Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 32   | 1 | 2 |

Section Multiplication of distributions by functions

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 of distributions by functions

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The product [\alpha T] of a distribution T on [{\bb R}^{n}] by a function α over [{\bb R}^{n}] will be defined by transposition:[\langle \alpha T, \varphi \rangle = \langle T, \alpha \varphi \rangle \quad \hbox{for all } \varphi \in {\scr D}.]In order that [\alpha T] be a distribution, the mapping [\varphi \,\longmapsto\, \alpha \varphi] must send [{\scr D}({\bb R}^{n})] continuously into itself; hence the multipliers α must be infinitely differentiable. The product of two general distributions cannot be defined. The need for a careful treatment of multipliers of distributions will become clear when it is later shown (Section[link]) that the Fourier transformation turns convolutions into multiplications and vice versa.

If T is a distribution of order m, then α needs only have continuous derivatives up to order m. For instance, δ is a distribution of order zero, and [\alpha \delta = \alpha ({\bf 0}) \delta] is a distribution provided α is continuous; this relation is of fundamental importance in the theory of sampling and of the properties of the Fourier transformation related to sampling (Sections[link],[link]). More generally, [D^{{\bf p}}\delta] is a distribution of order [|{\bf p}|], and the following formula holds for all [\alpha \in {\scr D}^{(m)}] with [m = |{\bf p}|]:[\alpha (D^{{\bf p}}\delta) = {\displaystyle\sum\limits_{{\bf q} \leq {\bf p}}} (-1)^{|{\bf p}-{\bf q}|} \pmatrix{{\bf p}\cr {\bf q}\cr} (D^{{\bf p}-{\bf q}} \alpha) ({\bf 0}) D^{\bf q}\delta.]

The derivative of a product is easily shown to be[\partial_{i}(\alpha T) = (\partial_{i}\alpha) T + \alpha (\partial_{i}T)]and generally for any multi-index p[D^{\bf p}(\alpha T) = {\displaystyle\sum\limits_{{\bf q}\leq {\bf p}}} \pmatrix{{\bf p}\cr {\bf q}\cr} (D^{{\bf p}-{\bf q}} \alpha) ({\bf 0}) D^{{\bf q}}T.]

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