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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 36   | 1 | 2 |

## Section 1.3.2.4.2.8. Differentiation

G. Bricognea

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#### 1.3.2.4.2.8. Differentiation

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Let us now suppose that and that is differentiable with . Integration by parts yieldsSince f′ is summable, f has a limit when , and this limit must be 0 since f is summable. Thereforewith the boundso that decreases faster than .

This result can be easily extended to several dimensions and to any multi-index m: if f is summable and has continuous summable partial derivatives up to order , thenand

Similar results hold for , with replaced by . Thus, the more differentiable f is, with summable derivatives, the faster and decrease at infinity.

The property of turning differentiation into multiplication by a monomial has many important applications in crystallography, for instance differential syntheses (Sections 1.3.4.2.1.9, 1.3.4.4.7.2, 1.3.4.4.7.5) and moment-generating functions [Section 1.3.4.5.2.1(c)].