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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{distribution} \begin{quote}% This entry is about the concept of distributional densities in functional analysis. For the concept in differential geometry and Lie theory see at \emph{[[distribution of subspaces]]}. \end{quote} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis} [[!include functional analysis - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak \noindent\hyperlink{Definitions}{Definition}\dotfill \pageref*{Definitions} \linebreak \noindent\hyperlink{TraditionalDefinition}{As continuous linear functionals}\dotfill \pageref*{TraditionalDefinition} \linebreak \noindent\hyperlink{CharacterizationByMonads}{As smooth linear functionals}\dotfill \pageref*{CharacterizationByMonads} \linebreak \noindent\hyperlink{operations_on_distributions}{Operations on distributions}\dotfill \pageref*{operations_on_distributions} \linebreak \noindent\hyperlink{inducing_operations_by_dual_extension}{Inducing operations by dual extension}\dotfill \pageref*{inducing_operations_by_dual_extension} \linebreak \noindent\hyperlink{MultiplicationsOfDistributions}{Multiplication of Distributions}\dotfill \pageref*{MultiplicationsOfDistributions} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak \noindent\hyperlink{variants}{Variants}\dotfill \pageref*{variants} \linebreak \noindent\hyperlink{in_synthetic_differential_geometry}{In synthetic differential geometry}\dotfill \pageref*{in_synthetic_differential_geometry} \linebreak \noindent\hyperlink{currents}{Currents}\dotfill \pageref*{currents} \linebreak \noindent\hyperlink{hyperfunctions_and_coulombeau_distributions}{Hyperfunctions and Coulombeau distributions}\dotfill \pageref*{hyperfunctions_and_coulombeau_distributions} \linebreak \noindent\hyperlink{distributions_from_nonstandard_analysis}{Distributions from nonstandard analysis}\dotfill \pageref*{distributions_from_nonstandard_analysis} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{traditional}{Traditional}\dotfill \pageref*{traditional} \linebreak \noindent\hyperlink{InTermsOfSmoothToposes}{In terms of smooth toposes}\dotfill \pageref*{InTermsOfSmoothToposes} \linebreak \hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea} In [[functional analysis]], the concept of \emph{distributional density}, usually just called \emph{distribution} for short, is a generalization of the concept of [[density]], hence of something that may be [[integration|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 \emph{[[generalized functions]]}, namely objects that may arise as potentially singular [[limit of a sequence|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 fields|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 \emph{[[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|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]] \begin{displaymath} \itexarray{ C^\infty_c(X) &\longrightarrow& \mathbb{R} \\ b &\mapsto& \int_{x \in X} b(x) dvol(x) } \,. \end{displaymath} 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 \begin{displaymath} \delta_{x_0} \;\colon\; b \mapsto b(x_0) \end{displaymath} 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 a sequence|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 \emph{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 \emph{[[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 [[partial function|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 [[derivative of a distribution|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]]''. \hypertarget{Definitions}{}\subsection*{{Definition}}\label{Definitions} We first recall the \begin{itemize}% \item \hyperlink{TraditionalDefinition}{Traditional definition}. \end{itemize} Then we consider the axiomatic reformulation in terms of [[monads]] following \hyperlink{Kock11}{Kock 11}. \begin{itemize}% \item \hyperlink{CharacterizationByMonads}{Characterization by monads} \end{itemize} \hypertarget{TraditionalDefinition}{}\subsubsection*{{As continuous linear