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

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. Bricognea

aMRC 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)[link]].

References

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








































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