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. 45
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The space of (equivalence classes of) square-integrable complex-valued functions f on the circle is contained in , since by the Cauchy–Schwarz inequality Thus all the results derived for hold for , a great simplification over the situation in or where neither nor was contained in the other.
However, more can be proved in , because is a Hilbert space (Section 1.3.2.2.4) for the inner product and because the family of functions constitutes an orthonormal Hilbert basis for .
The sequence of Fourier coefficients of belongs to the space of square-summable sequences: Conversely, every element of is the sequence of Fourier coefficients of a unique function in . The inner product makes into a Hilbert space, and the map from to established by the Fourier transformation is an isometry (Parseval/Plancherel): or equivalently: This is a useful property in applications, since (f, g) may be calculated either from f and g themselves, or from their Fourier coefficients and (see Section 1.3.4.4.6) for crystallographic applications).
By virtue of the orthogonality of the basis , the partial sum is the best mean-square fit to f in the linear subspace of spanned by , and hence (Bessel's inequality)