Editing Distribution (mathematics) (section)

===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>