Duke Mathematical Journal
- Duke Math. J.
- Volume 142, Number 2 (2008), 313-352.
Real Bargmann spaces, Fischer decompositions, and sets of uniqueness for polyharmonic functions
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 vanishes on distinct ellipsoids in the Euclidean space , 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 of all real-analytic functions defined on the ball of radius and center zero whose Taylor series of homogeneous polynomials converges compactly in . The main result states that for a given elliptic polynomial of degree and for sufficiently large radius , the following decomposition holds: for each function , there exist unique such that and . Another application of this result is the existence of polynomial solutions of the polyharmonic equation for polynomial data on certain classes of algebraic hypersurfaces
Duke Math. J., Volume 142, Number 2 (2008), 313-352.
First available in Project Euclid: 27 March 2008
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 31B30: Biharmonic and polyharmonic equations and functions
Secondary: 35A20: Analytic methods, singularities 14P99: None of the above, but in this section 12Y05: Computational aspects of field theory and polynomials
Render, Hermann. Real Bargmann spaces, Fischer decompositions, and sets of uniqueness for polyharmonic functions. Duke Math. J. 142 (2008), no. 2, 313--352. doi:10.1215/00127094-2008-008. https://projecteuclid.org/euclid.dmj/1206642157