Real Bargmann spaces, Fischer decompositions and Sets of uniqueness for polyharmonic functions

In this paper a positive answer is given to the following question of W.K. Hayman: if a polyharmonic entire function of order k vanishes on k distinct ellipsoids in the euclidean space Rn then it vanishes everywhere. Moreover a characterization of ellipsoids is given in terms of an extension property of solutions of entire data functions for the Dirichlet problem answering a question of D. Khavinson and H.S. Shapiro. These results are consequences from a more general result in the context of direct sum decompositions (Fischer decompositions) of polynomials or functions in the algebra A(BR) of all real-analytic functions defined on the ball BR of radius R and center 0 whose Taylor series of homogeneous polynomials converges compactly in BR. The main result states that for a given elliptic polynomial P of degree 2k and sufficiently large radius R > 0 the following decomposition holds: for each function f 2 A(BR) there exist unique q, r 2 A(BR) such that f = Pq + r and kr = 0. Another application of this result is the existence of polynomial solutions of the polyharmonic equation ku = 0 for polynomial data on certain classes of algebraic hypersurfaces. 2000 Mathematical Subject Classification. Primary: 31B30. Secondary: 35A20, 14P99, 12Y05