# Quadrature rules from finite orthogonality relations for Bernstein-Szego polynomials

## TL;DR

This paper constructs finite orthogonality relations for Bernstein-Szego polynomials by gluing two families, leading to Gauss-like quadrature rules for integrating rational functions with specific poles.

## Contribution

It introduces a novel method of creating finite orthogonality relations for Bernstein-Szego polynomials using a composite eigenbasis approach.

## Key findings

- Finite orthogonality relations for Bernstein-Szego polynomials established.
- Gauss-like quadrature rules derived for rational functions with prescribed poles.
- Eigenbasis construction linked to finite-dimensional Jacobi matrices.

## Abstract

We glue two families of Bernstein-Szego polynomials to construct the eigenbasis of an associated finite-dimensional Jacobi matrix. This gives rise to finite orthogonality relations for this composite eigenbasis of Bernstein-Szego polynomials. As an application, a number of Gauss-like quadrature rules are derived for the exact integration of rational functions with prescribed poles against the Chebyshev weight functions.