Orthogonal polynomials on the unit circle: Verblunsky coefficients with some restrictions imposed on a pair of related real sequences

TL;DR

This paper explores how restrictions like sign and periodicity on certain real sequences influence the Verblunsky coefficients and measures on the unit circle, revealing gaps in support and generating periodic coefficients.

Contribution

It introduces new conditions on sequences to analyze their impact on Verblunsky coefficients and the associated measures, including periodicity and support gaps.

Findings

Alternating sign sequences create gaps near z=-1 in measure support.

Periodic sequences with specific restrictions generate periodic Verblunsky coefficients.

Results connect positive chain sequences with periodicity in Verblunsky coefficients.

Abstract

It was shown recently that associated with a pair of real sequences $\{\{c_{n}\}_{n=1}^{\infty}, \{d_{n}\}_{n=1}^{\infty}\}$, with $\{d_{n}\}_{n=1}^{\infty}$ a positive chain sequence, there exists a unique nontrivial probability measure $\mu$ on the unit circle. The Verblunsky coefficients $\{\alpha_{n}\}_{n=0}^{\infty}$ associated with the orthogonal polynomials with respect to $\mu$ are given by the relation $$ \alpha_{n-1}=\overline{\tau}_{n-1}\left[\frac{1-2m_{n}-ic_{n}}{1-ic_{n}}\right], \quad n \geq 1, $$ where $\tau_0 = 1$, $\tau_{n}=\prod_{k=1}^{n}(1-ic_{k})/(1+ic_{k})$, $n \geq 1$ and $\{m_{n}\}_{n=0}^{\infty}$ is the minimal parameter sequence of $\{d_{n}\}_{n=1}^{\infty}$. In this manuscript we consider this relation and its consequences by imposing some restrictions of sign and periodicity on the sequences $\{c_{n}\}_{n=1}^{\infty}$ and $\{m_{n}\}_{n=1}^{\infty}$. When the sequence $ \{c_{n}\}_{n=1}^{\infty}$ is of alternating sign, we use information about the zeros of associated para-orthogonal polynomials to show that there is a gap in the support of the measure in the neighbourhood of $z= -1$. Furthermore, we show that it is possible to ge\-nerate periodic Verblunsky coefficients by choosing periodic sequences $\{c_{n}\}_{n=1}^{\infty}$ and $\{m_{n}\}_{n=1}^{\infty}$ with the additional restriction $c_{2n}=-c_{2n-1}, \, n\geq 1.$ We also give some results on periodic Verblunsky coefficients from the point of view of positive chain sequences. An example is provided to illustrate the results obtained.