Editing Distribution (mathematics) (section)

==Spaces of distributions==

{{See also|Spaces of test functions and distributions}}

For all <math>0 < k < \infty</math> and all <math>1 < p < \infty,</math> every one of the following canonical injections is continuous and has an [[Image of a function|image (also called the range)]] that is a [[Dense set|dense subset]] of its codomain:
<math display=block>\begin{matrix}
C_c^\infty(U) & \to & C_c^k(U) & \to & C_c^0(U) & \to & L_c^\infty(U) & \to & L_c^p(U) & \to & L_c^1(U) \\
\downarrow & &\downarrow && \downarrow \\
C^\infty(U) & \to & C^k(U) & \to & C^0(U) \\{}
\end{matrix}</math>
where the topologies on <math>L_c^q(U)</math> (<math>1 \leq q \leq \infty</math>) are defined as direct limits of the spaces <math>L_c^q(K)</math> in a manner analogous to how the topologies on <math>C_c^k(U)</math> were defined (so in particular, they are not the usual norm topologies). The range of each of the maps above (and of any composition of the maps above) is dense in its codomain.{{sfn|Trèves|2006|pp=150-160}}

Suppose that <math>X</math> is one of the spaces <math>C_c^k(U)</math> (for <math>k \in \{0, 1, \ldots, \infty\}</math>) or <math>L^p_c(U)</math> (for <math>1 \leq p \leq \infty</math>) or <math>L^p(U)</math> (for <math>1 \leq p < \infty</math>). Because the canonical injection <math>\operatorname{In}_X : C_c^\infty(U) \to X</math> is a continuous injection whose image is dense in the codomain, this map's [[Transpose of a linear map|transpose]] <math>{}^{t}\operatorname{In}_X : X'_b \to \mathcal{D}'(U) = \left(C_c^\infty(U)\right)'_b</math> is a continuous injection. This injective transpose map thus allows the [[continuous dual space]] <math>X'</math> of <math>X</math> to be identified with a certain vector subspace of the space <math>\mathcal{D}'(U)</math> of all distributions (specifically, it is identified with the image of this transpose map). This transpose map is continuous but it is {{em|not}} necessarily a [[topological embedding]].
A linear subspace of <math>\mathcal{D}'(U)</math> carrying a [[Locally convex topological vector space|locally convex]] topology that is finer than the [[subspace topology]] induced on it by <math>\mathcal{D}'(U) = \left(C_c^\infty(U)\right)'_b</math> is called '''{{em|a space of distributions}}'''.{{sfn|Trèves|2006|pp=240-252}}
Almost all of the spaces of distributions mentioned in this article arise in this way (for example, tempered distribution, restrictions, distributions of order <math>\leq</math> some integer, distributions induced by a positive Radon measure, distributions induced by an <math>L^p</math>-function, etc.) and any representation theorem about the continuous dual space of <math>X</math> may, through the transpose <math>{}^{t}\operatorname{In}_X : X'_b \to \mathcal{D}'(U),</math> be transferred directly to elements of the space <math>\operatorname{Im} \left({}^{t}\operatorname{In}_X\right).</math>

===Radon measures===

The inclusion map <math>\operatorname{In} : C_c^\infty(U) \to C_c^0(U)</math> is a continuous injection whose image is dense in its codomain, so the [[transpose]] <math>{}^{t}\operatorname{In} : (C_c^0(U))'_b \to \mathcal{D}'(U) = (C_c^\infty(U))'_b</math> is also a continuous injection.

Note that the continuous dual space <math>(C_c^0(U))'_b</math> can be identified as the space of [[Radon measure]]s, where there is a one-to-one correspondence between the continuous linear functionals <math>T \in (C_c^0(U))'_b</math> and integral with respect to a Radon measure; that is,

* if <math>T \in (C_c^0(U))'_b</math> then there exists a Radon measure <math>\mu</math> on {{mvar|U}} such that for all <math display=inline>f \in C_c^0(U), T(f) = \int_U f \, d\mu,</math> and
* if <math>\mu</math> is a Radon measure on {{mvar|U}} then the linear functional on <math>C_c^0(U)</math> defined by sending <math display=inline>f \in C_c^0(U)</math> to <math display=inline>\int_U f \, d\mu</math> is continuous.

