The Toeplitz–Carathéodory–Herglotz theorem

Bricogne, G.

doi:10.1107/97809553602060000551

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

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 43 | 1 | 2 |

Section 1.3.2.6.9.2. The Toeplitz–Carathéodory–Herglotz theorem

G. Bricogne^a

^a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, 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), Carathéodory (1911) and Herglotz (1911) 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.

References

Carathéodory, C. (1911). Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Functionen. Rend. Circ. Mat. Palermo, 32, 193–217.Google Scholar

Herglotz, G. (1911). Über Potenzreihen mit positiven, reellen Teil im Einheitskreis. Ber. Sächs. Ges. Wiss. Leipzig, 63, 501–511.Google Scholar

Toeplitz, O. (1911b). Über die Fouriersche Entwicklung positiver Funktionen. Rend. Circ. Mat. Palermo, 32, 191–192.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 43