Editing Distribution (mathematics) (section)

==Localization of distributions==

There is no way to define the value of a distribution in <math>\mathcal{D}'(U)</math> at a particular point of {{mvar|U}}. However, as is the case with functions, distributions on {{mvar|U}} restrict to give distributions on open subsets of {{mvar|U}}. Furthermore, distributions are {{em|locally determined}} in the sense that a distribution on all of {{mvar|U}} can be assembled from a distribution on an open cover of {{mvar|U}} satisfying some compatibility conditions on the overlaps. Such a structure is known as a [[Sheaf (mathematics)|sheaf]].

===Extensions and restrictions to an open subset===

Let <math>V \subseteq U</math> be open subsets of <math>\R^n.</math> 
Every function <math>f \in \mathcal{D}(V)</math> can be {{em|extended by zero}} from its domain {{mvar|V}} to a function on {{mvar|U}} by setting it equal to <math>0</math> on the [[Complement (set theory)|complement]] <math>U \setminus V.</math> This extension is a smooth compactly supported function called the {{em|trivial extension of <math>f</math> to <math>U</math>}} and it will be denoted by <math>E_{VU} (f).</math>
This assignment <math>f \mapsto E_{VU} (f)</math> defines the {{em|trivial extension}} operator 
<math>E_{VU} : \mathcal{D}(V) \to \mathcal{D}(U),</math> 
which is a continuous injective linear map. It is used to canonically identify <math>\mathcal{D}(V)</math> as a [[vector subspace]] of <math>\mathcal{D}(U)</math> (although {{em|not}} as a [[topological subspace]]). 
Its transpose ([[#Transpose of a linear operator|explained here]]) 
<math display=block>\rho_{VU} := {}^{t}E_{VU} : \mathcal{D}'(U) \to \mathcal{D}'(V),</math> 
is called the '''{{em|{{visible anchor|restriction map|text=restriction to <math>V</math> of distributions in <math>U</math>}}}}'''{{sfn|Trèves|2006|pp=245-247}} and as the name suggests, the image <math>\rho_{VU}(T)</math> of a distribution <math>T \in \mathcal{D}'(U)</math> under this map is a distribution on <math>V</math> called the '''restriction of <math>T</math> to <math>V.</math>''' The [[#Transpose of a linear operator|defining condition]] of the restriction <math>\rho_{VU}(T)</math> is:
<math display=block>\langle \rho_{VU} T, \phi \rangle = \langle T, E_{VU} \phi \rangle \quad \text{ for all } \phi \in \mathcal{D}(V).</math>
If <math>V \neq U</math> then the (continuous injective linear) trivial extension map <math>E_{VU} : \mathcal{D}(V) \to \mathcal{D}(U)</math> is {{em|not}} a topological embedding (in other words, if this linear injection was used to identify <math>\mathcal{D}(V)</math> as a subset of <math>\mathcal{D}(U)</math> then <math>\mathcal{D}(V)</math>'s topology would [[Comparison of topologies|strictly finer]] than the [[subspace topology]] that <math>\mathcal{D}(U)</math> induces on it; importantly, it would {{em|not}} be a [[topological subspace]] since that requires equality of topologies) and its range is also {{em|not}} dense in its [[codomain]] <math>\mathcal{D}(U).</math>{{sfn|Trèves|2006|pp=245-247}} Consequently if <math>V \neq U</math> then [[#restriction map|the restriction mapping]] is neither injective nor surjective.{{sfn|Trèves|2006|pp=245-247}} A distribution <math>S \in \mathcal{D}'(V)</math> is said to be '''{{em|extendible to {{mvar|U}}}}''' if it belongs to the range of the transpose of <math>E_{VU}</math> and it is called '''{{em|extendible}}''' if it is extendable to <math>\R^n.</math>{{sfn|Trèves|2006|pp=245-247}}

Unless <math>U = V,</math> the restriction to {{mvar|V}} is neither [[injective]] nor [[surjective]]. Lack of surjectivity follows since distributions can blow up towards the boundary of {{mvar|V}}. For instance, if <math>U = \R</math> and <math>V = (0, 2),</math> then the distribution
<math display=block>T(x) = \sum_{n=1}^\infty n \, \delta\left(x-\frac{1}{n}\right)</math>
is in <math>\mathcal{D}'(V)</math> but admits no extension to <math>\mathcal{D}'(U).</math>

