Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 36   | 1 | 2 |

Section Riemann–Lebesgue lemma

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 Riemann–Lebesgue lemma

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If [f \in L^{1} ({\bb R}^{n})], i.e. is summable, then [{\scr F}[\,f]] and [\bar{\scr F}[\,f]] exist and are continuous and essentially bounded:[\|{\scr F}[\,f]\|_{\infty} = \|\bar{\scr F}[\,f]\|_{\infty} \leq \|\,f\|_{1}.]In fact one has the much stronger property, whose statement constitutes the Riemann–Lebesgue lemma, that [{\scr F}[\,f] ({\boldxi})] and [\bar{\scr F}[\,f] ({\boldxi})] both tend to zero as [\|{\boldxi}\| \rightarrow \infty].

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