- Render, Hermann

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# Render, Hermann

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Render, Hermann

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- PublicationA Characterization of the Khavinson-Shapiro Conjecture Via Fischer OperatorsThe Khavinson-Shapiro conjecture states that ellipsoids are the only bounded domains in euclidean space satisfying the following property (KS): the solution of the Dirichlet problem for polynomial data is polynomial. In this paper we show that property (KS) for a domain Î© is equivalent to the surjectivity of a Fischer operator associated to the domain Î©.
214ScopusÂ© Citations 6 - PublicationPolyharmonic functions of infinite order on annular regionsPolyharmonic functions f of in nite order and type on annular regions are systematically studied. The rst main result states that the Fourier-Laplace coefficients fk;l (r) of a polyharmonic function f of in nite order and type 0 can be extended to analytic functions on the complex plane cut along the negative semiaxis. The second main result gives a constructive procedure via Fourier-Laplace series for the analytic extension of a polyharmonic function on annular region A(r0; r1) of in nite order and type less than 1=2r1 to the kernel of the harmonicity hull of the annular region. The methods of proof depend on an extensive investigation of Taylor series with respect to linear differential operators with constant coefficients.
305 - PublicationPolyharmonicity and algebraic support of measuresOur main result states that two signed measures Î¼ and Î½ with bounded support contained in the zero set of a polynomial P(Ï‡) are equal if they coincide on the subspace of all polynomials of polyharmonic degree NP where the natural number NP is explicitly computed by the properties of the polynomial P(Ï‡). The method of proof depends on a definition of a multivariate Markov transform which is another major objective of the present paper. The classical notion of orthogonal polynomial of second kind is generalized to the multivariate setting: it is a polyharmonic function which has similar features to those in the one-dimensional case.
327 - PublicationPositivity properties for the clamped plate boundary problem on the ellipse and stripThe positivity preserving property for the biharmonic operator with Dirichlet boundary condition is investigated. We discuss here the case where the domain is an ellipse (that may degenerate to a strip) and the data is a polynomial function. We provide various conditions for which the positivity is preserved.
ScopusÂ© Citations 1 406 - PublicationConvergence of polyharmonic splines on semi-regular grids Z x aZ^n for a to 0Let p,n âˆˆ N with 2 p â‰¥ n + 2 , and let I a be a polyharmonic spline of order p on the grid Z Ã— a Z n which satisfies the interpolating conditions I a ( j,am ) = d j ( am ) for j âˆˆ Z ,m âˆˆ Z n where the functions d j : R n â†’ R and the parameter a> 0 are given. Let B s ( R n ) be the set of all integrable functions f : R n â†’ C such that the integral k f k s := Z R n b f ( Î¾ ) (1 + | Î¾ | s ) dÎ¾ is finite. The main result states that for given Ïƒ â‰¥ 0 there exists a constant c> 0 such that whenever d j âˆˆ B 2 p ( R n ) âˆ© C ( R n ) ,j âˆˆ Z , satisfy k d j k 2 p â‰¤ D Â· (1 + | j | Ïƒ ) for all j âˆˆ Z there exists a polyspline S : R n +1 â†’ C of order p on strips such that | S ( t,y ) âˆ’ I a ( t,y ) |â‰¤ a 2 p âˆ’ 1 c Â· D Â· (1 + | t | Ïƒ ) for all y âˆˆ R n ,t âˆˆ R and all 0
161 - PublicationBernstein operators for exponential polynomialsLet L be a linear differential operator with constant coefficients of order n and complex eigenvalues Î» 0 ,...,Î» n . Assume that the set U n of all solutions of the equation Lf = 0 is closed under complex conjugation. If the length of the interval [ a,b ] is smaller than Ï€/M n , where M n := max {| Im Î» j | : j = 0 ,...,n } , then there exists a basis p n,k , k = 0 ,...n , of the space U n with the property that each p n,k has a zero of order k at a and a zero of order n âˆ’ k at b, and each p n,k is positive on the open interval ( a,b ) . Under the additional assumption that Î» 0 and Î» 1 are real and distinct, our first main result states that there exist points a = t 0
ScopusÂ© Citations 20 330 - PublicationThe approximation order of polysplinesWe show that the scaling spaces de ned by the polysplines of order p provide approximation order 2p: For that purpose we re ne the re- sults on one dimensional approximation order by L-splines obtained in [2].
211 - PublicationCauchy, Goursat and Dirichlet problems for holomorphic partial differential equationsn this paper we survey recent results about Fischer decomposi- tions of polynomials or entire functions and their applications to holomorphic partial di erential equations. We discuss Cauchy and Goursat problems for the polyharmonic operator. Special emphasis is given to the Khavinson-Shapiro conjecture concerning polynomial solvability of the Dirichlet problem.
657 - PublicationOn real-analytic recurrence relations for cardinal exponential B-splinesLet LN+1 be a linear differential operator of order N + 1 with constant coefficients and real eigenvalues Î» 1, ..., Î» N+1, let E( N+1) be the space of all Câˆž-solutions of LN+1 on the real line.We show that for N 2 and n = 2, ...,N, there is a recurrence relation from suitable subspaces Îµn to Îµn+1 involving real-analytic functions, and with ÎµN+1 = E(Î› N+1) if and only if contiguous eigenvalues are equally spaced.
ScopusÂ© Citations 2 276 - PublicationReal Bargmann spaces, Fischer decompositions and Sets of uniqueness for polyharmonic functionsIn 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
ScopusÂ© Citations 32 602