- Render, Hermann

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# Render, Hermann

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Render, Hermann

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- 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.
607 - 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.
268 - 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].
147 - 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.
252 - PublicationA reflection result for harmonic functions which vanish on a cylindrical surfaceSuppose that a harmonic function h on a finite cylinder U vanishes on the curved part A of the boundary. It was recently shown that h then has a harmonic continuation to the infinite strip bounded by the hyperplanes containing the flat parts of the boundary. This paper examines what can be said if the above function h is merely harmonic near A (and inside U). It is shown that h then has a harmonic extension to a larger domain formed by radial reflection.
259ScopusÂ© Citations 7 - PublicationOn the mixed Cauchy problem with data on singular conicsWe consider a problem of mixed Cauchy type for certain holomorphic partial differential operators with the principal part Q2p(D) essentially being the (complex) Laplace operator to a power, Î”p. We provide inital data on a singular conic divisor given by P = 0, where P is a homogeneous polynomial of degree 2p. We show that this problem is uniquely solvable if the polynomial P is elliptic, in a certain sense, with respect to the principal part Q2p(D).
197ScopusÂ© Citations 4 - 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
110 - PublicationShape preserving properties of generalized Bernstein operators on extended Chebyshev spacesWe study the existence and shape preserving properties of a generalized Bernstein operator B n fixing a strictly positive function f 0 , and a second function f 1 such that f 1 /f 0 is strictly increasing, within the framework of extended Chebyshev spaces U n . The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator B n : C [ a,b ] â†’ U n with strictly increasing nodes, fixing f 0 ,f 1 âˆˆ U n . If U n âŠ‚ U n +1 and U n +1 has a non-negative Bernstein basis, then there exists a Bernstein operator B n +1 : C [ a,b ] â†’ U n +1 with strictly increasing nodes, fixing f 0 and f 1 . In particular, if f 0 ,f 1 ,...,f n is a basis of U n such that the linear span of f 0 ,..,f k is an extended Chebyshev space over [ a,b ] for each k = 0 ,...,n , then there exists a Bernstein operator B n with increasing nodes fixing f 0 and f 1 . The second main result says that under the above assumptions the following inequalities hold B n f â‰¥ B n +1 f â‰¥ f for all ( f 0 ,f 1 )-convex functions f âˆˆ C [ a,b ] . Furthermore, B n f is ( f 0 ,f 1 )-convex for all ( f 0 ,f 1 )-convex functions f âˆˆ C [ a,b ] .
314ScopusÂ© Citations 40 - PublicationExtension results for harmonic functions which vanish on cylindrical surfacesThe Schwarz reflection principle applies to a harmonic function which continuously vanishes on a relatively open subset of a planar or spherical boundary surface. It yields a harmonic extension to a predefined larger domain and provides a simple formula for this extension. Although such a point-to-point reflection law is unavailable for other types of surface in higher dimensions, it is natural to investigate whether similar harmonic extension results still hold. This article describes recent progress on such results for the particular case of cylindrical surfaces, and concludes with several open questions.
339ScopusÂ© Citations 2 - PublicationHarmonic functions which vanish on a cylindrical surfaceSuppose that a harmonic function h on a finite cylinder vanishes on the curved part of the boundary. This paper answers a question of Khavinson by showing that h then has a harmonic continuation to the infinite strip bounded by the hyperplanes containing the flat parts of the boundary. The existence of this extension is established by an analysis of the convergence properties of a double series expansion of the Green function of an infinite cylinder beyond the domain itself.
242ScopusÂ© Citations 9