Through the injection <math>{}^{t}\operatorname{In} : (C_c^0(U))'_b \to \mathcal{D}'(U),</math> every Radon measure becomes a distribution on {{mvar|U}}. If <math>f</math> is a [[locally integrable]] function on {{mvar|U}} then the distribution <math display=inline>\phi \mapsto \int_U f(x) \phi(x) \, dx</math> is a Radon measure; so Radon measures form a large and important space of distributions.

The following is the theorem of the structure of distributions of [[Radon measure]]s, which shows that every Radon measure can be written as a sum of derivatives of locally <math>L^\infty</math> functions on {{mvar|U}}:

{{math theorem|name='''Theorem.'''{{sfn|Trèves|2006|pp=262–264}}|math_statement=
Suppose <math>T \in \mathcal{D}'(U)</math> is a Radon measure, where <math>U \subseteq \R^n,</math> let <math>V \subseteq U</math> be a neighborhood of the support of <math>T,</math> and let <math>I = \{p \in \N^n : |p| \leq n\}.</math> There exists a family <math>f=(f_p)_{p\in I}</math> of locally <math>L^\infty</math> functions on {{mvar|U}} such that <math>\operatorname{supp} f_p \subseteq V</math> for every <math>p\in I,</math> and
<math display=block>T = \sum_{p\in I} \partial^p f_p.</math>
Furthermore, <math>T</math> is also equal to a finite sum of derivatives of continuous functions on <math>U,</math> where each derivative has order <math>\leq 2 n.</math>
}}

====Positive Radon measures====

A linear function <math>T</math> on a space of functions is called '''{{em|positive}}''' if whenever a function <math>f</math> that belongs to the domain of <math>T</math> is non-negative (that is, <math>f</math> is real-valued and <math>f \geq 0</math>) then <math>T(f) \geq 0.</math> One may show that every positive linear functional on <math>C_c^0(U)</math> is necessarily continuous (that is, necessarily a Radon measure).{{sfn|Trèves|2006|p=218}} 
[[Lebesgue measure]] is an example of a positive Radon measure.

====Locally integrable functions as distributions====

One particularly important class of Radon measures are those that are induced locally integrable functions. The function <math>f : U \to \R</math> is called '''{{em|[[locally integrable]]}}''' if it is [[Lebesgue integration|Lebesgue integrable]] over every compact subset {{mvar|K}} of {{mvar|U}}. This is a large class of functions that includes all continuous functions and all [[Lp space]] <math>L^p</math> functions. The topology on <math>\mathcal{D}(U)</math> is defined in such a fashion that any locally integrable function <math>f</math> yields a continuous linear functional on <math>\mathcal{D}(U)</math> – that is, an element of <math>\mathcal{D}'(U)</math> – denoted here by <math>T_f,</math> whose value on the test function <math>\phi</math> is given by the Lebesgue integral:
<math display=block>\langle T_f, \phi \rangle = \int_U f \phi\,dx.</math>

Conventionally, one [[Abuse of notation|abuses notation]] by identifying <math>T_f</math> with <math>f,</math> provided no confusion can arise, and thus the pairing between <math>T_f</math> and <math>\phi</math> is often written
<math display=block>\langle f, \phi \rangle = \langle T_f, \phi \rangle.</math>

If <math>f</math> and <math>g</math> are two locally integrable functions, then the associated distributions <math>T_f</math> and <math>T_g</math> are equal to the same element of <math>\mathcal{D}'(U)</math> if and only if <math>f</math> and <math>g</math> are equal [[almost everywhere]] (see, for instance, {{harvtxt|Hörmander|1983|loc=Theorem 1.2.5}}). Similarly, every [[Radon measure]] <math>\mu</math> on <math>U</math> defines an element of <math>\mathcal{D}'(U)</math> whose value on the test function <math>\phi</math> is <math display=inline>\int\phi \,d\mu.</math> As above, it is conventional to abuse notation and write the pairing between a Radon measure <math>\mu</math> and a test function <math>\phi</math> as <math>\langle \mu, \phi \rangle.</math> Conversely, as shown in a theorem by Schwartz (similar to the [[Riesz representation theorem]]), every distribution which is non-negative on non-negative functions is of this form for some (positive) Radon measure.

