Distribution (mathematical analysis)

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 ${\displaystyle \phi }$ : U → R is said to have compact support if there exists a compact subset K of U such that ${\displaystyle \phi }$(x) = 0 for all x in U \ K. The elements of D(U) are the infinitely differentiable functions ${\displaystyle \phi }$ : 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 (${\displaystyle \phi }$_k) in D(U) is said to converge to ${\displaystyle \phi }$ ∈ D(U) if the following two conditions hold:

• There is a compact set K ⊂ U containing the supports of all ${\displaystyle \phi }$_k:

${\displaystyle {\bigcup }_k\mathrm{supp}({\phi }_k)\subset K.}$

• For each multi-index α, the sequence of partial derivatives ${\displaystyle {\mathrm{\partial }}^{\alpha }{\phi }_k}$ tends uniformly to ${\displaystyle {\mathrm{\partial }}^{\alpha }\phi }$.

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

${\displaystyle X={\bigcup }_i{X}_i}$

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 ${\displaystyle {\iota }_i}$ 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

${\displaystyle \mathrm{D}(U)={\bigcup }_i{\mathrm{D}}_{K_i}}$

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

${\displaystyle \Vert \phi {\Vert }_{\alpha }=\underset{x\in K_i}{max}\left|{\mathrm{\partial }}^{\alpha }\phi \right|,}$

i.e. the topology of uniform convergence of derivatives of arbitrary order. This makes each D_Ki 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

${\displaystyle \Vert \phi {\Vert }_{\alpha ,K_i}=\underset{x\in K_i}{max}\left|{\mathrm{\partial }}^{\alpha }\phi \right|.}$

However, this topology has the disadvantage of not being complete. On the other hand, because of the particular features of D_Ki's, a set this bounded with respect to τ if and only if it lies in some D_Ki's. The completeness of (D(U), τ) then follow from that of D_Ki'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).

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 ${\displaystyle \phi }$ a real (or complex) scalar T(${\displaystyle \phi }$) such that

${\displaystyle T(c_1{\phi }_1+c_2{\phi }_2)=c_{1}T({\phi }_1)+c_{2}T({\phi }_2)}$

for all test functions ${\displaystyle \phi }$₁, ${\displaystyle \phi }$₂ and scalars c₁, c₂. Moreover, T is continuous if and only if

${\displaystyle \underset{k\to \mathrm{\infty }}{lim}T({\phi }_k)=T(\underset{k\to \mathrm{\infty }}{lim}{\phi }_k)}$

for every convergent sequence ${\displaystyle \phi }$_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

${\displaystyle |T(\phi )|\le C_{K}sup\{|{\mathrm{\partial }}^{\alpha }\phi (x)||x\in K,|\alpha |\le N_K\}}$

for all test functions ${\displaystyle \phi }$ 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 ${\displaystyle \phi }$ in D(U) is denoted using angle brackets by

${\displaystyle \{\begin{array}{l}{\mathrm{D}}^{\prime }(U)\times \mathrm{D}(U)\to \mathbf{R}\\ (T,\phi )\mapsto ⟨T,\phi ⟩,\end{array}}$

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

A sequence of distributions (T_k) converges with respect to the weak-* topology on D′(U) to a distribution T if and only if

${\displaystyle ⟨T_k,\phi ⟩\to ⟨T,\phi ⟩}$

for every test function ${\displaystyle \phi }$ in D(U). For example, if f_k : R → R is the function

${\displaystyle f_k(x)=\{\begin{array}{ll}k& \text{if}0\le x\le 1/k\\ 0& \text{otherwise}\end{array}}$

and T_k is the distribution corresponding to f_k, then

${\displaystyle ⟨T_k,\phi ⟩=k{\int }_0^{1/k}\phi (x)dx\to \phi (0)=⟨\delta ,\phi ⟩}$

as k → ∞, so T_k → δ in D′(R). Thus, for large k, the function f_k can be regarded as an approximation of the Dirac delta distribution.

