nLab
distribution

Idea

In functional analysis, a distribution (or generalized function) is a functional on a space of functions which typically is not representable by a function itself. They are often used to give a notion of derivative of a function which has no derivative in the classical sense of Lebesgue integration theory, and to give an abstract framework in which one can describe fundamental solutions to partial differential equations.

Zoran: in ${ℝ}^n$ the distributions and generalized functions boil down to the same thing, so the terminology identifies them. But on manifold, the distributions (functionals on test functions) and generalized functions (functionals on test densities) do not agree. See Guillemin, Sternberg: Geometric asymptotics. While generalized functions pull back, distributions push forward.

Generalized functions were introduced by Sobolev in 1935, and independently (under the name distributions) by Laurent Schwartz in the 1940’s, who unaware of Sobolev’s work developed an extensive theory for them.

Definitions

Distributions come in various flavors, depending on what spaces of functions they act on. The functions they act on are called test functions; typically they are smooth functions on domains in Euclidean space satisfying some boundedness property.

The widest (and generally the default) notion is as follows. Let $U\subseteq {ℝ}^n$ be open. Let $C_c^{\mathrm{\infty }}(U)$ denote the vector space of smooth functions with compact support on $U$ (also called bump functions; these are the test functions in this case). Endow this space with the topology induced from the family of seminorms

$${\rho }_{K,\alpha }(f)=\underset{x\in K}{\sup }\mid {\partial }^{\alpha }f\mid$$\rho_{K, \alpha}(f) = \sup_{x \in K} |\partial^{\alpha} f|

where $K\subseteq U$ is compact and $\alpha =({\alpha }_1,\dots ,{\alpha }_n)$ is a multi-index and

$${\partial }^{\alpha }=\frac{{\partial }^{{\alpha }_1}}{\partial x^{{\alpha }_1}}\dots \frac{{\partial }^{{\alpha }_n}}{\partial x^{{\alpha }_n}}$$\partial^{\alpha} = \frac{\partial^{\alpha_1}}{\partial x^{\alpha_1}} \ldots \frac{\partial^{\alpha_n}}{\partial x^{\alpha_n}}

is the corresponding differential operator. The resulting TVS is locally convex and complete with respect to its uniformity; it is in fact an LF-space: an inductive limit of Fréchet spaces $C_c^{\mathrm{\infty }}(K)$ (each of which has empty interior as a subspace of $C_c^{\mathrm{\infty }}(U)$, so by the Baire category theorem, $C_c^{\mathrm{\infty }}(U)$ is not itself Fréchet).

A distribution on $U$ is a continuous linear functional

$$C_c^{\mathrm{\infty }}(U)\to ℝ$$C_c^{\infty}(U) \to \mathbb{R}

The space of distributions on $U$ is denoted $𝒟(U)$. There is an obvious bilinear pairing

$$𝒟(U)\times C_c^{\mathrm{\infty }}(U)\to ℝ:(S,\varphi )\mapsto S(\varphi )$$\mathcal{D}(U) \times C_c^{\infty}(U) \to \mathbb{R}: (S, \phi) \mapsto S(\phi)

given by evaluation; often one writes $⟨S,\varphi ⟩$ instead of $S(\varphi )$. The space of distributions can be given the weak $*$-topology, meaning the smallest topology rendering the maps

$$⟨-,\varphi ⟩:𝒟(U)\to ℝ$$\langle -, \phi\rangle: \mathcal{D}(U) \to \mathbb{R}

continuous for all test functions $\varphi$. As $C_c^{\mathrm{\infty }}(U)$ is reflexive, this agrees with the weak topology. Other natural topologies exist, such as uniform convergence on compact subsets of $C_c^{\mathrm{\infty }}(U)$ (in this case, this agrees with uniform convergence on bounded subsets which usually goes by the name of the strong topology).

