Division of distributions by functions

Bricogne, G.

doi:10.1107/97809553602060000551

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 33 | 1 | 2 |

Section 1.3.2.3.9.4. Division 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

1.3.2.3.9.4. Division of distributions by functions

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Given a distribution S on $[{\bb R}^{n}]$ and an infinitely differentiable multiplier function α, the division problem consists in finding a distribution T such that $[\alpha T = S]$ .

If α never vanishes, $[T = S/\alpha]$ is the unique answer. If [n = 1] , and if α has only isolated zeros of finite order, it can be reduced to a collection of cases where the multiplier is $[x^{m}]$ , for which the general solution can be shown to be of the form $[T = U + {\textstyle\sum\limits_{i=0}^{m-1}} c_{i}\delta^{(i)},]$ where U is a particular solution of the division problem $[x^{m} U = S]$ and the $[c_{i}]$ are arbitrary constants.

In dimension $[n \gt 1]$ , the problem is much more difficult, but is of fundamental importance in the theory of linear partial differential equations, since the Fourier transformation turns the problem of solving these into a division problem for distributions [see Hörmander (1963)].

References

Hörmander, L. (1963). Linear partial differential operators. Berlin: Springer-Verlag.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 33