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.
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 $dvol$ is fixed, then each other density is a function relative to $dvol$, 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 (i.e. of non-singular distributions). 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 $dvol$ on some smooth manifold $X$, then the operation of integration of bump functions (elements in the topological vector space $C^\infty_c(X)$) against $dvol$ yields the continuous linear functional
However, not every continuous linear functional on $C^\infty_c(X)$ arises this way. For example for $x_0 \in X$ any point, the simple evaluation map
is also a continuous linear functional on $C^\infty_c(X)$ (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 $x_0$. Therefore one defines a distributional density simply to be any continuous linear functional on $C^\infty_c(X)$.
Various immediate variants of this definition may be considered. For instance if the space of “test functions” $C^\infty_c(X)$ 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”.
We first recall the
Then we consider the axiomatic reformulation in terms of monads following Kock 11.
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.
(compactly supported test functions)
For $X \subset \mathbb{R}^n$ a smooth manifold given as an open subset of a Euclidean space, the topological vector space $C^\infty_c(X)$ of compactly supported test functions is the following
the underlying set is the set of bump functions, hence of smooth functions $X \to \mathbb{R}$ to the real numbers with compact support;
equipped with evident real vector space structure given by pointwise addition and pointwise multipication of functions.
equipped with the topology which is the metric topology induced from the family of seminorms
where $K \subseteq U$ is compact and $\alpha = (\alpha_1, \ldots, \alpha_n)$ is a multi-index and
is the corresponding differential operator.
The topological vector space $C^\infty_c(X)$ 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 $C_c^{\infty}(K)$ (each of which has empty interior as a subspace of $C_c^{\infty}(U)$, so by the Baire category theorem, $C_c^{\infty}(U)$ is not itself a Fréchet space).
(distribution on Euclidean space)
Let $X \subset \mathbb{R}^n$ be a smooth manifold given as an open subset of Euclidean space.
A distribution on $X$ is a continuous linear functional of the form
from the locally convex topological vector space of compactly supported test functions (def. 1) to the real numbers.
The space of distributions on $X$ is denoted $\mathcal{D}'(X)$ (see also remark 1). There is an obvious bilinear pairing
given by evaluation.
Often one writes $\langle S, \phi\rangle$ instead of $S(\phi)$. The space of distributions can be given the weak $*$-topology, meaning the smallest topology rendering the maps
continuous for all test functions $\phi$. As $C_c^\infty(U)$ is reflexive, this agrees with the weak topology.
See at locally convex topological vector space the section Continuous linear functionals for alternative characterizations of the continuity of distributions according to def. 2.
Notice that other natural topologies exist, such as uniform convergence on compact subsets of $C_c^\infty(U)$ (in this case, this agrees with uniform convergence on bounded subsets which usually goes by the name of the strong topology).
(notation)
On general grounds the symbols $D(X)$ or $\mathcal{D}(X)$ or similar would seem evident notation for the space of distributions on a smooth manifold $X$. However, Laurent Schwartz in his seminal work (Schwartz 50) used $\mathcal{D}(X)$ to denote the space $C^\infty_c(X)$ of compactly supported continuous functions, and then $\mathcal{D}'(X)$ for its linear continuous dual, hence for the space of distributions (see also Hör#mander 90, below def. 2.1.1).
If $f \colon X \to \mathbb{R}$ is locally integrable?, then for all test functions $\phi$ the Lebesgue integral
is defined; in this way a function $f$ locally integrable over $X$ may be regarded as a distribution on $X$ (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
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 $C^{\infty}(U)$ (test functions without compact support).
tempered distributions (usually on $U = \mathbb{R}^n$). These are functionals on what is called the Schwartz space $\mathcal{S}$: the space of smooth functions each of whose derivatives (of any order) decays rapidly (goes to zero more quickly than any negative power of $|x|$ as $|x| \to \infty$). The topology on Schwartz space is induced by the family of seminorms
where $\alpha$, $\beta$ are multi-indices.
Rapidly decaying distributions (usually on $U = \mathbb{R}^n$). 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).
(pullback of distributions along submersions)
If $X_1, X_2 \subset \mathbb{R}^n$ are two open subsets of Euclidean space, and if
is a submersion (i.e. its differential is a surjective function $d f_x \;\colon\; T_x X_1 \to T_{f(x)} X_2$ for all $x \in X_1$), then there is a unique continuous linear functional
between spaces of distributions (def. 2) which extends the pullback of functions in that on a distribution represented by a bump function $b$ it is given by precomposition
This is hence called the pullback of distributions.
(distributions on smooth manifolds)
Let $X$ be a smooth manifold. Then a distribution on $X$ is an equivalence class of
a choice of smooth atlas $\{\mathbb{R}^n \underoverset{\simeq}{\psi_i}{\longrightarrow} U_i \subset X\}_{i \in I}$;
for each $i \in I$ a distribution $\phi_i \;\colon\; C^\infty(\mathbb{R}^n)\to \mathbb{R}$ on the $i$th chart, as above;
such that for all pairs $(i,j) \in I \times I$ these component distributions are related on intersections of charts by pullback of distributions (def. 2) along the coordinate change maps:
$\phi_j = (\psi_i^{-1} \circ \psi_j)^\ast \phi_i$.
Since smooth functions on smooth manifolds are the subject of differential geometry, and since spaces of smooth functions are naturally themselves generalized smooth spaces, it makes sense to ask whether distribution theory is actually a native topic to differential geometry.
