Lower Bounds for Weighted Chebyshev and Orthogonal Polynomials

TL;DR

This paper establishes optimal lower bounds for weighted Chebyshev and orthogonal polynomials, extending classical results to broader classes of weights and sets, with implications for asymptotic behavior and polynomial extremal properties.

Contribution

It provides new asymptotic and non-asymptotic lower bounds for Widom factors on general compact sets and weights, extending classical bounds to more complex weight functions and sets.

Findings

Extended non-asymptotic lower bounds to regular compact sets and large class weights.

Extended Widom's asymptotic lower bounds to weights with zeros, including strong and infinitely many zeros.

Generalized Bernstein's asymptotics for weighted Chebyshev polynomials to broader weights and sets.

Abstract

We derive optimal asymptotic and non-asymptotic lower bounds on the Widom factors for weighted Chebyshev and orthogonal polynomials on compact subsets of the real line. In the Chebyshev case we extend the optimal non-asymptotic lower bound previously known only in a handful of examples to regular compact sets and a large class weights. Using the non-asymptotic lower bound, we extend Widom's asymptotic lower bound for weights bounded away from zero to a large class of weights with zeros including weights with strong zeros and infinitely many zeros. As an application of the asymptotic lower bound we extend Bernstein's 1931 asymptotics result for weighted Chebyshev polynomials on an interval to arbitrary Riemann integrable weights with finitely many zeros and to some continuous weights with infinitely many zeros. In the case of orthogonal polynomials, we derive optimal asymptotic and non-asymptotic lower bound on arbitrary regular compact sets for a large class of weights in the non-asymptotic case and for arbitrary Szeg\H{o} class weights in the asymptotic case, extending previously known bounds on finite gap and Parreau--Widom sets.