Polynomial expansions of positive harmonic functions in the unit ball

Abstract

It is well known that if h is harmonic in the open unit ball B of the Euclidean space N (where N ≥ 2), then there exist homogeneous harmonic polynomials Hn of degree n in N such that converges absolutely and locally uniformly to h in B (see e.g. Brelot[2], appendice). Further, it is easy to show that this series is unique and that each polynomial Hn is the sum of all the monomial terms of degree n in the multiple Taylor series of h about the origin 0. We call the polynomial expansion of h. Our aim is to obtain sharp upper and lower bounds for the individual terms Hn and for the partial sums of this expansion in the case where h > 0 in B.