Introduction

Bricogne, G.

doi:10.1107/97809553602060000551

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

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 34 | 1 | 2 |

Section 1.3.2.4.1. Introduction

G. Bricogne^a

^a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, 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). 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), Wiener (1933), Titchmarsh (1948), Katznelson (1968), Sneddon (1951, 1972), and Dym & McKean (1972) 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.]Google Scholar

Dym, H. & McKean, H. P. (1972). Fourier series and integrals. New York and London: Academic Press.Google Scholar

Katznelson, Y. (1968). An introduction to harmonic analysis. New York: John Wiley.Google Scholar

Sneddon, I. N. (1951). Fourier transforms. New York: McGraw-Hill.Google Scholar

Sneddon, I. N. (1972). The use of integral transforms. New York: McGraw-Hill.Google Scholar

Titchmarsh, E. C. (1948). Introduction to the theory of Fourier integrals. Oxford: Clarendon Press.Google Scholar

Wiener, N. (1933). The Fourier integral and certain of its applications. Cambridge University Press. [Reprinted by Dover Publications, New York, 1959.]Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 34