International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 39   | 1 | 2 |

Section 1.3.2.5.2.  [{\scr S}] as a test-function space

G. Bricognea

a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.2.5.2. [{\scr S}] as a test-function space

| top | pdf |

A notion of convergence has to be introduced in [{\scr S}({\bb R}^{n})] in order to be able to define and test the continuity of linear functionals on it.

A sequence [(\varphi_{j})] of functions in [{\scr S}] will be said to converge to 0 if, for any given multi-indices k and p, the sequence [({\bf x}^{{\bf k}}D^{{\bf p}} \varphi_{j})] tends to 0 uniformly on [{\bb R}^{n}].

It can be shown that [{\scr D}({\bb R}^{n})] is dense in [{\scr S}({\bb R}^{n})]. Translation is continuous for this topology. For any linear differential operator [P(D) = {\textstyle\sum_{\bf p}} a_{\bf p} D^{{\bf p}}] and any polynomial [Q({\bf x})] over [{\bb R}^{n}], [(\varphi_{j}) \rightarrow 0] implies [[Q({\bf x}) \times P(D)\varphi_{j}] \rightarrow 0] in the topology of [{\scr S}]. Therefore, differentiation and multiplication by polynomials are continuous for the topology on [{\scr S}].

The Fourier transformations [{\scr F}] and [\bar{\scr F}] are also continuous for the topology of [{\scr S}]. Indeed, let [(\varphi_{j})] converge to 0 for the topology on [{\scr S}]. Then, by Section 1.3.2.4.2[link], [\|(2\pi \boldxi)^{{\bf m}} D^{{\bf p}} ({\scr F}[\varphi_{j}])\|_{\infty} \leq \| D^{{\bf m}} [(2\pi {\bf x})^{{\bf p}} \varphi_{j}]\|_{1}.] The right-hand side tends to 0 as [j \rightarrow \infty] by definition of convergence in [{\scr S}], hence [\|\boldxi\|^{{\bf m}} D^{{\bf p}} ({\scr F}[\varphi_{j}]) \rightarrow 0] uniformly, so that [({\scr F}[\varphi_{j}]) \rightarrow 0] in [{\scr S}] as [j \rightarrow \infty]. The same proof applies to [\bar{\scr F}].








































to end of page
to top of page