International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 4247

Let be the subset of consisting of those points with (signed) integer coordinates; it is an ndimensional lattice, i.e. a free Abelian group on n generators. A particularly simple set of n generators is given by the standard basis of , and hence will be called the standard lattice in . Any other `nonstandard' ndimensional lattice Λ in is the image of this standard lattice by a general linear transformation.
If we identify any two points in whose coordinates are congruent modulo , i.e. differ by a vector in , we obtain the standard ntorus . The latter may be viewed as , i.e. as the Cartesian product of n circles. The same identification may be carried out modulo a nonstandard lattice Λ, yielding a nonstandard ntorus . The correspondence to crystallographic terminology is that `standard' coordinates over the standard 3torus are called `fractional' coordinates over the unit cell; while Cartesian coordinates, e.g. in ångströms, constitute a set of nonstandard coordinates.
Finally, we will denote by I the unit cube and by the subset
A distribution is called periodic with period lattice (or periodic) if for all (in crystallography the period lattice is the direct lattice).
Given a distribution with compact support , then is a periodic distribution. Note that we may write , where consists of Dirac δ's at all nodes of the period lattice .
Conversely, any periodic distribution T may be written as for some . To retrieve such a `motif' from T, a function ψ will be constructed in such a way that (hence has compact support) and ; then . Indicator functions (Section 1.3.2.2) such as or cannot be used directly, since they are discontinuous; but regularized versions of them may be constructed by convolution (see Section 1.3.2.3.9.7) as , with and η such that on and outside . Then the functionhas the desired property. The sum in the denominator contains at most nonzero terms at any given point x and acts as a smoothly varying `multiplicity correction'.
Throughout this section, `periodic' will mean `periodic'.
Let , and let [s] denote the largest integer . For , let be the unique vector with . If , then if and only if . The image of the map is thus modulo , or .
If f is a periodic function over , then implies ; we may thus define a function over by putting for any such that . Conversely, if is a function over , then we may define a function f over by putting , and f will be periodic. Periodic functions over may thus be identified with functions over , and this identification preserves the notions of convergence, local summability and differentiability.
Given , we may definesince the sum only contains finitely many nonzero terms; ϕ is periodic, and . Conversely, if we may define periodic by , and by putting with ψ constructed as above.
By transposition, a distribution defines a unique periodic distribution by ; conversely, periodic defines uniquely by .
We may therefore identify periodic distributions over with distributions over . We will, however, use mostly the former presentation, as it is more closely related to the crystallographer's perception of periodicity (see Section 1.3.4.1).
The content of this section is perhaps the central result in the relation between Fourier theory and crystallography (Section 1.3.4.2.1.1).
Let with r defined as in Section 1.3.2.6.2. Then , hence , so that : periodic distributions are tempered, hence have a Fourier transform. The convolution theorem (Section 1.3.2.5.8) is applicable, giving:and similarly for .
It is readily shown that Q is tempered and periodic, so that , while the periodicity of r implies thatSince the first factors have single isolated zeros at in , (see Section 1.3.2.3.9.4) and hence by periodicity ; convoluting with shows that . Thus we have the fundamental result: so thati.e., according to Section 1.3.2.3.9.3,
The righthand side is a weighted lattice distribution, whose nodes are weighted by the sample values of the transform of the motif at those nodes. Since , the latter values may be writtenBy the structure theorem for distributions with compact support (Section 1.3.2.3.9.7), is a derivative of finite order of a continuous function; therefore, from Section 1.3.2.4.2.8 and Section 1.3.2.5.8, grows at most polynomially as (see also Section 1.3.2.6.10.3 about this property). Conversely, let be a weighted lattice distribution such that the weights grow at most polynomially as . Then W is a tempered distribution, whose Fourier cotransform is periodic. If T is now written as for some , then by the reciprocity theoremAlthough the choice of is not unique, and need not yield back the same motif as may have been used to build T initially, different choices of will lead to the same coefficients because of the periodicity of .
The Fourier transformation thus establishes a duality between periodic distributions and weighted lattice distributions. The pair of relationsare referred to as the Fourier analysis and the Fourier synthesis of T, respectively (there is a discrepancy between this terminology and the crystallographic one, see Section 1.3.4.2.1.1). In other words, any periodic distribution may be represented by a Fourier series (ii), whose coefficients are calculated by (i). The convergence of (ii) towards T in will be investigated later (Section 1.3.2.6.10).
Let Λ denote the nonstandard lattice consisting of all vectors of the form , where the are rational integers and are n linearly independent vectors in . Let R be the corresponding lattice distribution: .
