Bernstein–Markov Property for Compact Sets in $\mathbb {C}^d$

Abstract

Given a compact set K in $\mathbb {C}^d.$ We concern with the Bernstein–Markov property of the pair $(K,\mu )$ where $\mu$ is a finite positive Borel measure with compact support K. In particular, we are able to give a class of $(K,\mu )$ having the Bernstein–Markov property with the measure $\mu$ satisfies a rather weak density condition. Using this result, we construct a pair $(K,\mu )$ satisfying the Bernstein–Markov property which is not covered by the known results in Bloom (Indiana Univ Math J 46:427–452, 1997) and Bloom and Levenberg (Trans Am Math Soc 351:4573–4567, 1999). Another main result of the note is a weak characterization of Bernstein–Markov property in terms of Chebyshev constants.