Zeros of quasi-paraorthogonal polynomials and positive quadrature

TL;DR

This paper explores the spectral properties of quasi-paraorthogonal polynomials on the unit circle, characterizes their zeros, and develops positive quadrature formulas with predetermined nodes, supported by numerical illustrations.

Contribution

It introduces a characterization of quasi-paraorthogonal polynomials on the unit circle as analogues of quasi-orthogonal polynomials on the real line, and analyzes their zeros for quadrature construction.

Findings

Characterization of quasi-paraorthogonal polynomials on the unit circle.

Development of positive quadrature formulas with fixed nodes.

Numerical illustrations demonstrating quadrature formulas.

Abstract

In this paper we illustrate that paraorthogonality on the unit circle $\mathbb{T}$ is the counterpart to orthogonality on $\mathbb{R}$ when we are interested in the spectral properties. We characterize quasi-paraorthogonal polynomials on the unit circle as the analogues of the quasi-orthogonal polynomials on $\mathbb{R}$. We analyze the possibilities of preselecting some of its zeros, in order to build positive quadrature formulas with prefixed nodes and maximal domain of validity. These quadrature formulas on the unit circle are illustrated numerically.