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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 37-38   | 1 | 2 |

Section 1.3.2.4.4.1. Definition and properties of [{\scr S}]

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

1.3.2.4.4.1. Definition and properties of [{\scr S}]

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The duality established in Sections 1.3.2.4.2.8[link] and 1.3.2.4.2.9[link] between the local differentiability of a function and the rate of decrease at infinity of its Fourier transform prompts one to consider the space [{\scr S}({\bb R}^{n})] of functions f on [{\bb R}^{n}] which are infinitely differentiable and all of whose derivatives are rapidly decreasing, so that for all multi-indices k and p[({\bf x}^{\bf k}D^{\bf p}f)({\bf x})\rightarrow 0 \ \hbox{ as } \ \|{\bf x}\|\rightarrow \infty.]The product of [f \in {\scr S}] by any polynomial over [{\bb R}^{n}] is still in [{\scr S}] ([{\scr S}] is an algebra over the ring of polynomials). Furthermore, [{\scr S}] is invariant under translations and differentiation.

If [f \in {\scr S}], then its transforms [{\scr F}[\,f]] and [\bar{\scr F}[\,f]] are

  • (i) infinitely differentiable because f is rapidly decreasing;

  • (ii) rapidly decreasing because f is infinitely differentiable;

hence [{\scr F}[\,f]] and [\bar{\scr F}[\,f]] are in [{\scr S}]: [{\scr S}] is invariant under [{\scr F}] and [\bar{\scr F}].

Since [L^{1} \supset {\scr S}] and [L^{2} \supset {\scr S}], all properties of [{\scr F}] and [\bar{\scr F}] already encountered above are enjoyed by functions of [{\scr S}], with all restrictions on differentiability and/or integrability lifted. For instance, given two functions f and g in [{\scr S}], then both fg and [f * g] are in [{\scr S}] (which was not the case with [L^{1}] nor with [L^{2}]) so that the reciprocity theorem inherited from [L^{2}][{\scr F}[\bar{\scr F}[\,f]] = f \quad\hbox{and}\quad \bar{\scr F}[{\scr F}[\,f]] = f]allows one to state the reverse of the convolution theorem first established in [L^{1}]:[\eqalign{{\scr F}[\,fg] &= {\scr F}[\,f] * {\scr F}[g]\cr \bar{\scr F}[\,fg] &= \bar{\scr F}[\,f] * \bar{\scr F}[g].}]








































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