Polyharmonic Functions on Trees

In this paper, we introduce and study polyharmonic functions on trees. We prove that the discrete version of the classical Riquier problem can be solved on trees. Next, we show that the discrete version of a characterization of harmonic functions due to Globevnik and Rudin holds for biharmonic functions on trees. Furthermore, on a homogeneous tree we characterize the polyharmonic functions in terms of integrals with respect to finitely-additive measures (distributions) of certain functions depending on the Poisson kernel. We define polymartingales on a homogeneous tree and show that the discrete version of a characterization of polyharmonic functions due to Almansi holds for polymartingales. We then show that on homogeneous trees there are 1-1 correspondences among the space of nth-order polyharmonic functions, the space of nth-order polymartingales, and the space of n-tuples of distributions. Finally, we study the boundary behavior of polyharmonic functions on homogeneous trees whose associated distributions satisfy various growth conditions.