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

TL;DR

This paper investigates the extreme zeros of para-orthogonal polynomials on the unit circle, providing bounds and conditions for gaps in the measure's support, with applications to spectral analysis.

Contribution

It introduces bounds for the extreme zeros of para-orthogonal polynomials based on measure parameters, and establishes conditions for support gaps at specific points on the unit circle.

Findings

Bounds for zeros near z=1 are derived.

Conditions for support gaps at z=1 are established.

Examples demonstrate the tightness and applications of bounds.

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.