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

Abstract

Given a non-trivial Borel measure $\mu$ on the unit circle $\mathbb T$, 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 $\mathbb T$. With a proper normalization they satisfy a three-term recurrence relation determined by two sequence of real coefficients, $\{c_n\}$ and $\{d_n\}$, where $\{d_n\}$ is additionally a positive chain sequence. Coefficients $(c_n,d_n)$ provide a parametrization of a family of measures related to $\mu$ 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 $(c_n,d_n)$-parametrization of the measure, and use this information to establish sufficient conditions for the existence of a gap in the support of $\mu$ at $z=1$. These results are easily reformulated in order to find gaps in the support of $\mu$ at any other $z\in \mathbb T$. We provide also some examples showing that the bounds are tight and illustrating their computational applications.