Let A be the nonsingular matrix whose successive columns are the coordinates of vectors in the standard basis of ; A will be called the period matrix of Λ, and the mapping will be denoted by A. According to Section 1.3.2.3.9.5 we havefor any , and hence . By Fourier transformation, according to Section 1.3.2.5.5,which we write:with
is a lattice distribution:associated with the reciprocal lattice whose basis vectors are the columns of . Since the latter matrix is equal to the adjoint matrix (i.e. the matrix of cofactors) of A divided by det A, the components of the reciprocal basis vectors can be written down explicitly (see Section 1.3.4.2.1.1 for the crystallographic case ).
A distribution T will be called Λperiodic if for all ; as previously, T may be written for some motif distribution with compact support. By Fourier transformation,so that is a weighted reciprocallattice distribution, the weight attached to node being times the value of the Fourier transform of the motif .
This result may be further simplified if T and its motif are referred to the standard period lattice by defining t and so that , , . Thenhenceso thatin nonstandard coordinates, whilein standard coordinates.
The reciprocity theorem may then be written:in nonstandard coordinates, or equivalently:in standard coordinates. It gives an ndimensional Fourier series representation for any periodic distribution over . The convergence of such series in will be examined in Section 1.3.2.6.10.
Let be a distribution with compact support (the `motif'). Its Fourier transform is analytic (Section 1.3.2.5.4) and may thus be used as a multiplier.
We may rephrase the preceding results as follows:
Thus the Fourier transformation establishes a duality between the periodization of a distribution by a period lattice Λ and the sampling of its transform at the nodes of lattice reciprocal to Λ. This is a particular instance of the convolution theorem of Section 1.3.2.5.8.
At this point it is traditional to break the symmetry between and which distribution theory has enabled us to preserve even in the presence of periodicity, and to perform two distinct identifications:
Let , so that . Let R be the lattice distribution associated to lattice Λ, with period matrix A, and let be associated to the reciprocal lattice . Then we may write:i.e.
This identity, which also holds for , is called the Poisson summation formula. Its usefulness follows from the fact that the speed of decrease at infinity of ϕ and are inversely related (Section 1.3.2.4.4.3), so that if one of the series (say, the lefthand side) is slowly convergent, the other (say, the righthand side) will be rapidly convergent. This procedure has been used by Ewald (1921) [see also Bertaut (1952), Born & Huang (1954)] to evaluate lattice sums (Madelung constants) involved in the calculation of the internal electrostatic energy of crystals (see Chapter 3.4 in this volume on convergence acceleration techniques for crystallographic lattice sums and Chapter 3.5 on modern extensions of the Ewald summation method).
When ϕ is a multivariate Gaussianthenand Poisson's summation formula for a lattice with period matrix A reads:or equivalentlywith
Let and be two Λperiodic distributions, the motifs and having compact support. The convolution does not exist, because S and T do not satisfy the support condition (Section 1.3.2.3.9.7). However, the three distributions R, and do satisfy the generalized support condition, so that their convolution is defined; then, by associativity and commutativity:
By Fourier transformation and by the convolution theorem:Let , and be the sets of Fourier coefficients associated to S, T and , respectively. Identifying the coefficients of for yields the forward version of the convolution theorem for Fourier series:
The backward version of the theorem requires that T be infinitely differentiable. The distribution is then well defined and its Fourier coefficients are given by
Toeplitz forms were first investigated by Toeplitz (1907, 1910, 1911a). They occur in connection with the `trigonometric moment problem' (Shohat & Tamarkin, 1943; Akhiezer, 1965) and probability theory (Grenander, 1952) and play an important role in several direct approaches to the crystallographic phase problem [see Sections 1.3.4.2.1.10, 1.3.4.5.2.2(e)]. Many aspects of their theory and applications are presented in the book by Grenander & Szegö (1958).
Let be realvalued, so that its Fourier coefficients satisfy the relations . The Hermitian form in complex variablesis called the nth Toeplitz form associated to f. It is a straightforward consequence of the convolution theorem and of Parseval's identity that may be written:
It was shown independently by Toeplitz (1911b), Carathéodory (1911) and Herglotz (1911) that a function is almost everywhere nonnegative if and only if the Toeplitz forms associated to f are positive semidefinite for all values of n.
This is equivalent to the infinite system of determinantal inequalitiesThe are called Toeplitz determinants. Their application to the crystallographic phase problem is described in Section 1.3.4.2.1.10.
The eigenvalues of the Hermitian form are defined as the real roots of the characteristic equation . They will be denoted by
It is easily shown that if for all x, then for all n and all . As these bounds, and the distribution of the within these bounds, can be made more precise by introducing two new notions.
