Limits for Szegö polynomials in frequency analysis

We characterize limits for orthogonal Szegö polynomials of fixed degree k, with respect to certain measures on the unit circle which are weakly convergent to a sum of m < k point masses. Such measures arise, for example, as a convolution of point masses with an approximate identity. It is readily seen that the underlying measures in two recently-proposed methods for estimating the m frequencies, θj , of a discrete-time trigonometric signal using Szegö polynomials are of this form. We prove existence of Szegö polynomial limits associated with a general class of weakly convergent measures, and prove that for convolution of point masses with the Poisson kernel, which underlies one of the recently-proposed methods, the limit has as a factor the Szegö polynomial with respect to a related measure, which we specify. Since m of the zeros approach the eiθj , this result uniquely characterizes the limit. A similar result is obtained for measures consisting of point masses with additive absolutely continuous part.  2004 Elsevier Inc. All rights reserved.