Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 37   | 1 | 2 |

Section The convolution theorem and the isometry property

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 The convolution theorem and the isometry property

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In [L^{2}], the convolution theorem (when applicable) and the Parseval/Plancherel theorem are not independent. Suppose that f, g, [f \times g] and [f * g] are all in [L^{2}] (without questioning whether these properties are independent). Then [f * g] may be written in terms of the inner product in [L^{2}] as follows:[(\,f * g)({\bf x}) = {\textstyle\int\limits_{{\bb R}^{n}}} f({\bf x} - {\bf y})g({\bf y}) \,\hbox{d}^{n}{\bf y} = {\textstyle\int\limits_{{\bb R}^{n}}} \overline{\breve{\bar{f}}({\bf y} - {\bf x})}g({\bf y}) \,\hbox{d}^{n}{\bf y},]i.e.[(\,f * g)({\bf x}) = (\tau_{\bf x}\,\breve{\bar{f}}, g).]

Invoking the isometry property, we may rewrite the right-hand side as[\eqalign{({\scr F}[\tau_{\bf x}\,\breve{\bar{f}}], {\scr F}[g]) &= (\exp (- 2\pi i{\bf x} \cdot {\boldxi}) \overline{{\scr F}[\,f]_{\boldxi}}, {\scr F}[g]_{\boldxi})\cr &= {\textstyle\int\limits_{{\bb R}^{n}}} ({\scr F}[\,f] \times {\scr F}[g])({\bf x})\cr &\quad \times \exp (+ 2\pi i{\bf x} \cdot {\boldxi}) \,\hbox{d}^{n}{\boldxi}\cr &= \bar{\scr F}[{\scr F}[\,f] \times {\scr F}[g]],}]so that the initial identity yields the convolution theorem.

To obtain the converse implication, note that[\eqalign{(\,f, g) &= {\textstyle\int\limits_{{\bb R}^{n}}} \overline{f({\bf y})}g({\bf y}) \,\hbox{d}^{n}{\bf y} = (\, \breve{\bar{f}} * g)({\bf 0})\cr &= \bar{\scr F}[{\scr F}[\,\breve{\bar{ f}}] \times {\scr F}[g]]({\bf 0})\cr &= {\textstyle\int\limits_{{\bb R}^{n}}} \overline{{\scr F}[\,f]({\boldxi})} {\scr F}[g]({\boldxi}) \,\hbox{d}^{n}{\boldxi} = ({\scr F}[\,f], {\scr F}[g]),}]where conjugate symmetry (Section[link]) has been used.

These relations have an important application in the calculation by Fourier transform methods of the derivatives used in the refinement of macromolecular structures (Section[link]).

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