Edwards' Theorem establishes duality between a convex cone in the space of continuous functions on a compact space and the set of representing or Jensen measures for this cone. In this paper we prove non-compact versions of this theorem.

The primary goal of this paper is to study the Dirichlet problem on a compact set K ⊂ R n . Initially we consider the space H(K) of functions on K which can be uniformly approximated by functions harmonic in a neighborhood of K as possible solutions. As in the classical theory, our Theorem 6.1 shows C(∂ f K) ∼ = H(K) for compact sets with ∂ f K closed, where ∂ f K is the fine boundary of K. However, in general a continuous solution cannot be expected even for continuous data on ∂ f K as illustrated by Theorem 6.1. Consequently, we show that the solution can be found in a class of finely harmonic functions. Moreover by Theorem 6.5, in complete analogy with the classical situation, this class is isometrically isomorphic to C b (∂ f K) for all compact sets K.

Abstract. We explore the relationship between convex and subharmonic functions on discrete sets. Our principal concern is to determine the setting in which a convex function is necessarily subharmonic. We initially consider the primary notions of convexity on graphs and show that more structure is needed to establish the desired result. To that end, we consider a notion of convexity defined on lattice-like graphs generated by normed abelian groups. For this class of graphs, we are able to prove that all convex functions are subharmonic.

In classical potential theory, one can solve the Dirichlet problem on unbounded domains such as the upper half plane. These domains have two types of boundary points; the usual finite boundary points and another point at infinity. W. Woess has solved a discrete version of the Dirichlet problem on the ends of graphs analogous to having multiple points at infinity and no finite boundary. Whereas C. Kiselman has solved a similar version of the Dirichlet problem on graphs analogous to bounded domains. In this work, we combine the two ideas to solve a version of the Dirichlet problem on graphs with finitely many ends and boundary points of the Kiselman type.Comment: 13 page

There are several equivalent ways to define continuous harmonic functions H(K) on a compact set K in R n . One may let H(K) be the unform closures of all functions in C(K) which are restrictions of harmonic functions on a neighborhood of K, or take H(K) as the subspace of C(K) consisting of functions which are finely harmonic on the fine interior of K. In [DG74] it was shown that these definitions are equivalent. Using a localization result of [BH78] one sees that a function h ∈ H(K) if and only if it is continuous and finely harmonic on on every fine connected component of the fine interior of K. Such collection of sets are usually called restoring.Another equivalent definition of H(K) was introduced in [P97] using the notion of Jensen measures which leads another restoring collection of sets. The main goal of this paper is to reconcile the results in [DG74] and [P97].To study these spaces, two notions of Green functions have previously been introduced. One by [P97] as the limit of Green functions on domains D j where the domains D j are decreasing to K, and alternatively following [F72, F75] one has the fine Green function on the fine interior of K. Our Theorem 3.2 shows that these are equivalent notions.In Section 4 a careful study of the set of Jensen measures on K, leads to an interesting extension result (Corollary 5.1) for superharmonic functions. This has a number of applications. In particular we show that the two restoring coverings are the same. We are also able to extend some results of [GL83] and [P97] to higher dimensions.