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

A compactly supported measure μ on the complex plane C is called a Jensen measure for 0 if $\text{log}\text{¦P(0)¦ ? ∝}\text{log}\text{¦P(z)¦dμ(z)}$ for every polynomial P. H²(μ) denotes the closure of the polynomials in L²(μ). We obtain the result that if μ is not the point mass at 0, then the functions in H²(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.