International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 34
|
Given a complex-valued function f on subject to suitable regularity conditions, its Fourier transform and Fourier cotransform are defined as follows: where 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 , while Parseval's theorem requires the Hilbert space structure of . After a brief review of these classical properties, the Fourier transformation will be examined in a space 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 ScholarDym, 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