The Carathéodory–Toeplitz problem with partial data Original Research Article

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.