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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 35   | 1 | 2 |

Section 1.3.2.4.1. 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

1.3.2.4.1. Introduction

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Given a complex-valued function f on [{\bb R}^{n}] subject to suitable regularity conditions, its Fourier transform [{\scr F}[\,f]] and Fourier cotransform [\bar{\scr F}[\,f]] are defined as follows:[\eqalign{{\scr F}[\,f] (\xi) &= {\textstyle\int\limits_{{\bb R}^{n}}} f({\bf x}) \exp (-2\pi i {\boldxi} \cdot {\bf x}) \,\hbox{d}^{n} {\bf x}\cr \bar{\scr F}[\,f] (\xi) &= {\textstyle\int\limits_{{\bb R}^{n}}} f({\bf x}) \exp (+2\pi i {\boldxi} \cdot {\bf x}) \,\hbox{d}^{n} {\bf x},}]where [{\boldxi} \cdot {\bf x} = {\textstyle\sum_{i=1}^{n}} \xi_{i}x_{i}] is the ordinary scalar product. The terminology and sign conventions given above are the standard ones in mathematics; those used in crystallography are slightly different (see Section 1.3.4.2.1.1[link]). These transforms enjoy a number of remarkable properties, whose natural settings entail different regularity assumptions on f: for instance, properties relating to convolution are best treated in [L^{1} ({\bb R}^{n})], while Parseval's theorem requires the Hilbert space structure of [L^{2} ({\bb R}^{n})]. After a brief review of these classical properties, the Fourier transformation will be examined in a space [{\scr S}({\bb R}^{n})] particularly well suited to accommodating the full range of its properties, which will later serve as a space of test functions to extend the Fourier transformation to distributions.

There exists an abundant literature on the `Fourier integral'. The books by Carslaw (1930)[link], Wiener (1933)[link], Titchmarsh (1948)[link], Katznelson (1968)[link], Sneddon (1951[link], 1972[link]), and Dym & McKean (1972)[link] are particularly recommended.

References

Carslaw, H. S. (1930). An Introduction to the Theory of Fourier's Series and Integrals. London: Macmillan. [Reprinted by Dover Publications, New York, 1950.]
Dym, H. & McKean, H. P. (1972). Fourier Series and Integrals. New York, London: Academic Press.
Katznelson, Y. (1968). An Introduction to Harmonic Analysis. New York: John Wiley.
Sneddon, I. N. (1951). Fourier Transforms. New York: McGraw-Hill.
Sneddon, I. N. (1972). The Use of Integral Transforms. New York: McGraw-Hill.
Titchmarsh, E. C. (1948). Introduction to the Theory of Fourier Integrals. Oxford: Clarendon Press.
Wiener, N. (1933). The Fourier Integral and Certain of its Applications. Cambridge University Press. [Reprinted by Dover Publications, New York, 1959.]








































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