Functions as distributions

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^p 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 ${\displaystyle \phi }$ is given by the Lebesgue integral:

${\displaystyle ⟨T_f,\phi ⟩={\int }_{U}f\phi dx.}$

Conventionally, one abuses notation by identifying T_f with f, provided no confusion can arise, and thus the pairing between T_f and ${\displaystyle \phi }$ is often written

${\displaystyle ⟨f,\phi ⟩=⟨T_f,\phi ⟩.}$

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 ${\displaystyle \phi }$ is ∫${\displaystyle \phi }$ dμ. As above, it is conventional to abuse notation and write the pairing between a Radon measure μ and a test function ${\displaystyle \phi }$ as ${\displaystyle ⟨\mu ,\phi ⟩}$. 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 ${\displaystyle \phi }$_i ∈ D(U) such that

${\displaystyle ⟨{\phi }_i,\psi ⟩\to ⟨T,\psi ⟩}$

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.

Operations on distributions

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^t : D(U) → D(U) such that

${\displaystyle {\int }_{U}A\phi (x)\cdot \psi (x)dx={\int }_U\phi (x)\cdot A^t\psi (x)dx\text{for all}\phi ,\psi \in D(U).}$

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

If such an operator A^t 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

${\displaystyle ⟨AT,\phi ⟩=⟨T,A^t\phi ⟩\text{for all}\phi \in D(U).}$

Differentiation

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

${\displaystyle A\phi =\frac{\mathrm{\partial }\phi }{\mathrm{\partial }{x}_k}.}$

If ${\displaystyle \phi }$ and ψ are in D(U), then an integration by parts gives

${\displaystyle {\int }_U\frac{\mathrm{\partial }\phi }{\mathrm{\partial }{x}_k}\psi dx=-{\int }_U\phi \frac{\mathrm{\partial }\psi }{\mathrm{\partial }{x}_k}dx,}$

so that A^t = −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

${\displaystyle ⟨\frac{\mathrm{\partial }T}{\mathrm{\partial }{x}_k},\phi ⟩=-⟨T,\frac{\mathrm{\partial }\phi }{\mathrm{\partial }{x}_k}⟩\text{for all}\phi \in D(U).}$

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 ∂^α is the associated partial derivative operator, then the partial derivative ∂^αT of the distribution T ∈ D′(U) is defined by

${\displaystyle ⟨{\mathrm{\partial }}^{\alpha }T,\phi ⟩=(-1{)}^{|\alpha |}⟨T,{\mathrm{\partial }}^{\alpha }\phi ⟩\text{ for all }\phi \in \mathrm{D}(U).}$

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.

Multiplication by a smooth function

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

${\displaystyle ⟨mT,\phi ⟩=⟨T,m\phi ⟩\text{for all}\phi \in D(U).}$

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${\displaystyle \phi }$ = m ${\displaystyle \phi }$), then

${\displaystyle {\int }_{U}M\phi (x)\cdot \psi (x)dx={\int }_{U}m(x)\phi (x)\cdot \psi (x)dx={\int }_U\phi (x)\cdot m(x)\psi (x)dx={\int }_U\phi (x)\cdot M\psi (x)dx,}$

so that M^t = 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

${\displaystyle m{\delta }^{\prime }=m(0){\delta }^{\prime }-m^{\prime }\delta =m(0){\delta }^{\prime }-m^{\prime }(0)\delta .}$

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

${\displaystyle PT={\sum }_{|\alpha |\le k}{p}_{\alpha }{\mathrm{\partial }}^{\alpha }T,}$

where the coefficients p_α are smooth functions on U. The action of the distribution PT on a test function ${\displaystyle \phi }$ is given by

${\displaystyle ⟨{\sum }_{|\alpha |\le k}{p}_{\alpha }{\mathrm{\partial }}^{\alpha }T,\phi ⟩=⟨T,{\sum }_{|\alpha |\le k}(-1{)}^{|\alpha |}{\mathrm{\partial }}^{\alpha }(p_{\alpha }\phi )⟩.}$

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.

Composition with a smooth function

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

${\displaystyle T\circ F\in {\mathrm{D}}^{\prime }(V).}$

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

${\displaystyle F^{\mathrm{♯}}:T\mapsto F^{\mathrm{♯}}T=T\circ F.}$

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 Rⁿ onto an open subset U of Rⁿ change of variables under the integral gives

${\displaystyle {\int }_V\phi \circ F(x)\psi (x)dx={\int }_U\phi (x)\psi \left(F^{-1}(x)\right)\left|detd{F}^{-1}(x)\right|dx.}$

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

${\displaystyle ⟨F^{\mathrm{♯}}T,\phi ⟩=⟨T,\left|detd(F^{-1})\right|\phi \circ F^{-1}⟩.}$

Localization of distributions

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.