If $f:U\to ℝ$ is locally integrable, then for all test functions $\varphi$ the Lebesgue integral

$$⟨f,\varphi ⟩={\int }_{U}f(x)\varphi (x)dx$$\langle f, \phi\rangle = \int_U f(x)\phi(x) d x

is defined; in this way a function $f$ locally integrable over $U$ may be regarded as a distribution on $U$ (explaining both the sense in which distributions are “generalized functions” and a reason for the angle-bracket notation for the evaluation pairing). In particular, there is an obvious inclusion

$$C_c^{\mathrm{\infty }}(U)\hookrightarrow 𝒟(U)$$C_c^{\infty}(U) \hookrightarrow \mathcal{D}(U)

and this inclusion turns out to be dense.

Other notions of spaces of distributions, each endowed with the weak $*$-topology, include

• Compactly supported distributions on $U$. These are functionals on $C^{\mathrm{\infty }}(U)$ (test functions without compact support).

• Rapidly decaying distributions (usually on $U={ℝ}^n$). These are functionals on the space of smooth functions each of whose partial derivatives (of any order) has “tempered” or moderate growth (i.e., bounded by polynomial growth).

• Tempered distributions (usually on $U={ℝ}^n$). These are functionals on so-called Schwartz space: the space of smooth functions each of whose derivatives (of any order) decays rapidly (goes to zero more quickly than any negative power of $\mid x\mid$ as $\mid x\mid \to \mathrm{\infty }$). The topology on Schwartz space is induced by the family of seminorms

  $${\rho }_{K,\alpha ,\beta }(\varphi )=\underset{x\in K}{\sup }\mid x^{\alpha }{\partial }^{\beta }\varphi \mid$$\rho_{K, \alpha, \beta}(\phi) = \sup_{x \in K} |x^\alpha \partial^\beta \phi|

  where $\alpha$, $\beta$ are multi-indices.

Operations on distributions

As $𝒟(U)$ is dual to $C_c^{\mathrm{\infty }}(U)$, each continuous linear operator on $C_c^{\mathrm{\infty }}(U)$ induces a corresponding linear operator on $𝒟(U)$ in the obvious way. Given

$$F:C_c^{\mathrm{\infty }}(U)\to C_c^{\mathrm{\infty }}(U)$$F\colon C_c^\infty(U) \to C_c^\infty(U)

we define

$$F^{*}:𝒟(U)\to 𝒟(U)$$F^*\colon \mathcal{D}(U) \to \mathcal{D}(U)

according to the usual formula for dualities

$$F^{*}S(\varphi )=S(F\varphi ).$$F^* S(\phi) = S(F \phi).

However, since there is an obvious inclusion $C_c^{\mathrm{\infty }}(U)\to 𝒟(U)$ induced by the standard inner product on $C_c^{\mathrm{\infty }}(U)$, what is more usually desired is not this dual operator but an extension operator. That is, instead of $F^{*}$ we want an operator $F^{†}:𝒟(U)\to 𝒟(U)$ with the property that for $\varphi \in C_c^{\mathrm{\infty }}(U)$ then $F^{†}(\varphi )=F(\varphi )$ (identifying $C_c^{\mathrm{\infty }}(U)$ with its image in $𝒟(U)$). Being slightly more careful, let us write $\iota :C_c^{\mathrm{\infty }}(U)\to 𝒟(U)$ for the inclusion induced by the inner product. Then we want $F^{†}(\iota \varphi )=\iota (F(\varphi ))$.

If the extension exists, we have

$$F^{†}(\iota \varphi )(\psi )=\iota (F(\varphi ))(\psi )=⟨F(\varphi ),\psi ⟩$$F^\dagger(\iota \phi)(\psi) = \iota(F(\phi))(\psi) = \langle F(\phi), \psi \rangle

Now suppose that $F$ has an adjoint, say $F^{+}$, with respect to the inner product. Note that this is not automatic since $C_c^{\mathrm{\infty }}(U)$ is not a Hilbert space. Moreover, even if $F$ extends to the Hilbert completion the Hilbertian adjoint may not work since it may not define a continuous linear map on the subspace $C_c^{\mathrm{\infty }}(U)$. But if $F^{+}$ does exist then we have

$$F^{†}(\iota \varphi )(\psi )=⟨F(\varphi ),\psi ⟩=⟨\varphi ,F^{+}(\psi )⟩$$F^\dagger(\iota \phi)(\psi) = \langle F(\phi), \psi \rangle = \langle \phi, F^+(\psi) \rangle

