Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 41   | 1 | 2 |

Section Multiplication and convolution

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 Multiplication and convolution

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Multiplier functions [\alpha ({\bf x})] for tempered distributions must be infinitely differentiable, as for ordinary distributions; furthermore, they must grow sufficiently slowly as [\|x\| \rightarrow \infty] to ensure that [\alpha \varphi \in {\scr S}] for all [\varphi \in {\scr S}] and that the map [\varphi \,\longmapsto\, \alpha \varphi] is continuous for the topology of [{\scr S}]. This leads to choosing for multipliers the subspace [{\scr O}_{M}] consisting of functions [\alpha \in {\scr E}] of polynomial growth. It can be shown that if f is in [{\scr O}_{M}], then the associated distribution [T_{f}] is in [{\scr S}\,'] (i.e. is a tempered distribution); and that conversely if T is in [{\scr S}\,', \mu * T] is in [{\scr O}_{M}] for all [\mu \in {\scr D}].

Corresponding restrictions must be imposed to define the space [{\scr O\,}_{\!C}'] of those distributions T whose convolution [S * T] with a tempered distribution S is still a tempered distribution: T must be such that, for all [\varphi \in {\scr S}, \theta ({\bf x}) = \langle T_{\bf y}, \varphi ({\bf x} + {\bf y})\rangle] is in [{\scr S}]; and such that the map [\varphi \,\longmapsto\, \theta] be continuous for the topology of [{\scr S}]. This implies that S is `rapidly decreasing'. It can be shown that if f is in [{\scr S}], then the associated distribution [T_{f}] is in [{\scr O\,}'_{\!C}]; and that conversely if T is in [{\scr O\,}'_{\!C}, \mu * T] is in [{\scr S}] for all [\mu \in {\scr D}].

The two spaces [{\scr O}_{M}] and [{\scr O\,}'_{\!C}] are mapped into each other by the Fourier transformation[\eqalign{{\scr F}({\scr O}_{M}) &= \bar{\scr F}({\scr O}_{M}) = {\scr O\,}'_{\!C} \cr {\scr F}({\scr O\,}'_{\!C}) &= \bar{\scr F}({\scr O\,}'_{\!C}) = {\scr O}_{M}}]and the convolution theorem takes the form[\eqalign{{\scr F}[\alpha S] &= {\scr F}[\alpha] * {\scr F}[S] \quad\, S \in {\scr S}\,', \alpha \in {\scr O}_{M},{\scr F}[\alpha] \in {\scr O\,}'_{\!C}\hbox{\semi}\cr {\scr F}[S * T] &= {\scr F}[S] \times {\scr F}[T] \quad S \in {\scr S}\,', T \in {\scr O\,}'_{\!C},{\scr F}[T] \in {\scr O}_{M}.}]The same identities hold for [\bar{\scr F}]. Taken together with the reciprocity theorem, these show that [{\scr F}] and [\bar{\scr F}] establish mutually inverse isomorphisms between [{\scr O}_{M}] and [{\scr O\,}'_{\!C}], and exchange multiplication for convolution in [{\scr S}\,'].

It may be noticed that most of the basic properties of [{\scr F}] and [\bar{\scr F}] may be deduced from this theorem and from the properties of δ. Differentiation operators [D^{\bf m}] and translation operators [\tau_{\bf a}] are convolutions with [D^{\bf m}\delta] and [\tau_{\bf a} \delta]; they are turned, respectively, into multiplication by monomials [(\pm 2\pi i{\boldxi})^{{\bf m}}] (the transforms of [D^{{\bf m}}\delta]) or by phase factors [\exp(\pm 2 \pi i{\boldxi} \cdot {\boldalpha})] (the transforms of [\tau_{\bf a}\delta]).

Another consequence of the convolution theorem is the duality established by the Fourier transformation between sections and projections of a function and its transform. For instance, in [{\bb R}^{3}], the projection of [f(x, y, z)] on the x, y plane along the z axis may be written[(\delta_{x} \otimes \delta_{y} \otimes 1_{z}) * f\hbox{\semi}]its Fourier transform is then[(1_{\xi} \otimes 1_{\eta} \otimes \delta_{\zeta}) \times {\scr F}[\,f],]which is the section of [{\scr F}[\,f]] by the plane [\zeta = 0], orthogonal to the z axis used for projection. There are numerous applications of this property in crystallography (Section[link]) and in fibre diffraction (Section[link]).

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