===Gluing and distributions that vanish in a set===

{{Math theorem
| name = Theorem{{sfn |Trèves|2006|pp=253-255}}
| math_statement = Let <math>(U_i)_{i \in I}</math> be a collection of open subsets of <math>\R^n.</math> For each <math>i \in I,</math> let <math>T_i \in \mathcal{D}'(U_i)</math> and suppose that for all <math>i, j \in I,</math> the restriction of <math>T_i</math> to <math>U_i \cap U_j</math> is equal to the restriction of <math>T_j</math> to <math>U_i \cap U_j</math> (note that both restrictions are elements of <math>\mathcal{D}'(U_i \cap U_j)</math>). Then there exists a unique <math display=inline>T \in \mathcal{D}'(\bigcup_{i \in I} U_i)</math> such that for all <math>i \in I,</math> the restriction of {{mvar|T}} to <math>U_i</math> is equal to <math>T_i.</math>
}}

Let {{mvar|V}} be an open subset of {{mvar|U}}. <math>T \in \mathcal{D}'(U)</math> is said to '''{{em|vanish in {{mvar|V}}}}''' if for all <math>f \in \mathcal{D}(U)</math> such that <math>\operatorname{supp}(f) \subseteq V</math> we have <math>Tf = 0.</math> {{mvar|T}} vanishes in {{mvar|V}} if and only if the restriction of {{mvar|T}} to {{mvar|V}} is equal to 0, or equivalently, if and only if {{mvar|T}} lies in the [[kernel (algebra)|kernel]] of the restriction map <math>\rho_{VU}.</math>

{{Math theorem
| name = Corollary{{sfn |Trèves|2006| pp=253-255}}
| math_statement = Let <math>(U_i)_{i \in I}</math> be a collection of open subsets of <math>\R^n</math> and let <math display=inline>T \in \mathcal{D}'(\bigcup_{i \in I} U_i).</math> <math>T = 0</math> if and only if for each <math>i \in I,</math> the restriction of {{mvar|T}} to <math>U_i</math> is equal to 0.
}}

{{Math theorem| name=Corollary{{sfn |Trèves|2006|pp=253-255}}| math_statement= The union of all open subsets of {{mvar|U}} in which a distribution {{mvar|T}} vanishes is an open subset of {{mvar|U}} in which {{mvar|T}} vanishes.}}

===Support of a distribution===

This last corollary implies that for every distribution {{mvar|T}} on {{mvar|U}}, there exists a unique largest subset {{mvar|V}} of {{mvar|U}} such that {{mvar|T}} vanishes in {{mvar|V}} (and does not vanish in any open subset of {{mvar|U}} that is not contained in {{mvar|V}}); the complement in {{mvar|U}} of this unique largest open subset is called {{em|the '''support''' of {{mvar|T}}}}.{{sfn|Trèves|2006|pp=253-255}} Thus
<math display=block> \operatorname{supp}(T) = U \setminus \bigcup \{V \mid \rho_{VU}T = 0\}.</math>

If <math>f</math> is a locally integrable function on {{mvar|U}} and if <math>D_f</math> is its associated distribution, then the support of <math>D_f</math> is the smallest closed subset of {{mvar|U}} in the complement of which <math>f</math> is [[almost everywhere]] equal to 0.{{sfn|Trèves|2006|pp=253-255}} If <math>f</math> is continuous, then the support of <math>D_f</math> is equal to the closure of the set of points in {{mvar|U}} at which <math>f</math> does not vanish.{{sfn|Trèves|2006| pp=253-255}} The support of the distribution associated with the [[Dirac measure]] at a point <math>x_0</math> is the set <math>\{x_0\}.</math>{{sfn|Trèves|2006|pp=253-255}} If the support of a test function <math>f</math> does not intersect the support of a distribution {{mvar|T}} then <math>Tf = 0.</math> A distribution {{mvar|T}} is 0 if and only if its support is empty. If <math>f \in C^\infty(U)</math> is identically 1 on some open set containing the support of a distribution {{mvar|T}} then <math>f T = T.</math> If the support of a distribution {{mvar|T}} is compact then it has finite order and there is a constant <math>C</math> and a non-negative integer <math>N</math> such that:{{sfn|Rudin|1991|pp=149-181}}
<math display=block>|T \phi| \leq C\|\phi\|_N := C \sup \left\{\left|\partial^\alpha \phi(x)\right| : x \in U, |\alpha| \leq N \right\} \quad \text{ for all } \phi \in \mathcal{D}(U).</math>

