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

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Abstract
In this article, a positive answer is given to the following question posed by Hayman [35, page 326]: 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 Khavinson and Shapiro [39, page 460]. 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 zero 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 for sufficiently large radius R>0, the following decomposition holds: for each function f∈A(BR), there exist unique q,r∈A(BR) such that f=Pq+r and Δkr=0. Another application of this result is the existence of polynomial solutions of the polyharmonic equation Δku=0 for polynomial data on certain classes of algebraic hypersurfaces

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