A lower bound for the minimum deviation of the Chebyshev polynomial on a compact real set

TL;DR

This paper establishes a sharp lower bound for the minimum deviation of Chebyshev polynomials on compact real sets, relating it to logarithmic capacity, with specific results for unions of intervals and subsets of the unit circle.

Contribution

It provides a new lower bound for Chebyshev polynomial deviation based on logarithmic capacity, including explicit bounds for unions of intervals and subsets of the unit circle.

Findings

Derived a sharp lower bound for Chebyshev polynomial deviation.

Connected deviation bounds to logarithmic capacity and elementary functions.

Extended results to subsets of the unit circle.

Abstract

In this paper, we give a sharp lower bound for the minimum deviation of the Chebyshev polynomial on a compact subset of the real line in terms of the corresponding logarithmic capacity. Especially if the set is the union of several real intervals, together with a lower bound for the logarithmic capacity derived recently by A.Yu.\,Solynin, one has a lower bound for the minimum deviation in terms of elementary functions of the endpoints of the intervals. In addition, analogous results for compact subsets of the unit circle are given.