Restriction

Let U and V be open subsets of Rⁿ with V ⊂ U. Let E_VU : D(V) → D(U) be the operator which extends by zero a given smooth function compactly supported in V to a smooth function compactly supported in the larger set U. Then the restriction mapping ρ_VU is defined to be the transpose of E_VU. Thus for any distribution T ∈ D′(U), the restriction ρ_VUT is a distribution in the dual space D′(V) defined by

${\displaystyle ⟨{\rho }_{VU}T,\phi ⟩=⟨T,E_{VU}\phi ⟩}$

for all test functions ${\displaystyle \phi }$ ∈ D(V).

Unless U = V, the restriction to V is neither injective nor surjective. Lack of surjectivity follows since distributions can blow up towards the boundary of V. For instance, if U = R and V = (0, 2), then the distribution

${\displaystyle T(x)=\sum _{n=1}^{\mathrm{\infty }}n\delta \left(x-\frac{1}{n}\right)}$

is in D′(V) but admits no extension to D′(U).

Support of a distribution

Let T ∈ D′(U) be a distribution on an open set U. Then T is said to vanish on an open set V of U if T lies in the kernel of the restriction map ρ_VU. Explicitly T vanishes on V if

${\displaystyle ⟨T,\phi ⟩=0}$

for all test functions ${\displaystyle \phi }$ ∈ C^∞(U) with support in V. Let V be a maximal open set on which the distribution T vanishes; i.e., V is the union of every open set on which T vanishes. The support of T is the complement of V in U. Thus

${\displaystyle \mathrm{supp}T=U\setminus \bigcup \left\{V\mid {\rho }_{VU}T=0\right\}.}$

The distribution T has compact support if its support is a compact set. Explicitly, T has compact support if there is a compact subset K of U such that for every test function ${\displaystyle \phi }$ whose support is completely outside of K, we have T(${\displaystyle \phi }$) = 0. Compactly supported distributions define continuous linear functionals on the space C^∞(U); the topology on C^∞(U) is defined such that a sequence of test functions ${\displaystyle \phi }$_k converges to 0 if and only if all derivatives of ${\displaystyle \phi }$_k converge uniformly to 0 on every compact subset of U. Conversely, it can be shown that every continuous linear functional on this space defines a distribution of compact support. The embedding of C_c^∞(U) into C^∞(U), where the spaces are given their respective topologies, is continuous and has dense image. Thus compactly supported distributions can be identified with those distributions that can be extended from C_c^∞(U) to C^∞(U).

Tempered distributions and Fourier transform

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

The space of test functions employed here, the so-called Schwartz space S(Rⁿ), is the function space of all infinitely differentiable functions that are rapidly decreasing at infinity along with all partial derivatives. Thus φ : Rⁿ → R is in the Schwartz space provided that any derivative of ${\displaystyle \phi }$, 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

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

for α, β multi-indices of size n. Then ${\displaystyle \phi }$ is a Schwartz function if all the values satisfy

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

The family of seminorms p_{α, β} defines a locally convex topology on the Schwartz space. When n is equal to 1, the seminorms are, in fact, norms on the Schwartz space. Otherwise, one can define a norm on S(Rⁿ) via

${\displaystyle \Vert \phi {\Vert }_k=\underset{|\alpha |+|\beta |\le k}{max}\underset{x\in {\mathbf{R}}^n}{sup}\left|x^{\alpha }{D}^{\beta }\phi (x)\right|}$ for k ≥ 1.

The Schwartz space is metrizable and complete. Because the Fourier transform changes differentiation by x^α into multiplication by x^α and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function.

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

${\displaystyle \underset{m\to \mathrm{\infty }}{lim}T({\phi }_m)=0.}$

is true whenever

${\displaystyle \underset{m\to \mathrm{\infty }}{lim}{p}_{\alpha ,\beta }({\phi }_m)=0}$

holds for all multi-indices α, β.

