Norms of positive definite Toeplitz matrices and detection of almost periodic components in random signals
V. M. Adamyan, José Luis Iserte, Igor M. Tkachenko, G. Verdú · Operators and Matrices · 2014
For positive definite Toeplitz matrices Q N = (b( jk)) N-1 j,k=0 generated by trigonometric moments b( j) of a non-negative measure dσ (θ ),θ ∈ [-π,π] , we note that the Hilbert-Schmidt norm Q N 2 and the maximal eigenvalue λ m (N) satisfy the following relationswhere {m α } is the set of jumps of σ (θ ) .Analogous relations hold for positive definite integral operators with difference kernels.The above relations are used in order to detect hidden almost periodic components in random signals.