Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 46   | 1 | 2 |

Section Classical [L^{1}] theory

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France Classical [L^{1}] theory

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The space [L^{1} ({\bb R} / {\bb Z})] consists of (equivalence classes of) complex-valued functions f on the circle which are summable, i.e. for which[\|\,f \|_{1} \equiv {\textstyle\int\limits_{0}^{1}}\, | \,f(x) | \,\hbox{d}x \,\lt\, + \infty.]It is a convolution algebra: If f and g are in [L^{1}], then [f * g] is in [L^{1}].

The mth Fourier coefficient [c_{m} (\,f)] of f,[c_{m} (\,f) = {\textstyle\int\limits_{0}^{1}}\, f(x) \exp (-2 \pi imx) \,\hbox{d}x]is bounded: [|c_{m} (\,f)| \leq \|\,f \|_{1}], and by the Riemann–Lebesgue lemma [c_{m} (\,f) \rightarrow 0] as [m \rightarrow \infty]. By the convolution theorem, [c_{m} (\,f * g) = c_{m} (\,f) c_{m} (g)].

The pth partial sum [S_{p}(\,f)] of the Fourier series of f,[S_{p}(\,f) (x) = {\textstyle\sum\limits_{|m|\leq p}} c_{m} (\,f) \exp (2 \pi imx),]may be written, by virtue of the convolution theorem, as [S_{p}(\,f) = D_{p} * f], where[D_{p} (x) = {\sum\limits_{|m|\leq p}} \exp (2 \pi imx) = {\sin [(2p + 1) \pi x] \over \sin \pi x}]is the Dirichlet kernel. Because [D_{p}] comprises numerous slowly decaying oscillations, both positive and negative, [S_{p}(\,f)] may not converge towards f in a strong sense as [p \rightarrow \infty]. Indeed, spectacular pathologies are known to exist where the partial sums, examined pointwise, diverge everywhere (Zygmund, 1959[link], Chapter VIII). When f is piecewise continuous, but presents isolated jumps, convergence near these jumps is marred by the Gibbs phenomenon: [S_{p}(\,f)] always `overshoots the mark' by about 9%, the area under the spurious peak tending to 0 as [p \rightarrow \infty] but not its height [see Larmor (1934)[link] for the history of this phenomenon].

By contrast, the arithmetic mean of the partial sums, also called the pth Cesàro sum,[C_{p}(\,f) = {1 \over p + 1} [S_{0}(\,f) + \ldots + S_{p}(\,f)],]converges to f in the sense of the [L^{1}] norm: [\|C_{p}(\,f) - f\|_{1} \rightarrow 0] as [p \rightarrow \infty]. 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 [p \rightarrow \infty]. It may be shown that[C_{p} (\,f) = F_{p} * f,]where[\eqalign{F_{p} (x) &= {\sum\limits_{|m| \leq p}} \left(1 - {|m| \over p + 1}\right) \exp (2 \pi imx) \cr &= {1 \over p + 1} \left[{\sin (p + 1) \pi x \over \sin \pi x}\right]^{2}}]is the Fejér kernel. [F_{p}] has over [D_{p}] the advantage of being everywhere positive, so that the Cesàro sums [C_{p} (\,f)] of a positive function f are always positive.

The de la Vallée Poussin kernel[V_{p} (x) = 2 F_{2p + 1} (x) - F_{p} (x)]has a trapezoidal distribution of coefficients and is such that [c_{m} (V_{p}) = 1] if [|m| \leq p + 1]; therefore [V_{p} * f] is a trigonometric polynomial with the same Fourier coefficients as f over that range of values of m.

The Poisson kernel[\eqalign{P_{r} (x) &= 1 + 2 {\sum\limits_{m = 1}^{\infty}} r^{m} \cos 2 \pi mx \cr &= {1 - r^{2} \over 1 - 2r \cos 2 \pi mx + r^{2}}}]with [0 \leq r \,\lt\, 1] gives rise to an Abel summation procedure [Tolstov (1962[link], p. 162); Whittaker & Watson (1927[link], p. 57)] since[(P_{r} * f) (x) = {\textstyle\sum\limits_{m \in {\bb Z}}} c_{m} (\,f) r^{|m|} \exp (2 \pi imx).]Compared with the other kernels, [P_{r}] has the disadvantage of not being a trigonometric polynomial; however, [P_{r}] is the real part of the Cauchy kernel (Cartan, 1961[link]; Ahlfors, 1966[link]):[P_{r} (x) = {\scr Re}\left[{1 + r \exp (2 \pi ix) \over 1 - r \exp (2 \pi ix)}\right]]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 [\alpha_{p} ({\bf x})] besides [D_{p}] of [F_{p}] which `tend towards δ' as [p \rightarrow \infty]. The convolution is performed by multiplying the Fourier coefficients of f by those of [\alpha_{p}], so that one forms the quantities[S'_{p} (\,f) (x) = {\textstyle\sum\limits_{|m| \leq p}} c_{m} (\alpha_{p}) c_{m} (\,f) \exp (2 \pi imx).]For instance the `sigma factors' of Lanczos (Lanczos, 1966[link], p. 65), defined by[\sigma_{m} = {\sin [m \pi / p] \over m \pi /p},]lead 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 with[\alpha_{p} = p\chi_{[-1/(2p), \, 1/(2p)]}{*} D_{p},]which is itself the convolution of a `rectangular pulse' of width [1/p] and of the Dirichlet kernel of order p.

A review of the summation problem in crystallography is given in Section[link].


Ahlfors, L. V. (1966). Complex Analysis. New York: McGraw-Hill.
Cartan, H. (1961). Théorie des Fonctions Analytiques. Paris: Hermann.
Lanczos, C. (1966). Discourse on Fourier Series. Edinburgh: Oliver & Boyd.
Larmor, J. (1934). The Fourier discontinuities: a chapter in historical integral calculus. Philos. Mag. 17, 668–678.
Tolstov, G. P. (1962). Fourier Series. Englewood Cliffs: Prentice-Hall.
Whittaker, E. T. & Watson, G. N. (1927). A Course of Modern Analysis, 4th ed. Cambridge University Press.
Zygmund, A. (1959). Trigonometric Series, Vols. 1 and 2. Cambridge University Press.

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