Chebyshev Polynomials on a System of Continua

TL;DR

This paper establishes precise estimates for the uniform norm of Chebyshev polynomials on complex sets composed of finitely many continua, especially when these are quasismooth arcs or Jordan domains.

Contribution

It provides exact estimates for Chebyshev polynomial norms on complex continua, extending previous results to quasismooth components and closed Jordan domains.

Findings

Exact estimates up to a constant factor for quasismooth arcs

Precise bounds for Chebyshev polynomials on Jordan domains

Extension of classical results to more general continua

Abstract

The estimates of the uniform norm of the Chebyshev polynomial associated with a compact set $K$ consisting of a finite number of continua in the complex plane are established. These estimates are exact (up to a constant factor) in the case where the components of $K$ are either quasismooth (in the sense of Lavrentiev) arcs or closed Jordan domains bounded by a quasismooth curve.