Positive trigonometric Quadrature Formulas and quadrature on the unit circle

TL;DR

This paper characterizes positive trigonometric quadrature formulas on the unit circle, linking them to orthogonal polynomials and their zeros, and provides asymptotic behavior of quadrature weights.

Contribution

It offers a complete description of positive quadrature formulas using orthogonal polynomials on the unit circle, including recurrence relations and zero interlacing properties.

Findings

Nodes polynomial generated by a simple recurrence

Interlacing properties of zeros of para-orthogonal polynomials

Asymptotic behavior of quadrature weights

Abstract

We give several descriptions of positive quadrature formulas which are exact for trigonometric -, respectively, Laurent polynomials of degree less or equal $n-1-m$, $0\leq m\leq n-1$. A complete and simple description is obtained with the help of orthogonal polynomials on the unit circle. In particular it is shown that the nodes polynomial can be generated by a simple recurrence relation. As a byproduct interlacing properties of zeros of para-orthogonal polynomials are obtained. Finally, asymptotics for the quadrature weights are presented.