Duke Mathematical Journal

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

Hermann Render

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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 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

Article information

Source
Duke Math. J., Volume 142, Number 2 (2008), 313-352.

Dates
First available in Project Euclid: 27 March 2008

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1206642157

Digital Object Identifier
doi:10.1215/00127094-2008-008

Mathematical Reviews number (MathSciNet)
MR2401623

Zentralblatt MATH identifier
1140.31004

Subjects
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

Citation

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


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