In particular we may ask how distributions in the functional analytic sense relate to the smooth linear functions on smooth spaces of smooth functions. Indeed, with respect to the natural formulation of differential geometry via functorial geometry (topos theory) in terms of diffeological spaces, smooth sets etc. it turns out that distributional densities are equivalently the smooth linear functionals on smooth spaces of smooth functions.
This is discussed at
As $\mathcal{D}'(U)$ is dual to $C_c^\infty(U)$, each continuous linear operator on $C_c^\infty(U)$ induces a corresponding linear operator on $\mathcal{D}'(U)$ in the obvious way. Given
we define
according to the usual formula for dualities
However, since there is an obvious inclusion $C_c^\infty(U) \to \mathcal{D}'(U)$ induced by the standard inner product on $C_c^\infty(U)$, what is more usually desired is not this dual operator but an extension operator. That is, instead of $F^*$ we want an operator $F^\dagger \colon \mathcal{D}'(U) \to \mathcal{D}'(U)$ with the property that for $\phi \in C_c^\infty(U)$ then $F^\dagger(\phi) = F(\phi)$ (identifying $C_c^\infty(U)$ with its image in $\mathcal{D}'(U)$). Being slightly more careful, let us write $\iota \colon C_c^\infty(U) \to \mathcal{D}'(U)$ for the inclusion induced by the inner product. Then we want $F^\dagger(\iota \phi) = \iota (F(\phi))$.
If the extension exists, we have
Now suppose that $F$ has an adjoint, say $F^+$, with respect to the inner product. Note that this is not automatic since $C_c^\infty(U)$ 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 $C_c^\infty(U)$. But if $F^+$ does exist then we have
In this case, the definition of $F^\dagger$ on the whole of $\mathcal{D}'(U)$ is obvious: simply take ${F^+}^*$. That is, the dual operator to the adjoint to $F$. In full, $F^\dagger \colon \mathcal{D}'(U) \to \mathcal{D}'(U)$ is defined via the formula
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 $C_c^\infty(U)$ but rather the associated bilinear pairing
This is to ensure that the inclusion $C_c^\infty(U) \to \mathcal{D}'(U)$ is complex linear and not conjugate linear. Otherwise extending operators becomes complex.
Two instances are of particular importance:
Multiplication by a smooth function $\theta$. 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:
where $\phi, \psi$ are arbitrary test functions. Thus we define $\theta \cdot S$ by
Differentiation. If $\partial^i$ is partial differentiation with respect to the $i^{th}$ coordinate, then for test functions $\psi$, $\phi$ we have
by simple integration by parts and the fact that $\phi$, $\psi$ are compactly supported. Thus differentiation is skew-adjoint and so we define the extension to distributions by
for all test functions $\phi$. In general,
where $|\alpha| = \alpha_1 + \ldots + \alpha_n$ 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”.
See at multiplication of distributions
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 $\mu: C_c(U) \to \mathbb{R}$ is essentially the same as a signed measure on $U$ (Riesz-Markov theorem), i.e., there is a unique signed measure $d m$ for which
Since the inclusion $i: C_c^\infty(U) \hookrightarrow C_c(U)$ is continuous, it follows that a measure $\mu$ defines a distribution by simple restriction along $i$:
Specializing further, consider any function of bounded variation on $U = \mathbb{R}$, say a bounded monotone increasing function $\alpha$. Then the Riemann-Stieltjes integral
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 $x \gt 0$, else 0. (“Heaviside”: what a great pun!) Here we have, for all $f \in C_c(\mathbb{R})$,
As a distribution, the Heaviside measure is the famous Dirac distribution. The long-standing intuitive practice among physicists and engineers is to write
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 $\mathbb{R}$ which does not arise from a measure, consider the derivative of the Dirac distribution. (As a functional, it maps a test function $\phi$ to $-\phi'(0)$.)
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 $S$ be a distribution on an open domain $U \subseteq \mathbb{R}^n$. Then, there exist a finite collection $A$ of multi-indices $\alpha$ and continuous functions $g_\alpha$ defined on $U$ for which
The theory of distributions (and more generally of microlocal analysis) is central in perturbative quantum field theory in its rigorous incarnation via causal perturbation theory/perturbative algebraic quantum field theory.
For example the reason for normal ordered products in Wick algebras is given by the Hörmander criterion on wave front sets for the product of distributions to be well defind, and renormalization is understood to be the freedom in choosing extension of distributions of the resulting products of Feynman propagators.
See also operator-valued distribution and 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:
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 $D T = \delta_0$. Then $u = f * T$ is smooth. The existence of a fundamental solution involves a theorem of Paley-Wiener type.
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 $\mathcal{Z}$ and $\mathcal{B}$ described there.
In $\mathbb{R}^n$ 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 $k$-forms in local coordinates they look like $\sum f_\alpha dx^{\alpha_1}\wedge \cdot \wedge dx^{\alpha_k}$. 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).
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 can be alternatively described using nonstandard analysis, see there.
Lawvere distribution (categorification of the concept of distributions)
See also hyperfunction, ultradistribution and references therein.
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
François Trèves, Topological Vector Spaces, Distributions and Kernels (Academic Press, New York, 1967)
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
and several chapters of the course
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
Discussion of distributions in terms morphisms out of internal homs in a smooth topos (distributions are the smooth linear functionals) is in
and for the Cahiers topos in
Anders Kock, Gonzalo Reyes, Some calculus with extensive quantities: wave equation, Theory and Applications of Categories , Vol. 11, 2003, No. 14, pp 321-336 (TAC)
Anders Kock, Gonzalo Reyes, Categorical distribution theory; heat equation (arXiv:math/0407242)
Anders Kock, Commutative monads as a theory of distributions (arxiv/1108.5952)
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
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