The convolution theorem and the isometry property

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. 36-37 | 1 | 2 |

Section 1.3.2.4.3.5. The convolution theorem and the isometry property

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

1.3.2.4.3.5. The convolution theorem and the isometry property

| top | pdf |

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 1.3.2.4.2.2) 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 1.3.4.4.7).

References

International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 36-37