Orthogonal trigonometric polynomials from Riemann-Hilbert view

TL;DR

This paper establishes fundamental theorems for orthogonal trigonometric polynomials using a Riemann-Hilbert approach, connecting them with orthogonal polynomials on the unit circle and deriving new identities and recursions.

Contribution

It introduces a Riemann-Hilbert framework that unifies the theory of OTP and OPUC, leading to new theorems, identities, and recurrence relations.

Findings

Proved Favard, Baxter, Geronimus, Rakhmanov, Szeg"o, and strong Szeg"o theorems for OTP.

Derived Szeg"o recursions and four-term recurrences for OTP and OPUC.

Established a mutual representation theorem linking OTP and OPUC via Riemann-Hilbert problems.

Abstract

In this work, some theorems are established for orthogonal trigonometric polynomials (OTP) including Favard, Baxter, Geronimus, Rakhmanov, Szeg\"o and the strong Szeg\"o theorems which are important in the theory of orthogonal polynomials on the unit circle (OPUC). All these results are based on the mutual representation theorem of OPUC and OTP which deduced by a Riemann-Hilbert problem simultaneously characterizing them. In addition, Szeg\"o recursions, four-term recurrences and some new identities for OPUC and OTP are also obtained by using their Riemann-Hilbert characterizations respectively.