Nonstationary Szegö theorem, band sequences and maximum entropy
The authors obtain generalizations of well-known results relating to positive definite extensions of banded Toeplitz matrices. These extensions arise from the author's study, in this paper, of ``m- sequences'', which are those sequences of linearly independent vectors in Hilbert space whose orthonormalizing matrix \(A=(a_{ij})\), given via the Gram-Schmidt process, is m-banded, i.e., satisfies \(a_{ij}=0\) for \(| i-j| >m\). The authors apply their results to obtain a method for finding among vectors \(\{v_ j\}\) for which the inner product \(<v_ i,v_ j>=a_{ij}\) those for which the distance from \(v_ k\) to the linear span of \(v_ 0,...,v_{k-1}\) is maximal for every \(k\geq 0\).
positive definite extensions of banded Toeplitz matrices
scientific article; zbMATH DE number 4063846