Extreme zeros in a sequence of para-orthogonal polynomials and bounds for the support of the measure

Abstract

Given a nontrivial Borel measure μ on the unit circle T{double-struck}, the corresponding reproducing (or Christoffel-Darboux) kernels with one of the variables fixed at z = 1 constitute a family of so-called para-orthogonal polynomials, whose zeros belong to T. With a proper normalization they satisfy a three-term recurrence relation determined by two sequences of real coefficients, {cn} and {dn}, where {dn} is additionally a positive chain sequence. Coefficients (cn, dn) provide a parametrization of a family of measures related to μ by addition of a mass point at z = 1. In this paper we estimate the location of the extreme zeros (those closest to z = 1) of the para-orthogonal polynomials from the (cn, dn)-parametrization of the measure, and use this information to establish sufficient conditions for the existence of a gap in the support of μ at z = 1. These results are easily reformulated in order to find gaps in the support of μ at any other z ∈ T{double-struck}. We provide also some examples showing that the bounds are tight and illustrate their computational applications.