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

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

Section 1.3.2.4.2.7. Riemann–Lebesgue lemma

G. Bricognea

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1.3.2.4.2.7. 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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