functionals}}\label{TraditionalDefinition} Distributions come in various flavors, depending on what spaces of functions they act on. The functions they act on are called \textbf{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. \begin{defn} \label{CompactlySupportedTestFunctions}\hypertarget{CompactlySupportedTestFunctions}{} \textbf{(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 \emph{compactly supported test functions} is the following \begin{enumerate}% \item the underlying set is the set of [[bump functions]], hence of [[smooth functions]] $X \to \mathbb{R}$ to the [[real numbers]] with [[compact support]]; \item equipped with evident [[real vector space]] structure given by pointwise addition and pointwise multipication of functions. \item equipped with the [[topological space|topology]] which is the [[metric topology]] induced from the family of [[seminorms]] \begin{displaymath} \rho_{K, \alpha}(f) = \sup_{x \in K} |\partial^{\alpha} f| \end{displaymath} where $K \subseteq U$ is [[compact topological space|compact]] and $\alpha = (\alpha_1, \ldots, \alpha_n)$ is a multi-index and \begin{displaymath} \partial^{\alpha} = \frac{\partial^{\alpha_1}}{\partial x^{\alpha_1}} \ldots \frac{\partial^{\alpha_n}}{\partial x^{\alpha_n}} \end{displaymath} is the corresponding [[differential operator]]. \end{enumerate} \end{defn} \begin{prop} \label{}\hypertarget{}{} The [[topological vector space]] $C^\infty_c(X)$ of compactly supported test functions (def. \ref{CompactlySupportedTestFunctions}) is [[locally convex topological vector space|locally convex]] and [[complete space|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 set|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]]). \end{prop} \begin{defn} \label{DistributionOnOpenSubsetOfEuclideanSpace}\hypertarget{DistributionOnOpenSubsetOfEuclideanSpace}{} \textbf{(distribution on Euclidean space)} Let $X \subset \mathbb{R}^n$ be a [[smooth manifold]] given as an [[open subset]] of [[Euclidean space]]. A \textbf{distribution on} $X$ is a [[continuous linear functional]] of the form \begin{displaymath} C^\infty_c(X) \longrightarrow \mathbb{R} \end{displaymath} from the [[locally convex space|locally convex]] [[topological vector space]] of compactly supported test functions (def. \ref{CompactlySupportedTestFunctions}) to the [[real numbers]]. The space of distributions on $X$ is denoted $\mathcal{D}'(X)$ (see also remark \ref{NotationForSpaceOfDistributions}). There is an obvious bilinear pairing \begin{displaymath} \itexarray{ \mathcal{D}'(X) \times C_c^{\infty}(X) &\longrightarrow& \mathbb{R} \\ (S, \phi) &\mapsto& S(\phi) } \end{displaymath} 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 \begin{displaymath} \langle -, \phi\rangle \;\colon\; \mathcal{D}'(U) \to \mathbb{R} \end{displaymath} continuous for all test functions $\phi$. As $C_c^\infty(U)$ is reflexive, this agrees with the weak topology. \end{defn} See at \emph{[[locally convex topological vector space]]} the section \emph{\href{locally+convex+space#ContinuousLinearFunctionals}{Continuous linear functionals}} for alternative characterizations of the continuity of distributions according to def. \ref{DistributionOnOpenSubsetOfEuclideanSpace}. 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 \emph{strong} topology). \begin{remark} \label{NotationForSpaceOfDistributions}\hypertarget{NotationForSpaceOfDistributions}{} \textbf{(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 (\hyperlink{Schwartz50}{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 \hyperlink{Hoermander90}{H\"o{}r\#mander 90, below def. 2.1.1}). \end{remark} \begin{example} \label{}\hypertarget{}{} If $f \colon X \to \mathbb{R}$ is [[locally integrable function|locally integrable]], then for all test functions $\phi$ the [[Lebesgue integral]] \begin{displaymath} \langle f, \phi\rangle = \int_X f(x)\phi(x) d x \end{displaymath} 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 \begin{displaymath} C_c^{\infty}(X) \hookrightarrow \mathcal{D}'(X) \end{displaymath} and this inclusion turns out to be [[dense