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1 April 2008 Real Bargmann spaces, Fischer decompositions, and sets of uniqueness for polyharmonic functions
Hermann Render
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Duke Math. J. 142(2): 313-352 (1 April 2008). DOI: 10.1215/00127094-2008-008


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 fA(BR), there exist unique q,rA(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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Hermann Render. "Real Bargmann spaces, Fischer decompositions, and sets of uniqueness for polyharmonic functions." Duke Math. J. 142 (2) 313 - 352, 1 April 2008.


Published: 1 April 2008
First available in Project Euclid: 27 March 2008

zbMATH: 1140.31004
MathSciNet: MR2401623
Digital Object Identifier: 10.1215/00127094-2008-008

Primary: 31B30
Secondary: 12Y05 , 14P99 , 35A20

Rights: Copyright © 2008 Duke University Press


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Vol.142 • No. 2 • 1 April 2008
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