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The Carathéodory–Toeplitz problem with partial data
Journal article   Peer reviewed

The Carathéodory–Toeplitz problem with partial data

Hugo J. Woerdeman
Linear algebra and its applications, v 342(1), 149
15 Feb 2002
url
https://doi.org/10.1016/S0024-3795(01)00450-5View
Published, Version of Record (VoR) Restricted

Abstract

Band method Carathéodory–Toeplitz problem Cyclostationary stochastic processes Diagonally connected patterns Stalactite-type patterns
We treat a positive extension problem in an algebra of periodic doubly infinite operator matrices. The given data have a stalactite-type pattern. The problem may be viewed as an operator-valued Carathéodory–Toeplitz problem with partially described data. Necessary and sufficient conditions are given for the existence of a solution, the set of all solutions is described and a maximum entropy result is obtained as well. In the matrix-valued case the problem was also considered by Alpay et al. [Linear Algebra Appl. 268 (1998) 247], where the problem was rephrased as an interpolation problem requiring symmetries and solutions were given in these terms. We shall also interpret our results in the context of cyclostationary stochastic processes.

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