 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 Kolmogorov & Fomin (1957), generalized functions originated in the work of Sergei Sobolev (1936) 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. Gårding (1997) comments that although the ideas in the transformative book by Schwartz (1951) 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 test functions. 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.

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 \(\varphi\) : R → R having two properties:

• \(\varphi\) is smooth (infinitely differentiable);
• \(\varphi\) 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(\(\varphi\)), it is conventional to write \(\langle T,\varphi \rangle\) for the value of T acting on a test function \(\varphi\). A simple example of a distribution is the Dirac delta δ, defined by

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

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.

Suppose that f : R → R is a locally integrable function. Then a corresponding distribution T_f may be defined by

\(\left\langle T_{f}, \varphi \right\rangle = \int_\mathbf{R} f(x) \varphi(x) \,dx \quad \text{for} \quad \varphi\in D(\mathbf{R}).\)

This integral is a real number which depends linearly and continuously on \(\varphi\). Conversely, the values of the distribution T_f 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 T_f. 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 R_μ may be defined by

\(\left\langle R_\mu, \varphi \right\rangle = \int_{\mathbf{R}} \varphi\, d\mu \quad \text{for} \quad \varphi\in D(\mathbf{R}).\)

This integral also depends linearly and continuously on \(\varphi\), so that R_μ is a distribution. If μ is absolutely continuous with respect to Lebesgue measure with density f and dμ = f dx, then this definition for R_μ is the same as the previous one for T_f, but if μ is not absolutely continuous, then R_μ 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

\(\int_{\mathbf{R}} \varphi\, dP = \varphi(0),\)

so that R_P = δ is the Dirac delta.

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 Schwartz (1954), and is usually referred to as the Schwartz' Impossibility Theorem.

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 \(T'_f = T_{f'}\). If \(\varphi\) is a test function, we can use integration by parts to see that

\(\left\langle f', \varphi\right\rangle = \int_{\mathbf{R}} f'\varphi \,dx = \Big[ f(x) \varphi(x) \Big]_{-\infty}^\infty - \int_{\mathbf{R}} f\varphi' \,dx = -\left\langle f, \varphi' \right\rangle\)

where the last equality follows from the fact that \(\varphi\) has compact support, so is zero outside of a bounded set. This suggests that if \(T\) is a distribution, we should define its derivative \(T'\) by

\(\left\langle T', \varphi \right\rangle = - \left\langle T, \varphi' \right\rangle.\)

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: Recall that the Dirac delta (so-called Dirac delta function) is the distribution defined by the equation

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

It is the derivative of the distribution corresponding to the Heaviside step function H: For any test function \(\varphi\),

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

so H′ = δ. Note, \(\varphi\)(∞) = 0 because \(\varphi\) has compact support by our definition of a test function. Similarly, the derivative of the Dirac delta is the distribution defined by the equation

\(\langle\delta',\varphi\rangle= -\varphi'(0).\)

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.

In the following, real-valued distributions on an open subset U of R will be formally defined. With minor modifications, one can also define complex-valued distributions, and one can replace R 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).

The space D(U) of test functions on U is defined as follows. A function \(\varphi\) : U → R is said to have compact support if there exists a compact subset K of U such that \(\varphi\)(x) = 0 for all x in U \ K. The elements of D(U) are the infinitely differentiable functions \(\varphi\) : 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 (\(\varphi\)_k) in D(U) is said to converge to \(\varphi\) ∈ D(U) if the following two conditions hold:

• There is a compact set K ⊂ U containing the supports of all \(\varphi\)_k:

\(\bigcup\nolimits_k \operatorname{supp}(\varphi_k)\subset K.\)

• For each multi-index α, the sequence of partial derivatives \(\partial^\alpha \varphi_k\) tends uniformly to \(\partial^\alpha\varphi\).

With this definition, D(U) becomes a complete locally convex topological vector space satisfying the Heine-Borel property.

This topology can be placed in the context of the following general construction: let

\(X = \bigcup\nolimits_i X_i\)

be a countable increasing union of locally convex topological vector spaces and ι_i : X_i → 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 \(\iota_i\) 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 ∩ X_i 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 X_i 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 X_i inherits from τ coincides with its original topology. When each X_i is a Fréchet space, (X, τ) is called an LF space.

Now let U be the union of U_i where {U_i} is a countable nested family of open subsets of U with compact closures K_i = U_i. Then we have the countable increasing union

\(\mathrm{D}(U) = \bigcup\nolimits_i \mathrm{D}_{K_i}\)

where D_Ki is the set of all smooth functions on U with support lying in K_i. On each D_Ki, consider the topology given by the seminorms

\(\| \varphi \|_\alpha = \max_{x \in K_i} \left |\partial^\alpha \varphi \right |,\)

\(\| \varphi \|_{\alpha, K_i} = \max_{x \in K_i} \left |\partial^\alpha \varphi \right | .\)

Condensed: the full section is in Wikipedia.

