It is shown that the Dirichlet problem for the slab (a,b)Ã—Rd with entire boundary data has an entire solution. The proof is based on a generalized Schwarz reflection principle. Moreover, it is shown that for a given entire harmonic function g the inhomogeneous difference equation h(t+1,y)âˆ’h(t,y)=g(t,y) has an entire harmonic solution h.

The main result of the paper states the following: Let Ïˆ be a polynomial
in n variables of degree t: Suppose that there exists a constant C > 0 such
that any polynomial f has a polynomial decomposition f = Ïˆ qf + hf with
khf = 0 and deg qf deg f + C: Then deg Ïˆ 2k. Here âˆ†k is the kth
iterate of the Laplace operator âˆ† : As an application, new classes of domains
in Rn are identi ed for which the Khavinson-Shapiro conjecture holds.