The growth of polynomials orthogonal on the unit circle with respect to a weight w that satisfies $w$,$1/w \in L^\infty(\mathbb{T})$

TL;DR

This paper investigates the growth behavior of orthonormal polynomials on the unit circle associated with weights that are essentially bounded and bounded away from zero, revealing conditions under which these polynomials can grow.

Contribution

It establishes new results on the growth of orthonormal polynomials for weights satisfying specific boundedness conditions on the unit circle.

Findings

Orthogonal polynomials can exhibit growth under certain weight conditions.

Growth is linked to the boundedness of the weight and its reciprocal.

Provides theoretical insights into polynomial behavior on the unit circle.

Abstract

We consider the weight w: 1<w<T on the unit circle and prove that the corresponding orthonormal polynomials can grow.