subset|dense]]. \end{example} Other notions of spaces of distributions, each endowed with the weak $*$-topology, include \begin{itemize}% \item [[compactly supported distributions]] on $U$. These are functionals on $C^{\infty}(U)$ (test functions without compact support). \item [[tempered distributions]] (usually on $U = \mathbb{R}^n$). These are functionals on what is called the \emph{[[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 \begin{displaymath} \rho_{K, \alpha, \beta}(\phi) = \sup_{x \in K} |x^\alpha \partial^\beta \phi| \end{displaymath} where $\alpha$, $\beta$ are multi-indices. \item 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). \end{itemize} \begin{prop} \label{PullbackOfDistributionAlongSubmersion}\hypertarget{PullbackOfDistributionAlongSubmersion}{} \textbf{([[pullback of distributions]] along [[submersions]])} If $X_1, X_2 \subset \mathbb{R}^n$ are two [[open subsets]] of [[Euclidean space]], and if \begin{displaymath} f \;\colon\; X_1 \overset{}{\longrightarrow} X_2 \end{displaymath} 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]] \begin{displaymath} f^\ast \;\colon\; \mathcal{D}'(X_2) \longrightarrow \mathcal{D}'(X_1) \end{displaymath} between spaces of distributions (def. \ref{DistributionOnOpenSubsetOfEuclideanSpace}) which extends the [[pullback of functions]] in that on a distribution represented by a [[bump function]] $b$ it is given by precomposition \begin{displaymath} f^\ast b = b \circ f \,. \end{displaymath} This is hence called the \emph{[[pullback of distributions]]}. \end{prop} (\hyperlink{Hoermander90}{H\"o{}rmander 90, theorem 6.1.2}) \begin{defn} \label{DistributionsOnSmoothManifolds}\hypertarget{DistributionsOnSmoothManifolds}{} \textbf{(distributions on smooth manifolds)} Let $X$ be a [[smooth manifold]]. Then a distribution on $X$ is an [[equivalence class]] of \begin{enumerate}% \item a choice of smooth [[atlas]] $\{\mathbb{R}^n \underoverset{\simeq}{\psi_i}{\longrightarrow} U_i \subset X\}_{i \in I}$; \item 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; \item 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. \ref{PullbackOfDistributionAlongSubmersion}) along the coordinate change maps: $\phi_j = (\psi_i^{-1} \circ \psi_j)^\ast \phi_i$. \end{enumerate} \end{defn} (\hyperlink{Hoermander90}{H\"o{}rmander 90, def. 6.3.3}) \hypertarget{CharacterizationByMonads}{}\subsubsection*{{As smooth linear functionals}}\label{CharacterizationByMonads} 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 analysis|functional analytic]] sense relate to the \emph{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 \begin{itemize}% \item \emph{[[distributions are the smooth linear functionals]]}. \end{itemize} \hypertarget{operations_on_distributions}{}\subsection*{{Operations on distributions}}\label{operations_on_distributions} \hypertarget{inducing_operations_by_dual_extension}{}\subsubsection*{{Inducing operations by dual extension}}\label{inducing_operations_by_dual_extension} 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 \begin{displaymath} F\colon C_c^\infty(U) \to C_c^\infty(U) \end{displaymath} we define \begin{displaymath} F^*\colon \mathcal{D}'(U) \to \mathcal{D}'(U) \end{displaymath} according to the usual formula for dualities \begin{displaymath} F^* S(\phi) = S(F \phi). \end{displaymath} 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 \textbf{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 \begin{displaymath} F^\dagger(\iota \phi)(\psi) = \iota(F(\phi))(\psi) = \langle F(\phi), \psi \rangle \end{displaymath} 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 \begin{displaymath} F^\dagger(\iota \phi)(\psi) = \langle F(\phi), \psi \rangle = \langle \phi, F^+(\psi) \rangle \end{displaymath} 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 \begin{displaymath} \langle F^\dagger(S),\phi\rangle = \langle S, F^+(\phi) \rangle \end{displaymath} If the ground field is $\mathbb{C}$ then this carries through essentially unchanged except for the fact that one does \textbf{not} use the inner product on $C_c^\infty(U)$ but