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

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

## Section 1.3.2.3.9.3. Multiplication of distributions by functions

G. Bricognea

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#### 1.3.2.3.9.3. Multiplication of distributions by functions

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The product of a distribution T on by a function α over will be defined by transposition:In order that be a distribution, the mapping must send 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 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, is a distribution of order , and the following formula holds for all with :

The derivative of a product is easily shown to beand generally for any multi-index p