Orthogonal polynomials and quadrature rules on the unit circle associated with perturbations of symmetric measures

Abstract

It was shown recently that given a pair of real sequences {{cn}n=1∞,{dn}n=1∞}, with {dn}n=1∞ a positive chain sequence, we can associate a unique nontrivial probability measure μ on the unit circle, and conversely. In this paper, we consider the set Q(c) of all nontrivial probability measures for which cn=(−1)nc, with c∈R. We show that every measure μ∈Q(c) can be obtained from a perturbation of a symmetric measure η on [−1,1]. Moreover, the sequence of orthogonal polynomials associated with μ can be given in terms of a perturbation of symmetric orthogonal polynomials associated with η. We also prove that every measure in Q(c) is quasi-symmetric, that is, there exists a complex valued function q(c)(z) satisfying dμ(z)=−q(c)(z)dμ(z¯), and such that q(c)(z)→1 when c→0. Quadrature rules with quasi-symmetric weights are also considered. Finally, some examples of orthogonal polynomials on the unit circle and its associated quasi-symmetric nontrivial probability measures are obtained.