We may now state an important theorem of Szegö (1915, 1920). Let , and put , . If m and M are finite, then for any continuous function defined in the interval [m, M] we haveIn other words, the eigenvalues of the and the values of f on a regular subdivision of ]0, 1[ are equally distributed.
Further investigations into the spectra of Toeplitz matrices may be found in papers by Hartman & Wintner (1950, 1954), Kac et al. (1953), Widom (1965), and in the notes by Hirschman & Hughes (1977).
The investigation of the convergence of Fourier series and of more general trigonometric series has been the subject of intense study for over 150 years [see e.g. Zygmund (1976)]. It has been a constant source of new mathematical ideas and theories, being directly responsible for the birth of such fields as set theory, topology and functional analysis.
This section will briefly survey those aspects of the classical results in dimension 1 which are relevant to the practical use of Fourier series in crystallography. The books by Zygmund (1959), Tolstov (1962) and Katznelson (1968) are standard references in the field, and Dym & McKean (1972) is recommended as a stimulant.
The space consists of (equivalence classes of) complexvalued functions f on the circle which are summable, i.e. for whichIt is a convolution algebra: If f and g are in , then is in .
The mth Fourier coefficient of f,is bounded: , and by the Riemann–Lebesgue lemma as . By the convolution theorem, .
The pth partial sum of the Fourier series of f,may be written, by virtue of the convolution theorem, as , whereis the Dirichlet kernel. Because comprises numerous slowly decaying oscillations, both positive and negative, may not converge towards f in a strong sense as . Indeed, spectacular pathologies are known to exist where the partial sums, examined pointwise, diverge everywhere (Zygmund, 1959, Chapter VIII). When f is piecewise continuous, but presents isolated jumps, convergence near these jumps is marred by the Gibbs phenomenon: always `overshoots the mark' by about 9%, the area under the spurious peak tending to 0 as but not its height [see Larmor (1934) for the history of this phenomenon].
By contrast, the arithmetic mean of the partial sums, also called the pth Cesàro sum,converges to f in the sense of the norm: as . If furthermore f is continuous, then the convergence is uniform, i.e. the error is bounded everywhere by a quantity which goes to 0 as . It may be shown thatwhereis the Fejér kernel. has over the advantage of being everywhere positive, so that the Cesàro sums of a positive function f are always positive.
The de la Vallée Poussin kernelhas a trapezoidal distribution of coefficients and is such that if ; therefore is a trigonometric polynomial with the same Fourier coefficients as f over that range of values of m.
The Poisson kernelwith gives rise to an Abel summation procedure [Tolstov (1962, p. 162); Whittaker & Watson (1927, p. 57)] sinceCompared with the other kernels, has the disadvantage of not being a trigonometric polynomial; however, is the real part of the Cauchy kernel (Cartan, 1961; Ahlfors, 1966):and hence provides a link between trigonometric series and analytic functions of a complex variable.
Other methods of summation involve forming a moving average of f by convolution with other sequences of functions besides of which `tend towards δ' as . The convolution is performed by multiplying the Fourier coefficients of f by those of , so that one forms the quantitiesFor instance the `sigma factors' of Lanczos (Lanczos, 1966, p. 65), defined bylead to a summation procedure whose behaviour is intermediate between those using the Dirichlet and the Fejér kernels; it corresponds to forming a moving average of f by convolution withwhich is itself the convolution of a `rectangular pulse' of width and of the Dirichlet kernel of order p.
A review of the summation problem in crystallography is given in Section 1.3.4.2.1.3.
The space of (equivalence classes of) squareintegrable complexvalued 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 squaresummable 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 meansquare fit to f in the linear subspace of spanned by , and hence (Bessel's inequality)
The use of distributions enlarges considerably the range of behaviour which can be accommodated in a Fourier series, even in the case of general dimension n where classical theories meet with even more difficulties than in dimension 1.
Let be a sequence of complex numbers with growing at most polynomially as , say . Then the sequence is in and even defines a continuous function and an associated tempered distribution . Differentiation of times then yields a tempered distribution whose Fourier transform leads to the original sequence of coefficients. Conversely, by the structure theorem for distributions with compact support (Section 1.3.2.3.9.7), the motif of a periodic distribution is a derivative of finite order of a continuous function; hence its Fourier coefficients will grow at most polynomially with as .
Thus distribution theory allows the manipulation of Fourier series whose coefficients exhibit polynomial growth as their order goes to infinity, while those derived from functions had to tend to 0 by virtue of the Riemann–Lebesgue lemma. The distributiontheoretic approach to Fourier series holds even in the case of general dimension n, where classical theories meet with even more difficulties (see Ash, 1976) than in dimension 1.
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