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. More generally, all functions that are products of polynomials with elements of L^p(Rⁿ) for p ≥ 1 are tempered distributions.

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 function space, and we can define the Fourier transform of the tempered distribution T by (FT)(Ψ) = T(Fψ) for every Schwartz function Ψ. FT 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

${\displaystyle F{\displaystyle \frac{dT}{dx}}=ixFT}$

and also with convolution: if T is a tempered distribution and Ψ is a slowly increasing infinitely differentiable function on Rⁿ, then ψT is again a tempered distribution and

${\displaystyle F(\psi T)=F\psi \ast FT}$

is the convolution of FT and Fψ. In particular, the Fourier transform of the constant function equal to 1 is the δ distribution.

Convolution

Under some circumstances, it is possible to define the convolution of a function with a distribution, or even the convolution of two distributions.

Convolution of a test function with a distribution

If f ∈ D(Rⁿ) is a compactly supported smooth test function, then convolution with f,

${\displaystyle \{\begin{array}{l}{C}_f:\mathrm{D}({\mathbf{R}}^n)\to \mathrm{D}({\mathbf{R}}^n)\\ C_f:g\mapsto f\ast g\end{array}}$

defines a linear operator which is continuous with respect to the LF space topology on D(Rⁿ).

Convolution of f with a distribution T ∈ D′(Rⁿ) can be defined by taking the transpose of C_f relative to the duality pairing of D(Rⁿ) with the space D′(Rⁿ) of distributions (Trèves 1967, Chapter 27). If f, g, ${\displaystyle \phi }$ ∈ D(Rⁿ), then by Fubini's theorem

${\displaystyle ⟨C_{f}g,\phi ⟩={\int }_{{\mathbf{R}}^n}\phi (x){\int }_{{\mathbf{R}}^n}f(x-y)g(y)dydx=⟨g,C_{\tilde{f}}\phi ⟩}$

where ${\displaystyle \tilde{f}(x)=f(-x)}$. Extending by continuity, the convolution of f with a distribution T is defined by

${\displaystyle ⟨f\ast T,\phi ⟩=⟨T,\tilde{f}\ast \phi ⟩}$

for all test functions ${\displaystyle \phi }$ ∈ D(Rⁿ).

An alternative way to define the convolution of a function f and a distribution T is to use the translation operator τ_x defined on test functions by

${\displaystyle {\tau }_x\phi (y)=\phi (y-x)}$

and extended by the transpose to distributions in the obvious way. The convolution of the compactly supported function f and the distribution T is then the function defined for each x ∈ Rⁿ by

${\displaystyle (f\ast T)(x)=⟨T,{\tau }_x\tilde{f}⟩.}$

It can be shown that the convolution of a smooth, compactly supported function and a distribution is a smooth function. If the distribution T has compact support as well, then f∗T is a compactly supported function, and the Titchmarsh convolution theorem (Hörmander 1983, Theorem 4.3.3) implies that

${\displaystyle \mathrm{ch}(\mathrm{supp}(f\ast T))=\mathrm{ch}\mathrm{supp}f+\mathrm{ch}\mathrm{supp}T}$

where ch denotes the convex hull and supp denotes the support.

Distribution of compact support

It is also possible to define the convolution of two distributions S and T on Rⁿ, provided one of them has compact support. Informally, in order to define S∗T where T has compact support, the idea is to extend the definition of the convolution ∗ to a linear operation on distributions so that the associativity formula

${\displaystyle S\ast (T\ast \phi )=(S\ast T)\ast \phi }$

continues to hold for all test functions ${\displaystyle \phi }$.

It is also possible to provide a more explicit characterization of the convolution of distributions (Trèves 1967, Chapter 27). Suppose that it is T that has compact support. For any test function ${\displaystyle \phi }$ in D(Rⁿ), consider the function

${\displaystyle \psi (x)=⟨T,{\tau }_{-x}\phi ⟩.}$

It can be readily shown that this defines a smooth function of x, which moreover has compact support. The convolution of S and T is defined by

${\displaystyle ⟨S\ast T,\phi ⟩=⟨S,\psi ⟩.}$

This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense:

${\displaystyle {\mathrm{\partial }}^{\alpha }(S\ast T)=({\mathrm{\partial }}^{\alpha }S)\ast T=S\ast ({\mathrm{\partial }}^{\alpha }T).}$

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

Distributions as derivatives of continuous functions

The formal definition of distributions exhibits them as a subspace of a very large space, namely the topological dual of D(U) (or S(R^d) for tempered distributions). 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. A precise version of this result, given below, holds for distributions of compact support, tempered distributions, and general distributions. Generally speaking, 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.