If {{mvar|T}} has compact support, then it has a unique extension to a continuous linear functional <math>\widehat{T}</math> on <math>C^\infty(U)</math>; this function can be defined by <math>\widehat{T} (f) := T(\psi f),</math> where <math>\psi \in \mathcal{D}(U)</math> is any function that is identically 1 on an open set containing the support of {{mvar|T}}.{{sfn|Rudin|1991|pp=149-181}}

If <math>S, T \in \mathcal{D}'(U)</math> and <math>\lambda \neq 0</math> then <math>\operatorname{supp}(S + T) \subseteq \operatorname{supp}(S) \cup \operatorname{supp}(T)</math> and <math>\operatorname{supp}(\lambda T) = \operatorname{supp}(T).</math> Thus, distributions with support in a given subset <math>A \subseteq U</math> form a vector subspace of <math>\mathcal{D}'(U).</math>{{sfn|Trèves|2006|pp=255-257}} Furthermore, if <math>P</math> is a differential operator in {{mvar|U}}, then for all distributions {{mvar|T}} on {{mvar|U}} and all <math>f \in C^\infty(U)</math> we have <math>\operatorname{supp} (P(x, \partial) T) \subseteq \operatorname{supp}(T)</math> and <math>\operatorname{supp}(fT) \subseteq \operatorname{supp}(f) \cap \operatorname{supp}(T).</math>{{sfn|Trèves|2006|pp=255-257}}

===Distributions with compact support===

====Support in a point set and Dirac measures====

For any <math>x \in U,</math> let <math>\delta_x \in \mathcal{D}'(U)</math> denote the distribution induced by the Dirac measure at <math>x.</math> For any <math>x_0 \in U</math> and distribution <math>T \in \mathcal{D}'(U),</math> the support of {{mvar|T}} is contained in <math>\{x_0\}</math> if and only if {{mvar|T}} is a finite linear combination of derivatives of the Dirac measure at <math>x_0.</math>{{sfn|Trèves|2006|pp=264-266}} If in addition the order of {{mvar|T}} is <math>\leq k</math> then there exist constants <math>\alpha_p</math> such that:{{sfn|Rudin|1991|p=165}}
<math display=block>T = \sum_{|p| \leq k} \alpha_p \partial^p \delta_{x_0}.</math>

Said differently, if {{mvar|T}} has support at a single point <math>\{P\},</math> then {{mvar|T}} is in fact a finite linear combination of distributional derivatives of the <math>\delta</math> function at {{mvar|P}}. That is, there exists an integer {{mvar|m}} and complex constants <math>a_\alpha</math> such that
<math display=block>T = \sum_{|\alpha|\leq m} a_\alpha \partial^\alpha(\tau_P\delta)</math>
where <math>\tau_P</math> is the translation operator.

====Distribution with compact support====

{{Math theorem|name=Theorem{{sfn|Rudin|1991|pp=149-181}}|math_statement=
Suppose {{mvar|T}} is a distribution on {{mvar|U}} with compact support {{mvar|K}}. There exists a continuous function <math>f</math> defined on {{mvar|U}} and a multi-index {{math|1=''p''}} such that
<math display=block>T = \partial^p f,</math>
where the derivatives are understood in the sense of distributions. That is, for all test functions <math>\phi</math> on {{mvar|U}},
<math display=block>T \phi = (-1)^{|p|} \int_{U} f(x) (\partial^p \phi)(x) \, dx.</math>
}}

