Multiplication of distributions by functions

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, p. 32 | 1 | 2 |

Section 1.3.2.3.9.3. Multiplication of distributions by 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

1.3.2.3.9.3. Multiplication of distributions by functions

| top | pdf |

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 1.3.2.5.8) 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 1.3.2.6.4, 1.3.2.6.6). 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 $[D^{\bf p}(\alpha T) = {\displaystyle\sum\limits_{{\bf q}\leq {\bf p}}} \pmatrix{{\bf p}\cr {\bf q}\cr} (D^{{\bf p}-{\bf q}} \alpha) ({\bf 0}) D^{{\bf q}}T.]$

References

International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 32