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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 39   | 1 | 2 |

## Section 1.3.2.5.1. Introduction

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

#### 1.3.2.5.1. Introduction

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It was found in Section 1.3.2.4.2 that the usual space of test functions is not invariant under and . By contrast, the space of infinitely differentiable rapidly decreasing functions is invariant under and , and furthermore transposition formulae such ashold for all . It is precisely this type of transposition which was used successfully in Sections 1.3.2.3.9.1 and 1.3.2.3.9.3 to define the derivatives of distributions and their products with smooth functions.

This suggests using instead of as a space of test functions ϕ, and defining the Fourier transform of a distribution T bywhenever T is capable of being extended from to while remaining continuous. It is this latter proviso which will be subsumed under the adjective `tempered'. As was the case with the construction of , it is the definition of a sufficiently strong topology (i.e. notion of convergence) in which will play a key role in transferring to the elements of its topological dual (called tempered distributions) all the properties of the Fourier transformation.

Besides the general references to distribution theory mentioned in Section 1.3.2.3.1 the reader may consult the books by Zemanian (1965, 1968). Lavoine (1963) contains tables of Fourier transforms of distributions.

### References

Lavoine, J. (1963). Transformation de Fourier des Pseudo-fonctions, avec Tables de Nouvelles Transformées. Paris: Editions du CNRS.
Zemanian, A. H. (1965). Distribution Theory and Transform Analysis. New York: McGraw-Hill.
Zemanian, A. H. (1968). Generalised Integral Transformations. New York: Interscience.