Tempered distributions

If f ∈ S′(Rⁿ) is a tempered distribution, then there exists a constant C > 0, and positive integers M and N such that for all Schwartz functions ${\displaystyle \phi }$ ∈ S(Rⁿ)

${\displaystyle ⟨f,\phi ⟩\le C{\sum }_{|\alpha |\le N,|\beta |\le M}\underset{x\in {\mathbf{R}}^n}{sup}\left|x^{\alpha }{D}^{\beta }\phi (x)\right|=C{\sum }_{|\alpha |\le N,|\beta |\le M}{p}_{\alpha ,\beta }(\phi ).}$

This estimate along with some techniques from functional analysis can be used to show that there is a continuous slowly increasing function F and a multi-index α such that

${\displaystyle f=D^{\alpha }F.}$

Restriction of distributions to compact sets

If f ∈ D′(Rⁿ), then for any compact set K ⊂ Rⁿ, there exists a continuous function F compactly supported in Rⁿ (possibly on a larger set than K itself) and a multi-index α such that f = D^αF on C_c^∞(K). This follows from the previously quoted result on tempered distributions by means of a localization argument.

Distributions with point support

If f has support at a single point {P}, then f is in fact a finite linear combination of distributional derivatives of the δ function at P. That is, there exists an integer m and complex constants a_α for multi-indices |α| ≤ m such that

${\displaystyle f={\sum }_{|\alpha |\le m}{a}_{\alpha }{D}^{\alpha }({\tau }_P\delta )}$

where τ_P is the translation operator.

General distributions

A version of the above theorem holds locally in the following sense (Rudin 1991). Let T be a distribution on U, then one can find for every multi-index α a continuous function g_α such that

${\displaystyle {\displaystyle T={\sum }_{\alpha }{D}^{\alpha }{g}_{\alpha }}}$

and that any compact subset K of U intersects the supports of only finitely many g_α; therefore, to evaluate the value of T for a given smooth function f compactly supported in U, we only need finitely many g_α; hence the infinite sum above is well-defined as a distribution. If the distribution T is of finite order, then one can choose g_α in such a way that only finitely many of them are nonzero.

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 Mikio Sato's algebraic analysis, using sheaf theory and several complex variables. This extends the range of symbolic methods that can be made rigorous mathematically, for example Feynman integrals.

Problem of multiplication

It is easy to define the product of a distribution with a smooth function, or more generally the product of two distributions whose singular supports are disjoint. With more effort it is possible to define a well-behaved product of several distributions provided their wave front sets at each point are compatible. A limitation of the theory of distributions (and hyperfunctions) is that there is no associative product of two distributions extending the product of a distribution by a smooth function, as has been proved by Laurent Schwartz in the 1950s. For example, if p.v. 1/x is the distribution obtained by the Cauchy principal value

${\displaystyle \left(\mathrm{p}.\mathrm{v}.\frac{1}{x}\right)[\phi ]=\underset{\epsilon \to 0^{+}}{lim}{\int }_{|x|\ge \epsilon }\frac{\phi (x)}{x}dx}$

for all ${\displaystyle \phi }$ ∈ S(R), and δ is the Dirac delta distribution then

${\displaystyle \left(\delta \times x\right)\times \mathrm{p}.\mathrm{v}.\frac{1}{x}=0}$

but

${\displaystyle \delta \times \left(x\times \mathrm{p}.\mathrm{v}.\frac{1}{x}\right)=\delta }$

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 divergences. 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 the Navier–Stokes equations of fluid dynamics.

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.

Inspired by Lyons' rough path theory, Martin Hairer proposed a consistent way of multiplying distributions with certain structure (regularity structures), available in many examples from stochastic analysis, notably stochastic partial differential equations. See also Gubinelli–Imkeller–Perkowski (2015) for a related development based on Bony's paraproduct from Fourier analysis.