For an unrelated notion in differential geometry see distribution of subspaces.
In functional analysis, a distribution (or generalized function) is a linear functional on a space of functions which typically is not representable by a function itself. In fact on a manifold, one may more properly distinguish the generalized functions from distributions, in the sense that generalized functions be functionals on densities, and distributions functionals on functions. They extend traditional spaces of Lebesgue integrable functions in such a way to have possibility to define generalized derivatives without exceptions and to treat fundamental solutions of linear partial differential equations as “functions”.
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.
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. Let $U \subseteq \mathbb{R}^n$ be open. Let $C_{c}^{\infty}(U)$ 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
where $K \subseteq U$ is compact and $\alpha = (\alpha_1, \ldots, \alpha_n)$ is a multi-index and
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 $C_c^{\infty}(K)$ (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 Fréchet).
A distribution on $U$ is a continuous linear functional
The space of distributions on $U$ is denoted $\mathcal{D}(U)$. 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. 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).
If $f: U \to \mathbb{R}$ is locally integrable, then for all test functions $\phi$ the Lebesgue integral
is defined; in this way a function $f$ locally integrable over $U$ 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
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).
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).
Tempered distributions (usually on $U = \mathbb{R}^n$). 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 $|x|$ as $|x| \to \infty$). The topology on Schwartz space is induced by the family of seminorms
where $\alpha$, $\beta$ are multi-indices.
(…)
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”.
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. In certain mathematical interpretations of quantum field theory, a quantum field is a (operator valued) distribution and the Lagrangian of the standard model of particle physics contains products of those.
The fact that there is no extension of multiplication to distributions is a famous no-go theorem of Laurent Schwartz.
Two heuristic explanations why multiplication is not possible:
In the construction of distributions we consider the algebra of compactly supported smooth functions, forget about the multiplication and see it as a TVS only, and then take the dual. The algebra structure therefore does not enter the construction in any way.
Let $H(x)$ be the Heaviside function, we clearly have
where the product on the right side is the product of classical functions. Applying differentiation and the product rule naivly results in a contradiction immediatly:
There is no product defined on the whole TVS of distributions, but some distributions may nevertheless be multiplied. A deeper explanation of this phenomenon needs the concept of wavefront sets. An exposition of QED that avoids divergences by carefully using only rigorously defined products of distributions is this:
J.F. Colombeau has developed a theory where multiplication is possible; see this brief Wikipedia article and for example these slides.
Briefly: Colombeau considers sequences of functions that converge to distributions (weakly) and defines the product of two distributions as the product of the sequences. This product is not independent of the chosen sequences, which means that the level of abstraction achieved by distribution theory is abandoned.
For further details see:
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
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.
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 chapter VII, section 3 of
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.
See Lawvere distribution.
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.
See also hyperfunction and references therein.
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
Modern accounts include
L.V. Hörmander, The analysis of linear partial differential operators, vol. I, Springer 1983
M. Kunzinger, R. Steinbauer, Foundations of a nonlinear distributional geometry, Acta Appl. Math. 71, 179-206 (2002)
M. Grosser, E. Farkas, M. Kunzinger, R. Steinbauer, On the foundations of nonlinear generalized functions I, II, Mem. Amer. Math. Soc. 153 (2001)
Lecture notes include
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
N. N. Bogolyubov, A. A. Logunov, I.T. Todorov, Introduction to axiomatic quantum field theory, Benjamin (1975)
In theorem 5.1.1 of
the convenient vector space of Schwartz distributions of compact support is characterized by a monadic construction. In generalization of this, an abstract formalization of distribiutions in category theory/universal algebra is discussed in
following the formalization of “intensive and extensive” in
See also