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 \(\varphi\) a real (or complex) scalar T(\(\varphi\)) such that

\(T(c_1\varphi_1 + c_2\varphi_2) = c_1 T(\varphi_1) + c_2 T(\varphi_2)\)

for all test functions \(\varphi\)₁, \(\varphi\)₂ and scalars c₁, c₂. Moreover, T is continuous if and only if

\(\lim_{k\to\infty}T(\varphi_k)= T\Bigl(\lim_{k\to\infty}\varphi_k\Bigr)\)

for every convergent sequence \(\varphi\)_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 C_K and a non-negative integer N_K such that

\(|T(\varphi)| \le C_K \sup \bigl\{ |\partial^\alpha\varphi(x)| \mathrel{\big|} x\in K, |\alpha|\leq N_K \bigr\}\)

for all test functions \(\varphi\) with support contained in K.

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. No matter which topology is chosen, D′(U) will be a non-metrizable, locally convex topological vector space.

The duality pairing between a distribution T in D′(U) and a test function \(\varphi\) in D(U) is denoted using angle brackets by

\(\begin{cases} \mathrm{D}'(U) \times \mathrm{D}(U) \to \mathbf{R} \\ (T, \varphi) \mapsto \langle T, \varphi \rangle, \end{cases}\)

so that ⟨T,\(\varphi\)⟩ = T(\(\varphi\)). One interprets this notation as the distribution T acting on the test function \(\varphi\) to give a scalar, or symmetrically as the test function \(\varphi\) acting on the distribution T.

\(\langle T_k, \varphi\rangle \to \langle T, \varphi\rangle\)

\(f_k(x) = \begin{cases} k & \text{if}\ 0\le x \le 1/k \\ 0 & \text{otherwise} \end{cases}\)

\(\langle T_k, \varphi\rangle = k\int_0^{1/k} \varphi(x)\, dx \to \varphi(0) = \langle \delta, \varphi\rangle\)

Condensed: the full section is in Wikipedia.

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 and all L 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), that is, an element of D′(U), denoted here by T_f, whose value on the test function \(\varphi\) is given by the Lebesgue integral:

\(\langle T_f,\varphi \rangle = \int_U f\varphi\,dx.\)

Conventionally, one abuses notation by identifying T_f with f, provided no confusion can arise, and thus the pairing between T_f and \(\varphi\) is often written

\(\langle f, \varphi\rangle = \langle T_f,\varphi\rangle.\)

If f and g are two locally integrable functions, then the associated distributions T_f and T_g are equal to the same element of D′(U) if and only if f and g are equal almost everywhere (see, for instance, Hörmander (1983, Theorem 1.2.5)). In a similar manner, every Radon measure μ on U defines an element of D′(U) whose value on the test function \(\varphi\) is ∫\(\varphi\) dμ. As above, it is conventional to abuse notation and write the pairing between a Radon measure μ and a test function \(\varphi\) as \(\langle \mu, \varphi \rangle\). Conversely, as shown in a theorem by Schwartz (similar to the Riesz representation theorem), every distribution which is non-negative on non-negative functions is of this form for some (positive) Radon measure.

The test functions are themselves locally integrable, and so define distributions. As such they are dense in D′(U) with respect to the topology on D′(U) in the sense that for any distribution T ∈ D′(U), there is a net \(\varphi\)_i ∈ D(U) such that

\(\langle\varphi_i,\psi\rangle\to \langle T,\psi\rangle\)

for all Ψ ∈ D(U). This fact follows from the Hahn-Banach theorem, since the dual of D′(U) with its weak-* topology is the space D(U). A stronger result of sequential density can be proven more constructively by a convolution argument.

Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if A : D(U) → D(U) is a linear mapping of vector spaces which is continuous with respect to the weak-* topology, then it is possible to extend A to a mapping A : D′(U) → D′(U) by passing to the limit. (This approach works for non-linear mappings as well, provided they are assumed to be uniformly continuous.)

In practice, however, it is more convenient to define operations on distributions by means of the transpose. If A : D(U) → D(U) is a continuous linear operator, then the transpose is an operator A : D(U) → D(U) such that

\(\int_U A\varphi(x)\cdot \psi(x) \,dx = \int_U \varphi(x) \cdot A^t\psi(x)\, dx\qquad \text{for all}\ \varphi,\psi\in D(U).\)

(For operators acting on spaces of complex-valued test functions, the transpose A differs from the adjoint A in that it does not include a complex conjugate.)

