Editing Distribution (mathematics) (section)

{{Short description|Mathematical term generalizing the concept of function}}
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{{about|generalized functions in mathematical analysis|the concept of distributions in probability theory|Probability distribution|other uses|Distribution (disambiguation)#In mathematics{{!}}Distribution § Mathematics}}
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{{Very long|date=February 2025}} 

'''Distributions''', also known as '''Schwartz distributions''' are a kind of [[generalized function]] in [[mathematical analysis]]. Distributions make it possible to [[derivative|differentiate]] functions whose derivatives do not exist in the classical sense. In particular, any [[locally integrable]] function has a [[distributional derivative]].

Distributions are widely used in the theory of [[partial differential equation]]s, where it may be easier to establish the existence of distributional solutions ([[weak solution]]s) than [[Solution of a differential equation|classical solutions]], or where appropriate classical solutions may not exist. Distributions are also important in [[physics]] and [[engineering]] where many problems naturally lead to differential equations whose solutions or initial conditions are singular, such as the [[Dirac delta function|Dirac delta]] function.

A [[Function (mathematics)|function]] <math>f</math> is normally thought of as {{em|[[Group action|acting]]}} on the {{em|points}} in the function [[Domain (function)|domain]] by "sending" a point <math>x</math> in the domain to the point <math>f(x).</math> Instead of acting on points, distribution theory reinterprets functions such as <math>f</math> as acting on {{em|test functions}} in a certain way. In applications to physics and engineering, '''{{em|test functions}}''' are usually [[Smooth function|infinitely differentiable]] [[Complex number|complex]]-valued (or [[Real number|real]]-valued) functions with [[Compact space|compact]] [[Support (mathematics)|support]] that are defined on some given non-empty [[Open set|open subset]] <math>U \subseteq \R^n</math>. ([[Bump function]]s are examples of test functions.) The set of all such test functions forms a [[vector space]] that is denoted by <math>C_c^\infty(U)</math> or <math>\mathcal{D}(U).</math>

Most commonly encountered functions, including all [[Continuous function|continuous]] maps <math>f : \R \to \R</math> if using <math>U := \R,</math> can be canonically reinterpreted as acting via "[[Integral|integration]] against a test function." Explicitly, this means that such a function <math>f</math> "acts on" a test function <math>\psi \in \mathcal{D}(\R)</math> by "sending" it to the [[Integral (mathematics)|number]] <math display=inline>\int_\R f \, \psi \, dx,</math> which is often denoted by <math>D_f(\psi).</math> This new action <math display=inline>\psi \mapsto D_f(\psi)</math> of <math>f</math> defines a [[Functional (mathematics)|scalar-valued map]] <math>D_f : \mathcal{D}(\R) \to \Complex,</math> whose domain is the space of test functions <math>\mathcal{D}(\R).</math> This [[Functional (mathematics)|functional]] <math>D_f</math> turns out to have the two defining properties of what is known as a {{em|distribution on <math>U = \R</math>}}: it is [[Linear form|linear]], and it is also [[Continuous function|continuous]] when <math>\mathcal{D}(\R)</math> is given a certain [[topology]] called {{em|the canonical LF topology}}. The action (the integration <math display=inline>\psi \mapsto \int_\R f \, \psi \, dx</math>) of this distribution <math>D_f</math> on a test function <math>\psi</math> can be interpreted as a weighted average of the distribution on the [[Support (mathematics)|support]] of the test function, even if the values of the distribution at a single point are not well-defined. Distributions like <math>D_f</math> that arise from functions in this way are prototypical examples of distributions, but there exist many distributions that cannot be defined by integration against any function. Examples of the latter include the [[Dirac delta function]] and distributions defined to act by integration of test functions <math display=inline>\psi \mapsto \int_U \psi d \mu</math> against certain [[Measure (mathematics)|measures]] <math>\mu</math> on <math>U.</math> Nonetheless, it is still always possible to [[#Decomposition of distributions as sums of derivatives of continuous functions|reduce any arbitrary distribution]] down to a simpler {{em|family}} of related distributions that do arise via such actions of integration.

More generally, a {{em|'''distribution''' on <math>U</math>}} is by definition a [[Linear form|linear functional]] on <math>C_c^\infty(U)</math> that is [[Continuous linear functional|continuous]] when <math>C_c^\infty(U)</math> is given a topology called the '''{{em|canonical LF topology}}'''. This leads to {{em|the}} space of (all) distributions on <math>U</math>, usually denoted by <math>\mathcal{D}'(U)</math> (note the [[Prime (symbol)|prime]]), which by definition is the [[Vector space|space]] of all distributions on <math>U</math> (that is, it is the [[continuous dual space]] of <math>C_c^\infty(U)</math>); it is these distributions that are the main focus of this article.

Definitions of the appropriate topologies on [[spaces of test functions and distributions]] are given in the article on [[spaces of test functions and distributions]]. This article is primarily concerned with the definition of distributions, together with their properties and some important examples.

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