====Test functions as distributions====

The test functions are themselves locally integrable, and so define distributions. The space of test functions <math>C_c^\infty(U)</math> is sequentially [[dense (topology)|dense]] in <math>\mathcal{D}'(U)</math> with respect to the strong topology on <math>\mathcal{D}'(U).</math>{{sfn|Trèves|2006|pp=300-304}} This means that for any <math>T \in \mathcal{D}'(U),</math> there is a sequence of test functions, <math>(\phi_i)_{i=1}^\infty,</math> that converges to <math>T \in \mathcal{D}'(U)</math> (in its strong dual topology) when considered as a sequence of distributions. Or equivalently,
<math display=block>\langle \phi_i, \psi \rangle \to \langle T, \psi \rangle \qquad \text{ for all } \psi \in \mathcal{D}(U).</math>

===Distributions with compact support===

The inclusion map <math>\operatorname{In}: C_c^\infty(U) \to C^\infty(U)</math> is a continuous injection whose image is dense in its codomain, so the [[Transpose of a linear map|transpose map]] <math>{}^{t}\operatorname{In}: (C^\infty(U))'_b \to \mathcal{D}'(U) = (C_c^\infty(U))'_b</math> is also a continuous injection. Thus the image of the transpose, denoted by <math>\mathcal{E}'(U),</math> forms a space of distributions.{{sfn|Trèves|2006|pp=255-257}}

The elements of <math>\mathcal{E}'(U) = (C^\infty(U))'_b</math> can be identified as the space of distributions with compact support.{{sfn|Trèves|2006|pp=255-257}} Explicitly, if <math>T</math> is a distribution on {{mvar|U}} then the following are equivalent,

* <math>T \in \mathcal{E}'(U).</math>
* The support of <math>T</math> is compact.
* The restriction of <math>T</math> to <math>C_c^\infty(U),</math> when that space is equipped with the subspace topology inherited from <math>C^\infty(U)</math> (a coarser topology than the canonical LF topology), is continuous.{{sfn|Trèves|2006|pp=255-257}}
* There is a compact subset {{mvar|K}} of {{mvar|U}} such that for every test function <math>\phi</math> whose support is completely outside of {{mvar|K}}, we have <math>T(\phi) = 0.</math>

Compactly supported distributions define continuous linear functionals on the space <math>C^\infty(U)</math>; recall that the topology on <math>C^\infty(U)</math> is defined such that a sequence of test functions <math>\phi_k</math> converges to 0 if and only if all derivatives of <math>\phi_k</math> converge uniformly to 0 on every compact subset of {{mvar|U}}. Conversely, it can be shown that every continuous linear functional on this space defines a distribution of compact support. Thus compactly supported distributions can be identified with those distributions that can be extended from <math>C_c^\infty(U)</math> to <math>C^\infty(U).</math>

===Distributions of finite order===

Let <math>k \in \N.</math> The inclusion map <math>\operatorname{In}: C_c^\infty(U) \to C_c^k(U)</math> is a continuous injection whose image is dense in its codomain, so the [[transpose]] <math>{}^{t}\operatorname{In}: (C_c^k(U))'_b \to \mathcal{D}'(U) = (C_c^\infty(U))'_b</math> is also a continuous injection. Consequently, the image of <math>{}^{t}\operatorname{In},</math> denoted by <math>\mathcal{D}'^{k}(U),</math> forms a space of distributions. The elements of <math>\mathcal{D}'^k(U)</math> are '''{{em|the distributions of order <math>\,\leq k.</math>}}'''{{sfn|Trèves|2006|pp=258-264}} The distributions of order <math>\,\leq 0,</math> which are also called '''{{em|distributions of order {{math|0}}}}''' are exactly the distributions that are Radon measures (described above).