====Distributions of finite order with support in an open subset====

{{Math theorem|name=Theorem{{sfn|Rudin|1991|pp=149-181}}|math_statement=
Suppose {{mvar|T}} is a distribution on {{mvar|U}} with compact support {{mvar|K}} and let {{mvar|V}} be an open subset of {{mvar|U}} containing {{mvar|K}}. Since every distribution with compact support has finite order, take {{mvar|N}} to be the order of {{mvar|T}} and define <math>P:=\{0,1,\ldots, N+2\}^n.</math> There exists a family of continuous functions <math>(f_p)_{p\in P}</math> defined on {{mvar|U}} '''with support in {{mvar|V}}''' such that
<math display=block>T = \sum_{p \in P} \partial^p f_p,</math>
where the derivatives are understood in the sense of distributions. That is, for all test functions <math>\phi</math> on {{mvar|U}},
<math display=block>T \phi = \sum_{p \in P} (-1)^{|p|} \int_U f_p(x) (\partial^p \phi)(x) \, dx.</math>
}}

===Global structure of distributions===

The formal definition of distributions exhibits them as a subspace of a very large space, namely the topological dual of <math>\mathcal{D}(U)</math> (or the [[Schwartz space]] <math>\mathcal{S}(\R^n)</math> for tempered distributions). It is not immediately clear from the definition how exotic a distribution might be. To answer this question, it is instructive to see distributions built up from a smaller space, namely the space of continuous functions. Roughly, any distribution is locally a (multiple) derivative of a continuous function. A precise version of this result, given below, holds for distributions of compact support, tempered distributions, and general distributions. Generally speaking, no proper subset of the space of distributions contains all continuous functions and is closed under differentiation. This says that distributions are not particularly exotic objects; they are only as complicated as necessary.

====Distributions as [[Sheaf (mathematics)|sheaves]]====

{{Math theorem|name=Theorem{{sfn|Trèves|2006|pp=258-264}}|math_statement=
Let {{mvar|T}} be a distribution on {{mvar|U}}.
There exists a sequence <math>(T_i)_{i=1}^\infty</math> in <math>\mathcal{D}'(U)</math> such that each {{mvar|T<sub>i</sub>}} has compact support and every compact subset <math>K \subseteq U</math> intersects the support of only finitely many <math>T_i,</math> and the sequence of partial sums <math>(S_j)_{j=1}^\infty,</math> defined by <math>S_j := T_1 + \cdots + T_j,</math> converges in <math>\mathcal{D}'(U)</math> to {{mvar|T}}; in other words we have:
<math display=block>T = \sum_{i=1}^\infty T_i.</math>
Recall that a sequence converges in <math>\mathcal{D}'(U)</math> (with its strong dual topology) if and only if it converges pointwise.
}}

====Decomposition of distributions as sums of derivatives of continuous functions====

By combining the above results, one may express any distribution on {{mvar|U}} as the sum of a series of distributions with compact support, where each of these distributions can in turn be written as a finite sum of distributional derivatives of continuous functions on {{mvar|U}}. In other words, for arbitrary <math>T \in \mathcal{D}'(U)</math> we can write:
<math display=block>T = \sum_{i=1}^\infty \sum_{p \in P_i} \partial^p f_{ip},</math>
where <math>P_1, P_2, \ldots</math> are finite sets of multi-indices and the functions <math>f_{ip}</math> are continuous.

{{Math theorem|name=Theorem{{sfn|Rudin|1991|pp=169-170}}|math_statement=
Let {{mvar|T}} be a distribution on {{mvar|U}}. For every multi-index {{mvar|p}} there exists a continuous function <math>g_p</math> on {{mvar|U}} such that

# any compact subset {{mvar|K}} of {{mvar|U}} intersects the support of only finitely many <math>g_p,</math> and
# <math>T = \sum\nolimits_p \partial^p g_p.</math>

Moreover, if {{mvar|T}} has finite order, then one can choose <math>g_p</math> in such a way that only finitely many of them are non-zero.
}}

Note that the infinite sum above is well-defined as a distribution. The value of {{mvar|T}} for a given <math>f \in \mathcal{D}(U)</math> can be computed using the finitely many <math>g_\alpha</math> that intersect the support of <math>f.</math>