nLab
distribution

This entry is about the concept of distributional densities in functional analysis. For the concept in differential geometry and Lie theory see at distribution of subspaces.

Contents

Idea

In functional analysis, the concept of distributional density, usually just called distribution for short, is a generalization of the concept of density, hence of something that may be integrated against a bump function to produce a number. If a non-degenerate background density/volume form dvoldvol is fixed, then each other density is a function relative to dvoldvol, and hence with such an identification understood distributional densities are generalized functions, namely objects that may arise as potentially singular limits of sequences of smooth functions. Famous examples of such are the delta distributions and the Heaviside distribution which behave like constant functions with an infinitely sharp spike or kink, respectively.

Distributional densities appear notably as fundamental solutions to linear partial differential equations (such as for the wave equation/Klein-Gordon equation, whose fundamental solutions are the propagators of free quantum fields), which is the context in which the concept was originally introduced. The study of their singularity structure (encoded by their singular support and their wave front set) is a fundamental tool in PDE theory (for instance in the propagation of singularities theorem), known as microlocal analysis. Distributions are also fundamental in the rigorous construction of perturbative quantum field theory, where they appear in the variant as operator-valued distributions.

Often distributions are considered by default just on open subsets of Euclidean space with its canonical volume form tacitly understood. But the concept of distributions makes sense more generally on general smooth manifolds (at least). If these are equipped with the structure of a (pseudo-) Riemannian manifold then the induced volume form again identifies distributions with generalized functions.

More in detail, given an actual density/volume form dvoldvol on some smooth manifold XX, then the operation of integration of bump functions (elements in the topological vector space C c (X)C^\infty_c(X)) against dvoldvol yields the continuous linear functional

C c (X) b xXb(x)dvol(x). \array{ C^\infty_c(X) &\longrightarrow& \mathbb{R} \\ b &\mapsto& \int_{x \in X} b(x) dvol(x) } \,.

However, not every continuous linear functional on C c (X)C^\infty_c(X) arises this way. For example for x 0Xx_0 \in X any point, the simple evaluation map

δ x 0:bb(x 0) \delta_{x_0} \;\colon\; b \mapsto b(x_0)

is also a continuous linear functional on C c (X)C^\infty_c(X) (the “delta distribution”). While this is not the integral against any bump function times a fixed density, it is the limit of integrations against any sequence of bump functions (times the fixed density) whose support narrows in on x 0x_0. Therefore one defines a distributional density simply to be any continuous linear functional on C c (X)C^\infty_c(X).

Various immediate variants of this definition may be considered. For instance if the space of “test functions” C c (X)C^\infty_c(X) is generalized to that of all smooth functions, then one speaks of compactly supported distributions, or if it is enlarged just to the Schwartz space of functions with all derivatives rapidly decreasing, then one speaks of tempered distributions. These are important as on them there is a a good concept of Fourier transform of distributions.

Most of the usual constructions of differential calculus generalize from smooth functions to distributions, notably there is a concept of derivative of distributions (defined by generalizing the formula for integration by parts). A key subtlety is that, however, some standard operations on functions become only partially defined on distributions, namely only when their singularity structure is compatible. In particular there is a concept of pullback of distributions and the product of distributions compatible with that of smooth functions, but defined only whenever the wave front sets of the distributions involved satisfy suitable compatibility conditions. Taking this subtlety into account for the operator-valued distributions appearing in perturbative quantum field theory is what leads to the concept of Wick algebras (“normal ordering”), see there for more.

Due to their potentially singular nature, there is more freedom in the extension of distributions than there is for smooth functions. Notably for extensions from the complement of a single point to that point the freedom is in choosing a point-supported distribution, and these are precisely the derivatives of delta-distributions. In the construction of time-ordered products of operator-valued distributions it is precisely this freedom in choosing point-extensions of distributions which in perturbative quantum field theory is known as “renormalization”.