In this case, the definition of $F^{†}$ on the whole of $𝒟(U)$ is obvious: simply take ${F^{+}}^{*}$. That is, the dual operator to the adjoint to $F$. In full, $F^{†}:𝒟(U)\to 𝒟(U)$ is defined via the formula

$$⟨F^{†}(S),\varphi ⟩=⟨S,F^{+}(\varphi )⟩$$\langle F^\dagger(S),\phi\rangle = \langle S, F^+(\phi) \rangle

If the ground field is $ℂ$ then this carries through essentially unchanged except for the fact that one does not use the inner product on $C_c^{\mathrm{\infty }}(U)$ but rather the associated bilinear pairing

$$(\varphi ,\psi )={\int }_U\varphi \psi$$(\phi,\psi) = \int_U \phi \psi

This is to ensure that the inclusion $C_c^{\mathrm{\infty }}(U)\to 𝒟(U)$ is complex linear and not conjugate linear. Otherwise extending operators becomes complex.

Two instances are of particular importance:

• Multiplication by a smooth function $\theta$. If $\theta$ is any smooth function on $U$ (not necessarily compactly supported), then we can define $\theta \cdot S$ by observing that this multiplication is self-adjoint:

  $$⟨\theta \cdot \varphi ,\psi ⟩=⟨\varphi ,\psi \cdot \theta ⟩$$\langle \theta \cdot \phi, \psi \rangle = \langle \phi, \psi \cdot \theta\rangle

  where $\varphi ,\psi$ are arbitrary test functions. Thus we define $\theta \cdot S$ by

  $$⟨\theta \cdot S,\psi ⟩=⟨S,\theta \cdot \psi$$\langle \theta \cdot S, \psi \rangle = \langle S, \theta \cdot \psi

• Differentiation. If ${\partial }^i$ is partial differentiation with respect to the $i^{\mathrm{th}}$ coordinate, then for test functions $\psi$, $\varphi$ we have

  $${\int }_U{\partial }^i(\psi )(x)\varphi (x)dx=-{\int }_U\psi (x){\partial }^i(\varphi )(x)dx$$\int_U \partial^i(\psi)(x) \phi(x)\; d x = -\int_U \psi(x) \partial^i(\phi)(x)\; d x

  by simple integration by parts and the fact that $\varphi$, $\psi$ are compactly supported. Thus differentiation is skew-adjoint and so we define the extension to distributions by

  $$⟨{\partial }^i(S),\varphi ⟩=-⟨S,{\partial }^i(\varphi )⟩$$\langle \partial^i(S), \phi\rangle = -\langle S, \partial^i(\phi) \rangle

  for all test functions $\varphi$. In general,

  $$⟨{\partial }^{\alpha }S,\varphi ⟩=(-1{)}^{\mid \alpha \mid }⟨S,{\partial }^{\alpha }\varphi ⟩$$\langle \partial^\alpha S, \phi\rangle = (-1)^{|\alpha|}\langle S, \partial^\alpha \phi \rangle

  where $\mid \alpha \mid ={\alpha }_1+\dots +{\alpha }_n$ is the total degree of the multi-index.

Thus derivatives of distributions are defined to all orders. Some examples are given in the following section.

Examples

As explained above, any locally integrable function on $U$ defines a distribution on $U$. Other examples may be produced fairly cheaply by restriction of functionals on various TVS which contain the test functions.

For instance: if $C_c(U)$ denotes the space of real-valued continuous functions with compact support in $U$ (topologized by uniform convergence on compacta), then a functional $\mu :C_c(U)\to ℝ$ is essentially the same as a signed measure on $U$ (Riesz-Markov theorem), i.e., there is a unique signed measure $dm$ for which

$$\mu (\varphi )={\int }_U\varphi dm.$$\mu(\phi) = \int_U \phi d m.

