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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 47   | 1 | 2 |

## Section 1.3.2.6.10.2. Classical theory

G. Bricognea

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#### 1.3.2.6.10.2. Classical theory

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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 inequalityThus 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 productand 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 productmakes 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)