Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 40   | 1 | 2 |

Section Definition and examples of tempered distributions

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France Definition and examples of tempered distributions

| top | pdf |

A distribution [T \in {\scr D}\,'({\bb R}^{n})] is said to be tempered if it can be extended into a continuous linear functional on [{\scr S}].

If [{\scr S}\,'({\bb R}^{n})] is the topological dual of [{\scr S}({\bb R}^{n})], and if [S \in {\scr S}^{\prime}({\bb R}^{n})], then its restriction to [{\scr D}] is a tempered distribution; conversely, if [T \in {\scr D}\,'] is tempered, then its extension to [{\scr S}] is unique (because [{\scr D}] is dense in [{\scr S}]), hence it defines an element S of [{\scr S}\,']. We may therefore identify [{\scr S}\,'] and the space of tempered distributions.

A distribution with compact support is tempered, i.e. [{\scr S}\,' \supset {\scr E}\,']. By transposition of the corresponding properties of [{\scr S}], it is readily established that the derivative, translate or product by a polynomial of a tempered distribution is still a tempered distribution.

These inclusion relations may be summarized as follows: since [{\scr S}] contains [{\scr D}] but is contained in [{\scr E}], the reverse inclusions hold for the topological duals, and hence [{\scr S}\,'] contains [{\scr E}\,'] but is contained in [{\scr D}\,'].

A locally summable function f on [{\bb R}^{n}] will be said to be of polynomial growth if [|\,f({\bf x})|] can be majorized by a polynomial in [\|{\bf x}\|] as [\|{\bf x}\| \rightarrow \infty]. It is easily shown that such a function f defines a tempered distribution [T_{f}] via[\langle T_{f}, \varphi \rangle = {\textstyle\int\limits_{{\bb R}^{n}}} f({\bf x}) \varphi ({\bf x}) \,\hbox{d}^{n} {\bf x}.]In particular, polynomials over [{\bb R}^{n}] define tempered distributions, and so do functions in [{\scr S}]. The latter remark, together with the transposition identity (Section[link]), invites the extension of [{\scr F}] and [\bar{\scr F}] from [{\scr S}] to [{\scr S}\,'].

to end of page
to top of page