Chapter 1.3 Fourier transforms in crystallography: theory, algorithms and applications

Reciprocal space

Second Online Edition (2010)

Part 1. General relationships and techniques

  1. G. Bricogne

Published Online: 1 JUN 2010

DOI: 10.1107/97809553602060000760

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Bricogne, G. 2010. Fourier transforms in crystallography: theory, algorithms and applications. International Tables for Crystallography. B:1:1.3:24–113.

Author Information

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

Publication History

  1. Published Online: 1 JUN 2010


In the first part of this chapter, the mathematical theory of the Fourier transformation is cast in the language of Schwartz’s theory of distributions, allowing Fourier transforms, Fourier series and discrete Fourier transforms to be treated together. Next the numerical computation of the discrete Fourier transform is discussed. One-dimensional algorithms are examined first, including the Cooley–Tukey algorithm, the Good (or prime factor) algorithm, the Rader algorithm and the Winograd algorithms. Multidimensional algorithms are then covered. The last part of the chapter surveys the crystallographic applications of Fourier transforms.


  • Fourier transforms;
  • topology;
  • distributions;
  • convergence;
  • convolution;
  • convolution theorem;
  • Heisenberg’s inequality;
  • Hardy’s theorem;
  • reciprocity theorem;
  • Sobolev spaces;
  • Fourier series;
  • Poisson summation formula;
  • Toeplitz forms;
  • Szegö’s theorem;
  • correlation;
  • correlation functions;
  • sampling;
  • Cooley–Tukey algorithm;
  • Good algorithm;
  • prime factor algorithm;
  • Chinese remainder theorem;
  • Rader algorithm;
  • Winograd algorithms;
  • multidimensional algorithms;
  • Nussbaumer–Quandalle algorithm;
  • structure factors;
  • Friedel’s law;
  • Patterson functions;
  • determinantal inequalities;
  • crystallographic groups;
  • Parseval’s theorem;
  • generalized multiplexing;
  • electron density;
  • molecular envelopes;
  • noncrystallographic symmetry;
  • molecular averaging;
  • Green’s theorem;
  • least-squares methods;
  • fast Fourier transforms;
  • helical symmetry;
  • fibres;
  • macromolecular crystallography;
  • probability theory