For <math>0 \neq k \in \N,</math> a '''{{em|distribution of order {{mvar|k}}}}''' is a distribution of order <math>\,\leq k</math> that is not a distribution of order <math>\,\leq k - 1</math>.{{sfn|Trèves|2006|pp=258-264}}

A distribution is said to be of '''{{em|finite order}}''' if there is some integer <math>k</math> such that it is a distribution of order <math>\,\leq k,</math> and the set of distributions of finite order is denoted by <math>\mathcal{D}'^{F}(U).</math> Note that if <math>k \leq l</math> then <math>\mathcal{D}'^k(U) \subseteq \mathcal{D}'^l(U)</math> so that <math>\mathcal{D}'^{F}(U) := \bigcup_{n=0}^\infty \mathcal{D}'^n(U)</math> is a vector subspace of <math>\mathcal{D}'(U)</math>, and furthermore, if and only if <math>\mathcal{D}'^{F}(U) = \mathcal{D}'(U).</math>{{sfn|Trèves|2006|pp=258-264}}

====Structure of distributions of finite order====

Every distribution with compact support in {{mvar|U}} is a distribution of finite order.{{sfn|Trèves|2006|pp=258-264}} Indeed, every distribution in {{mvar|U}} is {{em|locally}} a distribution of finite order, in the following sense:{{sfn|Trèves|2006|pp=258-264}} If {{mvar|V}} is an open and relatively compact subset of {{mvar|U}} and if <math>\rho_{VU}</math> is the restriction mapping from {{mvar|U}} to {{mvar|V}}, then the image of <math>\mathcal{D}'(U)</math> under <math>\rho_{VU}</math> is contained in <math>\mathcal{D}'^{F}(V).</math>

The following is the theorem of the structure of distributions of finite order, which shows that every distribution of finite order can be written as a sum of derivatives of [[Radon measure]]s:

{{math theorem|name=Theorem{{sfn|Trèves|2006|pp=258-264}}|math_statement=Suppose <math>T \in \mathcal{D}'(U)</math> has finite order and <math>I =\{p \in \N^n : |p| \leq k\}.</math> Given any open subset {{mvar|V}} of {{mvar|U}} containing the support of <math>T,</math> there is a family of Radon measures in {{mvar|U}}, <math>(\mu_p)_{p \in I},</math> such that for very <math>p \in I, \operatorname{supp}(\mu_p) \subseteq V</math> and
<math display=block>T = \sum_{|p| \leq k} \partial^p \mu_p.</math>}}

'''Example.''' (Distributions of infinite order) Let <math>U := (0, \infty)</math> and for every test function <math>f,</math> let <math display=block>S f := \sum_{m=1}^\infty (\partial^m f)\left(\frac{1}{m}\right).</math>

Then <math>S</math> is a distribution of infinite order on {{mvar|U}}. Moreover, <math>S</math> can not be extended to a distribution on <math>\R</math>; that is, there exists no distribution <math>T</math> on <math>\R</math> such that the restriction of <math>T</math> to {{mvar|U}} is equal to <math>S.</math>{{sfn|Rudin|1991|pp=177-181}}

===Tempered distributions and Fourier transform {{anchor|Tempered distribution}}===

{{Redirect|Tempered distribution|tempered distributions on semisimple groups|Tempered representation}}

Defined below are the '''{{em|tempered distributions}}''', which form a subspace of <math>\mathcal{D}'(\R^n),</math> the space of distributions on <math>\R^n.</math> This is a proper subspace: while every tempered distribution is a distribution and an element of <math>\mathcal{D}'(\R^n),</math> the converse is not true. Tempered distributions are useful if one studies the [[Fourier transform]] since all tempered distributions have a Fourier transform, which is not true for an arbitrary distribution in <math>\mathcal{D}'(\R^n).</math>

