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

摘要

Asymptotic forms of the Hilbert-Scmidt and Hilbert norms of positive definiteToeplitz matrices $Q_{N}=(b(j-k))_{j,k=0}^{N-1}$ as $N\to \infty $ aredetermined. Here $b(j)$ are consequent trigonometric moments of a generatingnon-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 withdifference kernels. Obtained results are applied to processing of stationary random signals, inparticular, neutron signals emitted by boiling water nuclear reactors.