Real Bargmann spaces, Fischer decompositions, and sets of uniqueness for polyharmonic functions
Render, Hermann
Duke Math. J., Tome 141 (2008) no. 1, p. 313-352 / Harvested from Project Euclid
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 $f=Pq+r$ 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
Publié le : 2008-04-01
Classification:  31B30,  35A20,  14P99,  12Y05
@article{1206642157,
     author = {Render, Hermann},
     title = {Real Bargmann spaces, Fischer decompositions, and sets of uniqueness for polyharmonic functions},
     journal = {Duke Math. J.},
     volume = {141},
     number = {1},
     year = {2008},
     pages = { 313-352},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1206642157}
}
Render, Hermann. Real Bargmann spaces, Fischer decompositions, and sets of uniqueness for polyharmonic functions. Duke Math. J., Tome 141 (2008) no. 1, pp.  313-352. http://gdmltest.u-ga.fr/item/1206642157/