Definition

We first recall the

Then we consider the axiomatic reformulation in terms of monads following Kock 11.

Traditional definition

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.

Definition

(compactly supported test functions)

For X nX \subset \mathbb{R}^n a smooth manifold given as an open subset of a Euclidean space, the topological vector space C c (X)C^\infty_c(X) of compactly supported test functions is the following

  1. the underlying set is the set of bump functions, hence of smooth functions XX \to \mathbb{R} to the real numbers with compact support;

  2. equipped with evident real vector space structure given by pointwise addition and pointwise multipication of functions.

  3. equipped with the topology which is the metric topology induced from the family of seminorms

    ρ K,α(f)=sup xK| αf|\rho_{K, \alpha}(f) = \sup_{x \in K} |\partial^{\alpha} f|

    where KUK \subseteq U is compact and α=(α 1,,α n)\alpha = (\alpha_1, \ldots, \alpha_n) is a multi-index and

    α= α 1x α 1 α nx α 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.

Proposition

The topological vector space C c (X)C^\infty_c(X) of compactly supported test functions (def. 1) 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 (K)C_c^{\infty}(K) (each of which has empty interior as a subspace of C c (U)C_c^{\infty}(U), so by the Baire category theorem, C c (U)C_c^{\infty}(U) is not itself a Fréchet space).

Definition

(distribution on Euclidean space)

Let X nX \subset \mathbb{R}^n be a smooth manifold given as an open subset of Euclidean space.

A distribution on XX is a continuous linear functional of the form

C c (X)C^\infty_c(X) \longrightarrow \mathbb{R}

from the topological vector space of compactly supported test functions (def. 1) to the real numbers.

The space of distributions on XX is denoted 𝒟(X)\mathcal{D}'(X) (see also remark 1). There is an obvious bilinear pairing

𝒟(X)×C c (X) (S,ϕ) S(ϕ) \array{ \mathcal{D}'(X) \times C_c^{\infty}(X) &\longrightarrow& \mathbb{R} \\ (S, \phi) &\mapsto& S(\phi) }

given by evaluation.

Often one writes S,ϕ\langle S, \phi\rangle instead of S(ϕ)S(\phi). The space of distributions can be given the weak **-topology, meaning the smallest topology rendering the maps

,ϕ:𝒟(U)\langle -, \phi\rangle \;\colon\; \mathcal{D}'(U) \to \mathbb{R}

continuous for all test functions ϕ\phi. As C c (U)C_c^\infty(U) is reflexive, this agrees with the weak topology.

Other natural topologies exist, such as uniform convergence on compact subsets of C c (U)C_c^\infty(U) (in this case, this agrees with uniform convergence on bounded subsets which usually goes by the name of the strong topology).

Remark

(notation)

On general grounds the symbols D(X)D(X) or 𝒟(X)\mathcal{D}(X) or similar would seem evident notation for the space of distributions on a smooth manifold XX. However, Laurent Schwartz in his seminal work (Schwartz 50) used 𝒟(X)\mathcal{D}(X) to denote the space C c (X)C^\infty_c(X) of compactly supported continuous functions, and then 𝒟(X)\mathcal{D}'(X) for its linear continuous dual, hence for the space of distributions (see also Hörmander 90, below def. 2.1.1).

Example

If f:Xf \colon X \to \mathbb{R} is locally integrable?, then for all test functions ϕ\phi the Lebesgue integral

f,ϕ= Xf(x)ϕ(x)dx\langle f, \phi\rangle = \int_X f(x)\phi(x) d x

is defined; in this way a function ff locally integrable over XX may be regarded as a distribution on XX (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 (X)𝒟(X)C_c^{\infty}(X) \hookrightarrow \mathcal{D}'(X)

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 UU. These are functionals on C (U)C^{\infty}(U) (test functions without compact support).

