Asymptotics for Szegö polynomial zeros

AbstractThe Wiener-Levinson method and algorithm, formulated here in terms of Szegö polynomials ρn(ψN,I;z) orthogonal on the unit circle, is used to find unknown frequencies ωj from anN-sample of a discrete time signal consisting of the superposition of sinusoidal waves with frequencies ω1,...,ω1. In a recent paper the authors (and W.J. Thron) have shown that zerosz(j, n, N, I) of ρn(ψN,I;z) converge asN→∞ to the critical points $$e^{i\omega _j } $$ ,j=1, 2,...,I, providedn≥n0(I)=2I+L, whereL is 0 or 1. The present paper gives results on the convergence of zerosz(j, n, N, I) to some of the $$e^{i\omega _j } $$ for the case in whichn≤n0(I), wheren is the degree of ρn(ψN,I;z).