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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 37-38   | 1 | 2 |

## Section 1.3.2.4.4.1. Definition and properties of

G. Bricognea

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#### 1.3.2.4.4.1. Definition and properties of

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The duality established in Sections 1.3.2.4.2.8 and 1.3.2.4.2.9 between the local differentiability of a function and the rate of decrease at infinity of its Fourier transform prompts one to consider the space of functions f on which are infinitely differentiable and all of whose derivatives are rapidly decreasing, so that for all multi-indices k and pThe product of by any polynomial over is still in ( is an algebra over the ring of polynomials). Furthermore, is invariant under translations and differentiation.

If , then its transforms and are

 (i) infinitely differentiable because f is rapidly decreasing; (ii) rapidly decreasing because f is infinitely differentiable;

hence and are in : is invariant under and .

Since and , all properties of and already encountered above are enjoyed by functions of , with all restrictions on differentiability and/or integrability lifted. For instance, given two functions f and g in , then both fg and are in (which was not the case with nor with ) so that the reciprocity theorem inherited from allows one to state the reverse of the convolution theorem first established in :