If such an operator A exists and is continuous on D(U), then the original operator A may be extended to D′(U) by defining AT for a distribution T as

\(\langle AT, \varphi\rangle = \langle T, A^t\varphi\rangle\qquad \text{for all}\ \varphi\in D(U).\)

Suppose A : D(U) → D(U) is the partial derivative operator

\(A\varphi = \frac{\partial\varphi}{\partial x_k}.\)

If \(\varphi\) and ψ are in D(U), then an integration by parts gives

\(\int_U \frac{\partial\varphi}{\partial x_k} \psi \, dx = -\int_U\varphi \frac{\partial\psi}{\partial x_k}\, dx,\)

so that A = −A. This operator is a continuous linear transformation on D(U). So, if T ∈ D′(U) is a distribution, then the partial derivative of T with respect to the coordinate x_k is defined by the formula

\(\left\langle \frac{\partial T}{\partial x_{k}}, \varphi \right\rangle = - \left\langle T, \frac{\partial \varphi}{\partial x_{k}} \right\rangle \qquad \text{for all}\ \varphi\in D(U).\)

With this definition, every distribution is infinitely differentiable, and the derivative in the direction x_k is a linear operator on D′(U).

More generally, if α = (α₁, ..., α_n) is an arbitrary multi-index and ∂ is the associated partial derivative operator, then the partial derivative ∂T of the distribution T ∈ D′(U) is defined by

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

Differentiation of distributions is a continuous operator on D′(U); this is an important and desirable property that is not shared by most other notions of differentiation.

If m : U → R is an infinitely differentiable function and T is a distribution on U, then the product mT is defined by

\(\langle mT, \varphi\rangle = \langle T, m\varphi \rangle\qquad \text{for all}\ \varphi\in D(U).\)

This definition coincides with the transpose definition since if M : D(U) → D(U) is the operator of multiplication by the function m (i.e., M\(\varphi\) = m \(\varphi\)), then

\(\int_U M\varphi(x)\cdot \psi(x)\,dx = \int_U m(x)\varphi(x)\cdot \psi(x)\,dx = \int_U \varphi(x)\cdot m(x)\psi(x)\,dx = \int_U \varphi(x)\cdot M\psi(x)\,dx,\)

so that M = M.

Under multiplication by smooth functions, D′(U) is a module over the ring C(U). With this definition of multiplication by a smooth function, the ordinary product rule of calculus remains valid. However, a number of unusual identities also arise. For example, if δ is the Dirac delta distribution on R, then mδ = m(0)δ, and if δ′ is the derivative of the delta distribution, then

\(m\delta' = m(0)\delta' - m'\delta = m(0)\delta' - m'(0)\delta.\,\)

These definitions of differentiation and multiplication also make it possible to define the operation of a linear differential operator with smooth coefficients on a distribution. A linear differential operator P takes a distribution T ∈ D′(U) to another distribution PT given by a sum of the form

\(PT = \sum\nolimits_{|\alpha|\le k} p_\alpha \partial^\alpha T,\)

where the coefficients p_α are smooth functions on U. The action of the distribution PT on a test function \(\varphi\) is given by

\(\left\langle \sum\nolimits_{|\alpha|\le k} p_\alpha \partial^\alpha T,\varphi\right\rangle = \left\langle T,\sum\nolimits_{|\alpha|\le k} (-1)^{|\alpha|} \partial^\alpha(p_\alpha\varphi)\right\rangle.\)

The minimum integer k for which such an expansion holds for every distribution T is called the order of P. The space D′(U) is a D-module with respect to the action of the ring of linear differential operators.

Let T be a distribution on an open set U ⊂ R. Let V be an open set in R, and F : V → U. Then provided F is a submersion, it is possible to define

\(T\circ F \in \mathrm{D}'(V).\)

This is the composition of the distribution T with F, and is also called the pullback of T along F, sometimes written

\(F^\sharp : T\mapsto F^\sharp T = T\circ F.\)

The pullback is often denoted F*, although this notation should not be confused with the use of '*' to denote the adjoint of a linear mapping.

The condition that F be a submersion is equivalent to the requirement that the Jacobian derivative dF(x) of F is a surjective linear map for every x ∈ V. A necessary (but not sufficient) condition for extending F to distributions is that F be an open mapping. The inverse function theorem ensures that a submersion satisfies this condition.

If F is a submersion, then F is defined on distributions by finding the transpose map. Uniqueness of this extension is guaranteed since F is a continuous linear operator on D(U). Existence, however, requires using the change of variables formula, the inverse function theorem (locally) and a partition of unity argument.

In the special case when F is a diffeomorphism from an open subset V of R onto an open subset U of R change of variables under the integral gives

\(\int_V\varphi\circ F(x) \psi(x)\,dx = \int_U\varphi(x) \psi \left (F^{-1}(x) \right ) \left |\det dF^{-1}(x) \right |\,dx.\)

In this particular case, then, F is defined by the transpose formula:

\(\bigl\langle F^\sharp T,\varphi \bigr\rangle = \left\langle T, \left |\det d(F^{-1}) \right | \varphi\circ F^{-1} \right\rangle.\)

There is no way to define the value of a distribution in D′(U) at a particular point of U. However, as is the case with functions, distributions on U restrict to give distributions on open subsets of U. Furthermore, distributions are locally determined in the sense that a distribution on all of U can be assembled from a distribution on an open cover of U satisfying some compatibility conditions on the overlap. Such a structure is known as a sheaf.