The structure of the Naimark dilation and Gaussian stationary processes
The main concern of this paper is the study of positive definite block Toeplitz forms over \({\mathbb{N}}\). In the paper ''On the structure of Naimark dilation'' by \textit{T. Constantinescu} which appeared in J. Oper. Theory 12, 159-175 (1984), the analysis of the structure of the Naimark dilation of such a form is done using so-called ''choice sequences'' (a generalization to the operator case of the classical notion of ''Schur sequence''). This paper uses this analysis in order to compute some angles in Gaussian stationary processes. This is possible because such a process is completely defined by its correlation matrix, which is a positive definite Toeplitz form. The fact that choice sequences are useful for angle computations is shown in Section 4. In Section 5, this is applied for obtaining a criterium for informational regularity of the process in terms of the associated choice sequence. More explicit results are obtained for the scalar case in Section 6, where the connections with the entropy and Szegö limit theorems are also presented. Recently, these results were extended to the non-Toeplitz case.
scientific article; zbMATH DE number 3895689
scientific article