Asymptotic estimates of the norms of positive definite Toeplitz matrices and detection of quasi-periodic components of stationary random signals

Abstract

Asymptotic forms of the Hilbert-Scmidt and Hilbert norms of positive definite Toeplitz matrices $Q_{N}=(b(j-k))_{j,k=0}^{N-1}$ as $N\to \infty $ are determined. Here $b(j)$ are consequent trigonometric moments of a generating non-negative mesure $d\sigma (\theta)$ on $[ -\pi ,\pi ] $. It is proven that $\sigma (\theta)$ is continuous if and only if any of those contributions is $o(N)$. Analogous criteria are given for positive integral operators with difference kernels. Obtained results are applied to processing of stationary random signals, in particular, neutron signals emitted by boiling water nuclear reactors.