Editing Distribution (mathematics) (section)

====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>