Differentiation

Bricogne, G.

doi:10.1107/97809553602060000551

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 35-36 | 1 | 2 |

Section 1.3.2.4.2.8. Differentiation

G. Bricogne^a

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

1.3.2.4.2.8. Differentiation

| top | pdf |

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, 1.3.4.4.7.2, 1.3.4.4.7.5) and moment-generating functions [Section 1.3.4.5.2.1(c )].

References

International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 35-36