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

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 ${\mathbb{R}}^n$, 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 $A(B_R)$ of all real-analytic functions defined on the ball $B_R$ of radius $R$ and center zero whose Taylor series of homogeneous polynomials converges compactly in $B_R$. 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 $f\in A(B_R)$, there exist unique $q,r\in A(B_R)$ such that $f=\mathrm{Pq}+r$ and ${\Delta }^{k}r=0$. Another application of this result is the existence of polynomial solutions of the polyharmonic equation ${\Delta }^{k}u=0$ for polynomial data on certain classes of algebraic hypersurfaces