  • Rapidly decaying distributions (usually on U= nU = \mathbb{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= nU = \mathbb{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 |x||x| as |x||x| \to \infty). The topology on Schwartz space is induced by the family of seminorms

    ρ K,α,β(ϕ)=sup xK|x α βϕ|\rho_{K, \alpha, \beta}(\phi) = \sup_{x \in K} |x^\alpha \partial^\beta \phi|

    where α\alpha, β\beta are multi-indices.

Proposition

(pullback of distributions along submersion)

If X 1,X 2 nX_1, X_2 \subset \mathbb{R}^n are two open subsets of Euclidean space, and if

f:X 1X 2 f \;\colon\; X_1 \overset{}{\longrightarrow} X_2

is a submersion (i.e. its differential is a surjective function df x:T xX 1T f(x)X 2d f_x \;\colon\; T_x X_1 \to T_{f(x)} X_2 for all xX 1x \in X_1), then there is a unique continuous linear functional

f *:𝒟(X 2)𝒟(X 1) f^\ast \;\colon\; \mathcal{D}'(X_2) \longrightarrow \mathcal{D}'(X_1)

between spaces of distributions (def. 2) which extends the pullback of functions in that on a distribution represented by a bump function bb it is given by precomposition

f *b=bf. f^\ast b = b \circ f \,.

This is hence called the pullback of distributions.

(Hörmander 90, theorem 6.1.2)

Definition

(distributions of smooth manifolds)

Let XX be a smooth manifold. Then a distribution on XX is an equivalence class of

  1. a choice of smooth atlas { nψ iU iX} iI\{\mathbb{R}^n \underoverset{\simeq}{\psi_i}{\longrightarrow} U_i \subset X\}_{i \in I};

  2. for each iIi \in I a distribution ϕ i:C ( n)\phi_i \;\colon\; C^\infty(\mathbb{R}^n)\to \mathbb{R} on the iith chart, as above;

  3. such that for all pairs (i,j)I×I(i,j) \in I \times I these component distributions are related on intersections of charts by pullback of distributions (def. 2) along the coordinate change maps:

    ϕ j=(ψ i 1ψ j) *ϕ i\phi_j = (\psi_i^{-1} \circ \psi_j)^\ast \phi_i .

(Hörmander 90, def. 6.3.3)

Characterization by monads

(…) Kock 11 (…)

Operations on distributions

Inducing operations by dual extension

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

F:C c (U)C c (U) F\colon C_c^\infty(U) \to C_c^\infty(U)

we define

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

according to the usual formula for dualities

F *S(ϕ)=S(Fϕ). F^* S(\phi) = S(F \phi).

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

If the extension exists, we have

F (ιϕ)(ψ)=ι(F(ϕ))(ψ)=F(ϕ),ψ F^\dagger(\iota \phi)(\psi) = \iota(F(\phi))(\psi) = \langle F(\phi), \psi \rangle

Now suppose that FF has an adjoint, say F +F^+, with respect to the inner product. Note that this is not automatic since C c (U)C_c^\infty(U) is not a Hilbert space. Moreover, even if FF 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 (U)C_c^\infty(U). But if F +F^+ does exist then we have

F (ιϕ)(ψ)=F(ϕ),ψ=ϕ,F +(ψ) F^\dagger(\iota \phi)(\psi) = \langle F(\phi), \psi \rangle = \langle \phi, F^+(\psi) \rangle

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

F (S),ϕ=S,F +(ϕ) \langle F^\dagger(S),\phi\rangle = \langle S, F^+(\phi) \rangle

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

(ϕ,ψ)= Uϕψ (\phi,\psi) = \int_U \phi \psi

This is to ensure that the inclusion C c (U)𝒟(U)C_c^\infty(U) \to \mathcal{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 θ\theta. If θ\theta is any smooth function on UU (not necessarily compactly supported), then we can define θS\theta \cdot S by observing that this multiplication is self-adjoint:

