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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 45   | 1 | 2 |

Section 1.3.2.6.9.2. The Toeplitz–Carathéodory–Herglotz theorem

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

1.3.2.6.9.2. The Toeplitz–Carathéodory–Herglotz theorem

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It was shown independently by Toeplitz (1911b)[link], Carathéodory (1911)[link] and Herglotz (1911)[link] that a function [f \in L^{1}] is almost everywhere non-negative if and only if the Toeplitz forms [T_{n} [\,f]] associated to f are positive semidefinite for all values of n.

This is equivalent to the infinite system of determinantal inequalities[D_{n} = \det \pmatrix{c_{0} &c_{-1} &\cdot &\cdot &c_{-n}\cr c_{1} &c_{0} &c_{-1} &\cdot &\cdot\cr \cdot &c_{1} &\cdot &\cdot &\cdot\cr \cdot &\cdot &\cdot &\cdot &c_{-1}\cr c_{n} &\cdot &\cdot &c_{1} &c_{0}\cr} \geq 0 \quad \hbox{for all } n.]The [D_{n}] are called Toeplitz determinants. Their application to the crystallographic phase problem is described in Section 1.3.4.2.1.10[link].

References

Carathéodory, C. (1911). Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Functionen. Rend. Circ. Mat. Palermo, 32, 193–217.
Herglotz, G. (1911). Über Potenzreihen mit positiven, reellen Teil im Einheitskreis. Ber. Sächs. Ges. Wiss. Leipzig, 63, 501–511.
Toeplitz, O. (1911b). Über die Fouriersche Entwicklung positiver Funktionen. Rend. Circ. Mat. Palermo, 32, 191–192.








































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