Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

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

Section 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 Introduction

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It was found in Section[link] that the usual space of test functions [{\scr D}] is not invariant under [{\scr F}] and [\bar{\scr F}]. By contrast, the space [{\scr S}] of infinitely differentiable rapidly decreasing functions is invariant under [{\scr F}] and [\bar{\scr F}], and furthermore transposition formulae such as[\langle {\scr F}[\,f], g\rangle = \langle \,f, {\scr F}[g]\rangle]hold for all [f, g \in {\scr S}]. It is precisely this type of transposition which was used successfully in Sections[link] and[link] to define the derivatives of distributions and their products with smooth functions.

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

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


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.

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