    θϕ,ψ=ϕ,ψθ\langle \theta \cdot \phi, \psi \rangle = \langle \phi, \psi \cdot \theta\rangle

    where ϕ,ψ\phi, \psi are arbitrary test functions. Thus we define θS\theta \cdot S by

    θS,ψ=S,θψ\langle \theta \cdot S, \psi \rangle = \langle S, \theta \cdot \psi
  • Differentiation. If i\partial^i is partial differentiation with respect to the i thi^{th} coordinate, then for test functions ψ\psi, ϕ\phi we have

    U i(ψ)(x)ϕ(x)dx= Uψ(x) i(ϕ)(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 ϕ\phi, ψ\psi are compactly supported. Thus differentiation is skew-adjoint and so we define the extension to distributions by

    i(S),ϕ=S, i(ϕ)\langle \partial^i(S), \phi\rangle = -\langle S, \partial^i(\phi) \rangle

    for all test functions ϕ\phi. In general,

    αS,ϕ=(1) |α|S, αϕ\langle \partial^\alpha S, \phi\rangle = (-1)^{|\alpha|}\langle S, \partial^\alpha \phi \rangle

    where |α|=α 1++α n|\alpha| = \alpha_1 + \ldots + \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 section “examples”.

Multiplication of Distributions

See at multiplication of distributions

Examples

As explained above, any locally integrable function on UU defines a distribution on UU. 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)C_c(U) denotes the space of real-valued continuous functions with compact support in UU (topologized by uniform convergence on compacta), then a functional μ:C c(U)\mu: C_c(U) \to \mathbb{R} is essentially the same as a signed measure on UU (Riesz-Markov theorem), i.e., there is a unique signed measure dmd m for which

μ(ϕ)= Uϕdm.\mu(\phi) = \int_U \phi d m.

Since the inclusion i:C c (U)C c(U)i: C_c^\infty(U) \hookrightarrow C_c(U) is continuous, it follows that a measure μ\mu defines a distribution by simple restriction along ii:

C c (U)iC c(U)μC_c^\infty(U) \overset{i}{\to} C_c(U) \overset{\mu}{\to} \mathbb{R}

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

f(x)dα(x)\int_{\mathbb{R}} f(x) d\alpha(x)

is defined for all functions ff with compact support; this provides a measure dαd\alpha and hence a distribution.

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

f,dH= 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)=δ 0(x)dxd H(x) = \delta_0(x) d x

where of course the function H(x)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)H(x) is itself the derivative of a continuous function: G(x)=max{x,0}G(x) = \max\{x, 0\}.

For an example of a distribution on \mathbb{R} which does not arise from a measure, consider the derivative of the Dirac distribution. (As a functional, it maps a test function ϕ\phi to ϕ(0)-\phi'(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 SS be a distribution on an open domain U nU \subseteq \mathbb{R}^n. Then, there exist a finite collection AA of multi-indices α\alpha and continuous functions g αg_\alpha defined on UU for which

S= αA αg αS = \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 mm at position aa (where one would use the distribution mδ(xa)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.

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

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

A proof is given in these notes by Helgason. The basic idea is to prove there exists a fundamental solution of DD, i.e., a distribution TT such that DT=δ 0D T = \delta_0. Then u=f*Tu = f * T is smooth. The existence of a fundamental solution involves a theorem of Paley-Wiener type.

Variants

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 (Moerdijk-Reyes 91).

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

Currents

In n\mathbb{R}^n the distributions and generalized functions boil down to the same thing, so the terminology identifies them. But on a manifold, the distributions/generalized densities (functionals on test functions) and generalized functions (functionals on test densities) do not agree. See V. Guillemin, S. Sternberg: Geometric asymptotics (free online). While generalized functions pull back, distributions/generalized densities push forward (under some conditions, though).

