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