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 1.3.2.4.2.8. Differentiation

G. Bricognea

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1.3.2.4.2.8. Differentiation

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Let us now suppose that [n = 1] and that [f \in L^{1} ({\bb R})] is differentiable with [f' \in L^{1} ({\bb R})]. Integration by parts yields[\eqalign{{\scr F}[\,f'] (\xi) &= {\textstyle\int\limits_{-\infty}^{+\infty}} f' (x) \exp (-2\pi i\xi \cdot x) \,\hbox{d}x\cr &= [\,f(x) \exp (-2\pi i\xi \cdot x)]_{-\infty}^{+\infty}\cr &\quad + 2\pi i\xi {\textstyle\int\limits_{-\infty}^{+\infty}} f(x) \exp (-2\pi i\xi \cdot x) \,\hbox{d}x.}]Since f′ is summable, f has a limit when [x \rightarrow \pm \infty], and this limit must be 0 since f is summable. Therefore[{\scr F}[\,f'] (\xi) = (2\pi i\xi) {\scr F}[\,f] (\xi)]with the bound[\|2\pi \xi {\scr F}[\,f]\|_{\infty} \leq \|\,f'\|_{1}]so that [|{\scr F}[\,f] (\xi)|] decreases faster than [1/|\xi| \rightarrow \infty].

This result can be easily extended to several dimensions and to any multi-index m: if f is summable and has continuous summable partial derivatives up to order [|{\bf m}|], then[{\scr F}[D^{{\bf m}} f] ({\boldxi}) = (2\pi i{\boldxi})^{{\bf m}} {\scr F}[\,f] ({\boldxi})]and[\|(2\pi {\boldxi})^{{\bf m}} {\scr F}[\,f]\|_{\infty} \leq \|D^{{\bf m}} f\|_{1}.]

Similar results hold for [\bar{\scr F}], with [2\pi i{\boldxi}] replaced by [-2\pi i{\boldxi}]. Thus, the more differentiable f is, with summable derivatives, the faster [{\scr F}[\,f]] and [\bar{\scr F}[\,f]] decrease at infinity.

The property of turning differentiation into multiplication by a monomial has many important applications in crystallography, for instance differential syntheses (Sections 1.3.4.2.1.9[link], 1.3.4.4.7.2[link], 1.3.4.4.7.5[link]) and moment-generating functions [Section 1.3.4.5.2.1[link](c[link])].








































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