rather the associated bilinear pairing \begin{displaymath} (\phi,\psi) = \int_U \phi \psi \end{displaymath} 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: \begin{itemize}% \item 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: \begin{displaymath} \langle \theta \cdot \phi, \psi \rangle = \langle \phi, \psi \cdot \theta\rangle \end{displaymath} where $\phi, \psi$ are arbitrary test functions. Thus we define $\theta \cdot S$ by \begin{displaymath} \langle \theta \cdot S, \psi \rangle = \langle S, \theta \cdot \psi \end{displaymath} \item Differentiation. If $\partial^i$ is partial differentiation with respect to the $i^{th}$ coordinate, then for test functions $\psi$, $\phi$ we have \begin{displaymath} \int_U \partial^i(\psi)(x) \phi(x)\; d x = -\int_U \psi(x) \partial^i(\phi)(x)\; d x \end{displaymath} by simple integration by parts and the fact that $\phi$, $\psi$ are compactly supported. Thus differentiation is \emph{skew}-adjoint and so we define the extension to distributions by \begin{displaymath} \langle \partial^i(S), \phi\rangle = -\langle S, \partial^i(\phi) \rangle \end{displaymath} for all test functions $\phi$. In general, \begin{displaymath} \langle \partial^\alpha S, \phi\rangle = (-1)^{|\alpha|}\langle S, \partial^\alpha \phi \rangle \end{displaymath} where $|\alpha| = \alpha_1 + \ldots + \alpha_n$ is the total degree of the multi-index. \end{itemize} Thus derivatives of distributions are defined to all orders. Some examples are given in the section ``examples''. \hypertarget{MultiplicationsOfDistributions}{}\subsubsection*{{Multiplication of Distributions}}\label{MultiplicationsOfDistributions} See at \emph{[[multiplication of distributions]]} \hypertarget{examples}{}\subsection*{{Examples}}\label{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 $\mu: C_c(U) \to \mathbb{R}$ 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 \begin{displaymath} \mu(\phi) = \int_U \phi d m. \end{displaymath} 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 $i$: \begin{displaymath} C_c^\infty(U) \overset{i}{\to} C_c(U) \overset{\mu}{\to} \mathbb{R} \end{displaymath} Specializing further, consider any function of bounded variation on $U = \mathbb{R}$, say a bounded monotone increasing function $\alpha$. Then the Riemann-Stieltjes integral \begin{displaymath} \int_{\mathbb{R}} f(x) d\alpha(x) \end{displaymath} 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 $f \in C_c(\mathbb{R})$, \begin{displaymath} \langle f, d H \rangle = \int_{\mathbb{R}} f(x) d H(x) = f(0) \end{displaymath} As a distribution, the Heaviside measure is the famous \textbf{[[Dirac distribution]]}. The long-standing intuitive practice among physicists and engineers is to write \begin{displaymath} d H(x) = \delta_0(x) d x \end{displaymath} 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 \emph{every} distribution arises as a linear combination of derivatives of continuous functions: \textbf{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 \begin{displaymath} S = \sum_{\alpha \in A} \partial^\alpha g_\alpha \end{displaymath} \hypertarget{applications}{}\subsection*{{Applications}}\label{applications} 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 H\"o{}rmander 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 \emph{[[operator-valued distribution]]} and \emph{[[Wightman axioms]]}. A brief survey of applications of distribution theory to perturbative quantum field theory may be found \href{http://www1.jinr.ru/Archive/Pepan/v-31-7a/I_ktp_04_t.pdf}{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: \begin{itemize}% \item \textbf{Theorem (Ehrenpreis, Malgrange):} Let $D$ be a linear differential operator on $\mathbb{R}^n$ with constant coefficients. Given a compactly supported smooth function $f$ on $\mathbb{R}^n$, there exists a smooth solution $u$ to the equation $D u = f$. \end{itemize} A proof is given in these \href{http://www-math.mit.edu/~helgason/hormander.pdf}{notes} by Helgason. The basic idea is to prove there exists a \textbf{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. \hypertarget{variants}{}\subsection*{{Variants}}\label{variants} \hypertarget{in_synthetic_differential_geometry}{}\subsubsection*{{In synthetic differential geometry}}\label{in_synthetic_differential_geometry} There is another point of view on distributions: that they \emph{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 object]]s and infinitely large integers. This is discussed in (\hyperlink{MoerdijkReyes91}{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. \hypertarget{currents}{}\subsubsection*{{Currents}}\label{currents} 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 \emph{densities}) do not agree. See V. Guillemin, S. Sternberg: \emph{Geometric asymptotics} (\href{http://www.ams.org/online_bks/surv14}{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 \textbf{currents}. They are useful e.g. in the study of higher dimensional residua in higher dimensional complex geometry (cf. \emph{Principles of algebraic geometry} by Griffiths and Harris) and in geometric measure theory (cf. the monograph by Federer). \hypertarget{hyperfunctions_and_coulombeau_distributions}{}\subsubsection*{{Hyperfunctions and Coulombeau distributions}}\label{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 \emph{[[hyperfunctions]]} and the theory of \textbf{Coulombeau distributions}. \hypertarget{distributions_from_nonstandard_analysis}{}\subsubsection*{{Distributions from nonstandard analysis}}\label{distributions_from_nonstandard_analysis} Distributions can be alternatively described using [[nonstandard analysis]], see there. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[non-singular distribution]] \item [[order of a distribution]] \item [[support of a distribution]] \begin{itemize}% \item [[point-supported distribution]] \end{itemize} \item [[scaling degree of a distribution]] \item [[product of a distribution with a smooth function]] \item [[derivative of a distribution]] \item [[generalized solution of a partial differential equation]] \item [[extension of distributions]] \item [[tensor product of distributions]] \item [[product of distributions]] \item [[convolution product of distributions]] \item [[Hörmander topology]] \item [[Fourier transform of distributions]] \item [[Lawvere distribution]] ([[categorification]] of the concept of distributions) \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} See also \emph{[[hyperfunction]]}, [[ultradistribution]] and references therein. \hypertarget{traditional}{}\subsubsection*{{Traditional}}\label{traditional} Generalized functions were introduced by S. L. Sobolev in 1935, and independently (under the name \emph{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 \begin{itemize}% \item [[Laurent Schwartz]], \emph{Th\'e{}orie des distributions}, 1--2 , Hermann (1950--1951) \item [[I. M. Gel'fand]], G.E. Shilov, \emph{Generalized functions}, 1--5 , Acad. Press (1966--1968) transl. from . . , . . , . 1-3, .:, 1958; 1: , 2: , 3: \item E. Magenes, G. Stampacchia, \emph{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 \textbf{24}, Springer 2011 \href{http://dx.doi.org/10.1007/978-3-642-10967-6}{doi} \item [[François Trèves]], \emph{Topological Vector Spaces, Distributions and Kernels} (Academic Press, New York, 1967) \end{itemize} Modern accounts include \begin{itemize}% \item [[Lars Hörmander]], \emph{The analysis of linear partial differential operators}, vol. I, Springer 1983, 1990 \item Walter Rudin, chapter 6 of \emph{Functional analysis}, McGraw-Hill, 1991 \item M. Grosser, E. Farkas, M. Kunzinger, R. Steinbauer, \emph{On the foundations of nonlinear generalized functions I, II}, Mem. Amer. Math. Soc. \textbf{153} (2001) \item M. Kunzinger, R. Steinbauer, \emph{Foundations of a nonlinear distributional geometry}, Acta Appl. Math. \textbf{71}, 179-206 (2002) \end{itemize} Lecture notes include \begin{itemize}% \item Hasse Carlsson, \emph{Lecture notes on distributions} (\href{http://www.math.chalmers.se/~hasse/distributioner_eng.pdf}{pdf}) \end{itemize} and several chapters of the course \begin{itemize}% \item Erik P. van den