Orthogonal Trigonometric Polynomials: Riemann-Hilbert Analysis and Relations with OPUC

TL;DR

This paper develops a Riemann-Hilbert approach to analyze orthogonal trigonometric polynomials, revealing their asymptotic behavior, relations with OPUC, and deriving key formulas and properties of zeros.

Contribution

It introduces a novel Riemann-Hilbert analysis for OTP and establishes new connections with OPUC, including recurrence and Christoffel-Darboux formulas.

Findings

Asymptotic formulas for OTP with positive analytic weights

Relations between OTP and OPUC established

Derived four-term recurrence and Christoffel-Darboux formulas

Abstract

In this paper, we study the theory of orthogonal trigonometric polynomials (OTP). We obtain asymptotics of OTP with positive and analytic weight functions by Riemann-Hilbert approach and find they have relations with orthogonal polynomials on the unit circle (OPUC). By the relations and the theory of OPUC, we also get four-terms recurrent formulae, Christoffel-Darboux formula and some properties of zeros for orthogonal trigonometric polynomials.