Szego polynomials applied to frequency analysis

Abstract This paper is concerned with the problem of determining unknown frequencies ω1,…,ωI, using an observed discrete time signal xN={xN(m)} arising from a continuous waveform that is the superposition of a finite number of sinusoidal waves with well-defined frequencies ωj ( periods p j =2πn/ω j , n ϵ Z ) , j=1,…,I. We investigate the Wiener-Levinson method formulated here in terms of Szegő polynomials ϱn(ψN; z) orthogonal on the unit circle with respect to a distribution function ψN(θ) defined by the N observed values of the signal. We prove that if n0 denotes the number of critical points eiwj, then for every n ⩾ n0 and N ⩾ 1, the zeros z(j, n, N) of ϱn(ψN; z) can be arranged so that limN → ∞z(j, n, N) = eiwj for each of the frequencies wj. This result confirms one of the main parts of the conjecture given by Jones, Njastad and Saff (this journal (1990)) on the convergence of zeros of the Szegő polynomials. A related result on the convergence of corresponding two-point Pade approximants is also given.