Real Bargmann spaces, Fischer decompositions and Sets of uniqueness for polyharmonic functions
|Title:||Real Bargmann spaces, Fischer decompositions and Sets of uniqueness for polyharmonic functions||Authors:||Render, Hermann||Permanent link:||http://hdl.handle.net/10197/5474||Date:||Apr-2008||Online since:||2014-03-20T15:48:50Z||Abstract:||In this paper a positive answer is given to the following question of W.K. Hayman: 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 D. Khavinson and H.S. Shapiro. 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 0 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 sufficiently large radius R > 0 the following decomposition holds: for each function f 2 A(BR) there exist unique q, r 2 A(BR) such that f = Pq + r and kr = 0. Another application of this result is the existence of polynomial solutions of the polyharmonic equation ku = 0 for polynomial data on certain classes of algebraic hypersurfaces. 2000 Mathematical Subject Classification. Primary: 31B30. Secondary: 35A20, 14P99, 12Y05||Type of material:||Journal Article||Publisher:||Duke University Press||Journal:||Duke Math. J.||Volume:||142||Issue:||2||Start page:||313||End page:||352||Copyright (published version):||2008 Duke University Press||Keywords:||Polyharmonic function; Harmonic divisor; Almansi theorem; Fischer pair; Direct sum decomposition; Bargmann space; Fock space; Dirichlet problem; Real Nullstellensatz||DOI:||10.1215/00127094-2008-008||Language:||en||Status of Item:||Peer reviewed|
|Appears in Collections:||Mathematics and Statistics Research Collection|
Show full item record
Page view(s) 5072
This item is available under the Attribution-NonCommercial-NoDerivs 3.0 Ireland. No item may be reproduced for commercial purposes. For other possible restrictions on use please refer to the publisher's URL where this is made available, or to notes contained in the item itself. Other terms may apply.