Green's function, Jensen measures, and bounded point evaluations

Abstract

A compactly supported measure μ on the complex plane C is called a Jensen measure for 0 if log ¦P(0)¦ ⩽ ∝ log¦P(z)¦dμ(z) for every polynomial P. H 2( μ) denotes the closure of the polynomials in L 2( μ). We obtain the result that if μ is not the point mass at 0, then the functions in H 2( μ) are analytic on an open set which contains 0 and whose closure contains the support of μ. The primary tool used to obtain this result is a generalized Green's function for a measure, and we also derive some of its properties.