====Schwartz space====

The [[Schwartz space]] <math>\mathcal{S}(\R^n)</math> is the space of all smooth functions that are [[rapidly decreasing]] at infinity along with all partial derivatives. Thus <math>\phi:\R^n\to\R</math> is in the Schwartz space provided that any derivative of <math>\phi,</math> multiplied with any power of <math>|x|,</math> converges to 0 as <math>|x| \to \infty.</math> These functions form a complete TVS with a suitably defined family of [[seminorm]]s. More precisely, for any [[multi-indices]] <math>\alpha</math> and <math>\beta</math> define
<math display=block>p_{\alpha, \beta}(\phi) = \sup_{x \in \R^n} \left|x^\alpha \partial^\beta \phi(x) \right|.</math>

Then <math>\phi</math> is in the Schwartz space if all the values satisfy
<math display=block>p_{\alpha, \beta}(\phi) < \infty.</math>

The family of seminorms <math>p_{\alpha,\beta}</math> defines a [[locally convex]] topology on the Schwartz space. For <math>n = 1,</math> the seminorms are, in fact, [[Norm (mathematics)|norms]] on the Schwartz space. One can also use the following family of seminorms to define the topology:{{sfn|Trèves|2006|pp=92-94}}
<math display=block>|f|_{m,k} = \sup_{|p|\le m} \left(\sup_{x \in \R^n} \left\{(1 + |x|)^k \left|(\partial^\alpha f)(x) \right|\right\}\right), \qquad k,m \in \N.</math>

Otherwise, one can define a norm on <math>\mathcal{S}(\R^n)</math> via
<math display=block>\|\phi\|_k = \max_{|\alpha| + |\beta| \leq k} \sup_{x \in \R^n} \left| x^\alpha \partial^\beta \phi(x)\right|, \qquad k \ge 1.</math>

The Schwartz space is a [[Fréchet space]] (that is, a [[Complete topological vector space|complete]] [[Metrizable topological vector space|metrizable]] locally convex space). Because the [[Fourier transform]] changes <math>\partial^\alpha</math> into multiplication by <math>x^\alpha</math> and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function.

A sequence <math>\{f_i\}</math> in <math>\mathcal{S}(\R^n)</math> converges to 0 in <math>\mathcal{S}(\R^n)</math> if and only if the functions <math>(1 + |x|)^k (\partial^p f_i)(x)</math> converge to 0 uniformly in the whole of <math>\R^n,</math> which implies that such a sequence must converge to zero in <math>C^\infty(\R^n).</math>{{sfn|Trèves|2006|pp=92–94}}

<math>\mathcal{D}(\R^n)</math> is dense in <math>\mathcal{S}(\R^n).</math> The subset of all analytic Schwartz functions is dense in <math>\mathcal{S}(\R^n)</math> as well.{{sfn|Trèves|2006|p=160}}

The Schwartz space is [[Nuclear space|nuclear]], and the tensor product of two maps induces a canonical surjective TVS-isomorphisms
<math display=block>\mathcal{S}(\R^m)\ \widehat{\otimes}\ \mathcal{S}(\R^n) \to \mathcal{S}(\R^{m+n}),</math>
where <math>\widehat{\otimes}</math> represents the completion of the [[injective tensor product]] (which in this case is identical to the completion of the [[projective tensor product]]).{{sfn|Trèves|2006|p=531}}

====Tempered distributions====

The inclusion map <math>\operatorname{In}: \mathcal{D}(\R^n) \to \mathcal{S}(\R^n)</math> is a continuous injection whose image is dense in its codomain, so the [[transpose]] <math>{}^{t}\operatorname{In}: (\mathcal{S}(\R^n))'_b \to \mathcal{D}'(\R^n)</math> is also a continuous injection. Thus, the image of the transpose map, denoted by <math>\mathcal{S}'(\R^n),</math> forms a space of distributions.

