The Nevai Condition

TL;DR

This paper investigates the Nevai condition for orthogonal polynomials, proving its validity for certain classes and providing counterexamples where it fails, thereby advancing understanding of spectral properties.

Contribution

It establishes the Nevai condition for the Nevai class on finite gap sets and presents a counterexample on a regular measure on [-2,2].

Findings

Nevai condition holds for finite gap sets within the spectrum.

Counterexample shows failure of the Nevai condition on some measures.

Provides insights into spectral behavior of orthogonal polynomials.

Abstract

We study Nevai's condition that for orthogonal polynomials on the real line, $K_n(x,x_0)^2 K_n(x_0,x_0)^{-1} d\rho (x)\to\delta_{x_0}$ where $K_n$ is the CD kernel. We prove that it holds for the Nevai class of a finite gap set uniformly on the spectrum and we provide an example of a regular measure on $[-2,2]$ where it fails on an interval.