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Crystals, X-rays and ProteinsComprehensive Protein Crystallography$
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Dennis Sherwood and Jon Cooper

Print publication date: 2010

Print ISBN-13: 9780199559046

Published to Oxford Scholarship Online: January 2011

DOI: 10.1093/acprof:oso/9780199559046.001.0001

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Fourier transforms and convolutions

Fourier transforms and convolutions

(p.133) 5 Fourier transforms and convolutions
Crystals, X-rays and Proteins

Dennis Sherwood

Jon Cooper

Oxford University Press

This chapter revisits the essential mathematics of integral calculus and introduces the important concept of the Fourier transform, the properties of which are elegantly demonstrated by sketching the curves of various functions. It demonstrates the essential principles of diffraction by determining the Fourier transforms of regularly repeating patterns which can be represented mathematically by Dirac delta functions — the very important concept of Fourier space (or reciprocal space) follows from this discussion. This section leads into a description of another highly important mathematical concept, the convolution. Convolutions allow two functions to be combined and provide an extremely elegant mathematical description of the crystalline state as well as an insight into one of crystallography's most important structure-solving tools, the Patterson function.

Keywords:   integrals, curve sketching, Fourier transform, inverse transform, real space, Fourier space, reciprocal space, delta functions, convolutions, the Patterson function

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