Entropy function and orthogonal polynomials

Abstract

We give a simple proof of a classical theorem by A.Máté, 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.

14 pages, 2 open problems

References in corpus (4)

• A spectral Szego theorem on the real line
• Szego condition and scattering for one-dimensional Dirac operators
• Zero sets, entropy, and pointwise asymptotics of orthogonal polynomials
• Pointwise convergence of the non-linear Fourier transform