More generally one can study generalized differential kk-forms in local coordinates they look like f αdx α 1dx α k\sum f_\alpha dx^{\alpha_1}\wedge \cdot \wedge dx^{\alpha_k}. Usually they are called currents. They are useful e.g. in the study of higher dimensional residua in higher dimensional complex geometry (cf. Principles of algebraic geometry by Griffiths and Harris) and in geometric measure theory (cf. the monograph by Federer).

Hyperfunctions and Coulombeau distributions

Sometimes one considers larger spaces of distributions, where worse singularities than in Schwarz theory are allowed. Most well known are the theory of hyperfunctions and the theory of Coulombeau distributions.

Distributions from nonstandard analysis

Distributions can be alternatively described using nonstandard analysis, see there.

References

See also hyperfunction, ultradistribution and references therein.

Traditional

Generalized functions were introduced by S. L. 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. For an infinite-dimensional variant used in the foundation of Feynman path integral see also Connes distribution.

The original articles include

  • Laurent Schwartz, Théorie des distributions, 1–2 , Hermann (1950–1951)

  • I. M. Gel'fand, G.E. Shilov, Generalized functions, 1–5 , Acad. Press (1966–1968) transl. from И. М. Гельфанд, Г. Е. Шилов Обобщенные функции, вып. 1-3, М.:Физматгиз, 1958; 1: Обобщенные функции и действия над ними, 2: Пространства основных обобщенных функций, 3: Некоторые вопросы теории дифференциальных уравнений

  • E. Magenes, G. Stampacchia, Teoria delle distribuzioni, Lectures Given at a Summer School of the Centro Internazionale Matematico Estivo (C.i.m.e.) Held in Saltino (Firenza) Italy, September 1-9, 1961; C.I.M.E., Ed. Cremonese, Roma, 1961; reprinted as CIME 24, Springer 2011 doi

Modern accounts include

  • Lars Hörmander, The analysis of linear partial differential operators, vol. I, Springer 1983, 1990

  • Walter Rudin, chapter 6 of Functional analysis, McGraw-Hill, 1991

  • M. Grosser, E. Farkas, M. Kunzinger, R. Steinbauer, On the foundations of nonlinear generalized functions I, II, Mem. Amer. Math. Soc. 153 (2001)

  • M. Kunzinger, R. Steinbauer, Foundations of a nonlinear distributional geometry, Acta Appl. Math. 71, 179-206 (2002)

Lecture notes include

  • Hasse Carlsson, Lecture notes on distributions (pdf)

and several chapters of the course

  • Erik P. van den Ban, Analysis on manifolds (2009) (web - especially lectures 1-3)

Applications of distributions in physics are discussed in

  • V. S. Vladimirov, Generalized functions in mathematical physics. Moskva, Nauka 1980, Mir 1979; Equations of mathematical physics, Mir 1984

  • Nikolay Bogolyubov, A. A. Logunov, I.T. Todorov, Introduction to axiomatic quantum field theory, Benjamin (1975)

Application of distributions in perturbative quantum field theory is discussed in

For more on this see the references at perturbative AQFT.

See also

References on Colombeau algebra include

  • J. F. Colombeau, New generalized functions and multiplications of distributions, North Holland, Amsterdam (1984); Elementary introduction in new generalized functions, North Holland (1985)
  • N. Djapić, S. Pilipović, Microlocal analysis of Colombeau’s generalized functions on a manifold, Indag. Math. N.S. 7, 293–309 (1996)
  • Stevan Pilipović, Milica Žigić, Suppleness of the sheaf of algebras of generalized functions on manifolds, J. Math. Anal. Appl. 379:2 (2011) 482–486, arxiv/1101.4552, MR2784335, doi

In terms of smooth toposes

Discussion of distributions in terms morphisms out of internal homs in a smooth topos is in

and for the Cahiers topos in

using results of

and following the general conception of “intensive and extensive” in

Similar sheaf theoretic discussion of distributions as morphisms of smooth spaces is in

category: analysis

Revised on September 10, 2017 05:07:20 by David Roberts (218.215.33.193)