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

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

Section 1.3.2.6.10.1. Classical theory

G. Bricognea

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1.3.2.6.10.1. Classical theory

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The space consists of (equivalence classes of) complex-valued 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.

References

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.