Biharmonic Extensions on Infinite Trees

Abstract

In the investigation of harmonic and potential functions on the Euclidean spaces, the Runge-type approximation theorem and Laurent decomposition theorem for harmonic functions are important. Their extensions to subharmonic functions are also crucial. In this note, we investigate various aspects of these results in the context of discrete potential theory on infinite trees. Given an infinite tree T with positive potentials, we prove that for a harmonic function h outside a finite set, there exists a harmonic function H on T such that h − H is bounded outside a finite set. Developing other results based on this theorem, we investigate in detail biharmonic functions on T and study their properties. The thrust is to extend these results to the study of discrete biharmonic and bisuperharmonic functions on infinite trees. This is always true in Rⁿ , n ≥ 5 because the fundamental solution of ∆² on this case tends to 0 at infinity. Based on this property we also define the notion of a tapered biharmonic space.