Entropy function and orthogonal polynomials

TL;DR

This paper presents a new real-variable proof of a classical theorem on orthogonal polynomials on the unit circle using entropy estimates, extends a Fourier series convergence theorem, and discusses open problems in the field.

Contribution

It introduces a novel real-variable approach with entropy estimates and extends existing theorems on orthogonal polynomials and Fourier series convergence.

Findings

Provided a simple proof of the asymptotic behavior theorem for orthogonal polynomials

Extended Freud's theorem on averaged Fourier series convergence

Discussed open problems in orthogonal polynomials on the unit circle

Abstract

We give a simple proof of a classical theorem by A.M\'at\'e, P.Nevai, and V.Totik on asymptotic behavior of orthogonal polynomials on the unit circle. It is based on a new real-variable approach involving an entropy estimate for the orthogonality measure. Our second result is an extension of a theorem by G.Freud on averaged convergence of Fourier series. We also discuss some related open problems in the theory of orthogonal polynomials on the unit circle.