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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 32-33   | 1 | 2 |

## Section 1.3.2.3.9.4. Division of distributions by functions

G. Bricognea

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#### 1.3.2.3.9.4. Division of distributions by functions

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Given a distribution S on and an infinitely differentiable multiplier function α, the division problem consists in finding a distribution T such that .

If α never vanishes, is the unique answer. If , and if α has only isolated zeros of finite order, it can be reduced to a collection of cases where the multiplier is , for which the general solution can be shown to be of the formwhere U is a particular solution of the division problem and the are arbitrary constants.

In dimension , 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.