Transforms of δ-functions

Bricogne, G.

doi:10.1107/97809553602060000551

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

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 39-40 | 1 | 2 |

Section 1.3.2.5.6. Transforms of δ-functions

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

1.3.2.5.6. Transforms of δ-functions

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Since δ has compact support, $[{\scr F}[\delta_{\bf x}]_{\boldxi} = \langle \delta_{\bf x}, \exp (-2\pi i{\boldxi} \cdot {\bf x})\rangle = 1_{\boldxi},\quad i.e.\ {\scr F}[\delta] = 1.]$ It is instructive to show that conversely $[{\scr F}[1] = \delta]$ without invoking the reciprocity theorem. Since $[\partial_{j} 1 = 0]$ for all $[j = 1, \ldots, n]$ , it follows from Section 1.3.2.3.9.4 that $[{\scr F}[1] = c\delta]$ ; the constant c can be determined by using the invariance of the standard Gaussian G established in Section 1.3.2.4.3: $[\langle {\scr F}[1]_{\bf x}, G_{\bf x}\rangle = \langle 1_{\boldxi}, G_{\boldxi}\rangle = 1\hbox{;}]$ hence [c = 1] . Thus, $[{\scr F}[1] = \delta]$ .

The basic properties above then read (using multi-indices to denote differentiation): $[\eqalign{{\scr F}[\delta_{\bf x}^{({\bf m})}]_{\boldxi} = (2\pi i{\boldxi})^{{\bf m}}, \quad &{\scr F}[{\bf x}^{{\bf m}}]_{\boldxi} = (-2\pi i)^{-|{\bf m}|} \delta_{\boldxi}^{({\bf m})}\hbox{;} \cr {\scr F}[\delta_{\bf a}]_{\boldxi} = \exp (-2\pi i{\bf a} \cdot {\boldxi}), \quad &{\scr F}[\exp (2\pi i{\boldalpha} \cdot {\bf x})]_{\boldxi} = \delta_{\boldalpha},}]$ with analogous relations for $[\bar{\scr F}]$ , i becoming −i. Thus derivatives of δ are mapped to monomials (and vice versa), while translates of δ are mapped to `phase factors' (and vice versa).

References

International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 39-40