Since the inclusion $i:C_c^{\mathrm{\infty }}(U)\hookrightarrow C_c(U)$ is continuous, it follows that a measure $\mu$ defines a distribution by simple restriction along $i$:

$$C_c^{\mathrm{\infty }}(U)\stackrel{i}{\to }{C}_c(U)\stackrel{\mu }{\to }ℝ$$C_c^\infty(U) \overset{i}{\to} C_c(U) \overset{\mu}{\to} \mathbb{R}

Specializing further, consider any function of bounded variation on $U=ℝ$, say a bounded monotone increasing function $\alpha$. Then the Riemann-Stieltjes integral

$${\int }_{ℝ}f(x)d\alpha (x)$$\int_{\mathbb{R}} f(x) d\alpha(x)

is defined for all functions $f$ with compact support; this provides a measure $d\alpha$ and hence a distribution.

A prototypical example of this is provided by the Heaviside function: $H(x)=1$ if $x>0$, else 0. (“Heaviside”: what a great pun!) Here we have, for all $f\in C_c(ℝ)$,

$$⟨f,dH⟩={\int }_{ℝ}f(x)dH(x)=f(0)$$\langle f, d H \rangle = \int_{\mathbb{R}} f(x) d H(x) = f(0)

As a distribution, the Heaviside measure is the famous Dirac distribution. The long-standing intuitive practice among physicists and engineers is to write

$$dH(x)={\delta }_0(x)dx$$d H(x) = \delta_0(x) d x

where of course the function $H(x)$ doesn’t have a derivative in the classical sense (i.e., as a function), but as a distribution, it does. Meanwhile, $H(x)$ is itself the derivative of a continuous function: $G(x)=\max \{x,0\}$.

For an example of a distribution on $ℝ$ which does not arise from a measure, consider the derivative of the Dirac distribution. (As a functional, it maps a test function $\varphi$ to $-\varphi \prime (0)$.)

These examples are by no means curiosities. A fairly deep theorem is that every distribution arises as a linear combination of derivatives of continuous functions:

Theorem: Let $S$ be a distribution on an open domain $U\subseteq {ℝ}^n$. Then, there exist a finite collection $A$ of multi-indices $\alpha$ and continuous functions $g_{\alpha }$ defined on $U$ for which

$$S=\sum _{\alpha \in A}{\partial }^{\alpha }{g}_{\alpha }$$S = \sum_{\alpha \in A} \partial^\alpha g_\alpha

Applications

Distributions rigorously address a need long-felt by physicists to mathematically represent objects such as point particles of mass $m$ at position $a$ (where one would use the distribution $m\delta (x-a)$). They thus appear in accounts of quantum theory which attempt to achieve mathematical rigor. An example of this tendency can be seen in axiomatic formulations of quantum field theory such as the Wightman axioms.

A brief survey of applications of distribution theory to perturbative quantum field theory may be found here.

However, distributions fail to address some uses to which physicists would like to put them (as in path integrals), since there is no good way to multiply distributions in a way that extends multiplication of functions. The researcher most prominently associated with a program to repair this and other defects is J.F. Colombeau; see this brief Wikipedia article and for example these slides.

Within mathematics, distributions are quite commonplace; for example, de Rham appropriated them for his theory of current?s. Distribution theory has also long been used in the theory of partial differential equations. Here is a sample theorem:

• Theorem (Ehrenpreis, Malgrange): Let $D$ be a linear differential operator on ${ℝ}^n$ with constant coefficients. Given a compactly supported smooth function $f$ on ${ℝ}^n$, there exists a smooth solution $u$ to the equation $Du=f$.

A proof is given in these notes by Helgason. The basic idea is to prove there exists a fundamental solution of $D$, i.e., a distribution $T$ such that $DT={\delta }_0$. Then $u=f*T$ is smooth. The existence of a fundamental solution involves a theorem of Paley-Wiener type.

In synthetic differential geometry

There is another point of view on distributions: that they can be modeled by actual functions provided that one admits infinite and infinitesimal quantities of the type used in Robinson nonstandard analysis. One particular approach is to formulate axiomatically the theory of distributions so that it can be interpreted in smooth toposes that model the axioms of synthetic differential geometry and support a suitable notion of invertible infinitesimal objects and infinitely large integers.

This is discussed in chapter VII, section 3 of

• Ieke Moerdijk, Gonzalo Reyes, Models for Smooth Infinitesimal Analysis

which closely mirrors the original treatment in Robinson’s book Non-standard Analysis. Examples of models that support these axioms are the toposes $𝒵$ and $ℬ$ described there.