The space <math>\mathcal{S}'(\R^n)</math> is called the space of {{em|tempered distributions}}. It is the [[continuous dual space]] of the Schwartz space. Equivalently, a distribution <math>T</math> is a tempered distribution if and only if
<math display=block>\left(\text{ for all } \alpha, \beta \in \N^n: \lim_{m\to \infty} p_{\alpha, \beta} (\phi_m) = 0 \right) \Longrightarrow \lim_{m\to \infty} T(\phi_m)=0.</math>

The derivative of a tempered distribution is again a tempered distribution. Tempered distributions generalize the bounded (or slow-growing) locally integrable functions; all distributions with compact support and all [[square-integrable]] functions are tempered distributions. More generally, all functions that are products of polynomials with elements of [[Lp space]] <math>L^p(\R^n)</math> for <math>p \geq 1</math> are tempered distributions.

The {{em|tempered distributions}} can also be characterized as {{em|slowly growing}}, meaning that each derivative of <math>T</math> grows at most as fast as some [[polynomial]]. This characterization is dual to the {{em|rapidly falling}} behaviour of the derivatives of a function in the Schwartz space, where each derivative of <math>\phi</math> decays faster than every inverse power of <math>|x|.</math> An example of a rapidly falling function is <math>|x|^n\exp (-\lambda |x|^\beta)</math> for any positive <math>n, \lambda, \beta.</math>

====Fourier transform====

To study the Fourier transform, it is best to consider complex-valued test functions and complex-linear distributions. The ordinary [[continuous Fourier transform]] <math>F : \mathcal{S}(\R^n) \to \mathcal{S}(\R^n)</math> is a TVS-[[automorphism]] of the Schwartz space, and the '''{{em|Fourier transform}}''' is defined to be its [[transpose]] <math>{}^{t}F : \mathcal{S}'(\R^n) \to \mathcal{S}'(\R^n),</math> which (abusing notation) will again be denoted by <math>F.</math> So the Fourier transform of the tempered distribution <math>T</math> is defined by <math>(FT)(\psi) = T(F \psi)</math> for every Schwartz function <math>\psi.</math> <math>FT</math> is thus again a tempered distribution. The Fourier transform is a TVS isomorphism from the space of tempered distributions onto itself. This operation is compatible with differentiation in the sense that
<math display=block>F \dfrac{dT}{dx} = ixFT</math>
and also with convolution: if <math>T</math> is a tempered distribution and <math>\psi</math> is a {{em|slowly increasing}} smooth function on <math>\R^n,</math> <math>\psi T</math> is again a tempered distribution and
<math display=block>F(\psi T) = F \psi * FT</math>
is the convolution of <math>FT</math> and <math>F \psi.</math> In particular, the Fourier transform of the constant function equal to 1 is the <math>\delta</math> distribution.

====Expressing tempered distributions as sums of derivatives====

If <math>T \in \mathcal{S}'(\R^n)</math> is a tempered distribution, then there exists a constant <math>C > 0,</math> and positive integers <math>M</math> and <math>N</math> such that for all [[Schwartz function]]s <math>\phi \in \mathcal{S}(\R^n)</math>
<math display=block>\langle T, \phi \rangle \le C\sum\nolimits_{|\alpha|\le N, |\beta|\le M}\sup_{x \in \R^n} \left|x^\alpha \partial^\beta \phi(x) \right|=C\sum\nolimits_{|\alpha|\le N, |\beta|\le M} p_{\alpha, \beta}(\phi).</math>

This estimate, along with some techniques from [[functional analysis]], can be used to show that there is a continuous slowly increasing function <math>F</math> and a multi-index <math>\alpha</math> such that
<math display=block>T = \partial^\alpha F.</math>

====Restriction of distributions to compact sets====

If <math>T \in \mathcal{D}'(\R^n),</math> then for any compact set <math>K \subseteq \R^n,</math> there exists a continuous function <math>F</math>compactly supported in <math>\R^n</math> (possibly on a larger set than {{mvar|K}} itself) and a multi-index <math>\alpha</math> such that <math>T = \partial^\alpha F</math> on <math>C_c^\infty(K).</math>