Ban, \emph{Analysis on manifolds} (2009) (\href{https://www.staff.science.uu.nl/~ban00101/anman2009/anman2009.html}{web} - especially lectures 1-3) \end{itemize} Applications of distributions in [[physics]] are discussed in \begin{itemize}% \item V. S. Vladimirov, \emph{Generalized functions in mathematical physics}. Moskva, Nauka 1980, Mir 1979; \emph{Equations of mathematical physics}, Mir 1984 \item [[Nikolay Bogolyubov]], A. A. Logunov, I.T. Todorov, \emph{Introduction to axiomatic quantum field theory}, Benjamin (1975) \end{itemize} Application of distributions in [[perturbative quantum field theory]] is discussed in \begin{itemize}% \item [[Günter Scharf]] \emph{[[Finite Quantum Electrodynamics -- The Causal Approach]]}, Springer 1995 (\href{http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0844.53052&format=complete}{ZMATH entry}) \end{itemize} For more on this see the references at \emph{[[perturbative AQFT]]}. See also \begin{itemize}% \item Springer online [[eom]], \emph{\href{http://eom.springer.de/g/g043810.htm}{generalized function}} \end{itemize} References on [[Colombeau algebra]] include \begin{itemize}% \item J. F. Colombeau, \emph{New generalized functions and multiplications of distributions}, North Holland, Amsterdam (1984); \emph{Elementary introduction in new generalized functions}, North Holland (1985) \item N. Djapi, S. Pilipovi, \emph{Microlocal analysis of Colombeau's generalized functions on a manifold}, Indag. Math. N.S. 7, 293--309 (1996) \item Stevan Pilipovi, Milica igi, \emph{Suppleness of the sheaf of algebras of generalized functions on manifolds}, J. Math. Anal. Appl. \textbf{379}:2 (2011) 482--486, \href{http://arxiv.org/abs/1101.4552}{arxiv/1101.4552}, \href{http://www.ams.org/mathscinet-getitem?mr=2784335}{MR2784335}, \href{http://dx.doi.org/10.1016/j.jmaa.2010.12.060}{doi} \end{itemize} \hypertarget{InTermsOfSmoothToposes}{}\subsubsection*{{In terms of smooth toposes}}\label{InTermsOfSmoothToposes} Discussion of distributions in terms morphisms out of [[internal homs]] in a [[smooth topos]] ([[distributions are the smooth linear functionals]]) is in \begin{itemize}% \item [[Ieke Moerdijk]], [[Gonzalo Reyes]], around prop. 3.6 of \emph{[[Models for Smooth Infinitesimal Analysis]]}, Springer 1991 \end{itemize} and for the [[Cahiers topos]] in \begin{itemize}% \item [[Anders Kock]], [[Gonzalo Reyes]], \emph{Some calculus with extensive quantities: wave equation}, Theory and Applications of Categories , Vol. 11, 2003, No. 14, pp 321-336 (\href{http://www.tac.mta.ca/tac/volumes/11/14/11-14abs.html}{TAC}) \item [[Anders Kock]], [[Gonzalo Reyes]], \emph{Categorical distribution theory; heat equation} (\href{https://arxiv.org/abs/math/0407242}{arXiv:math/0407242}) \item [[Anders Kock]], \emph{Commutative monads as a theory of distributions} (\href{http://arxiv.org/abs/1108.5952}{arxiv/1108.5952}) \end{itemize} using results of \begin{itemize}% \item [[Alfred Frölicher]], [[Andreas Kriegl]], section 5 of \emph{Linear spaces and differentiation theory}, Wiley 1988 (\href{http://www.fuw.edu.pl/~kostecki/scans/froelicherkriegl1988.pdf}{pdf}) \end{itemize} and following the general conception of ``[[intensive and extensive]]'' in \begin{itemize}% \item [[William Lawvere]], Introduction to \emph{[[Categories in Continuum Physics]], Lectures given at a Workshop held at SUNY, Buffalo 1982. Lecture Notes in Mathematics 1174. 1986} \end{itemize} Similar [[sheaf theory|sheaf theoretic]] discussion of distributions as morphisms of [[smooth spaces]] is in \begin{itemize}% \item [[Frédéric Paugam]], section 3.2 of \emph{Towards the mathematics of quantum field theory}, 2012 (\href{https://webusers.imj-prg.fr/~frederic.paugam/documents/enseignement/master-mathematical-physics.pdf}{pdf}) \end{itemize} category: analysis [[!redirects distributions]] [[!redirects Schwartz distribution]] [[!redirects Schwartz distributions]] [[!redirects linear distribution]] [[!redirects linear distributions]] [[!redirects distributional density]] [[!redirects distributional densities]] \end{document}