- Render, Hermann

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# Render, Hermann

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Render, Hermann

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- PublicationPadÃ© approximation for a multivariate Markov transformMethods of PadÃ¨ approximation are used to analyse a multivariate Markov transform which has been recently introduced by the authors. The first main result is a characterization of the rationality of the Markov transform via Hankel determinants. The second main result is a cubature formula for a special class of measure.
239ScopusÂ© Citations 3 - 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.
340ScopusÂ© Citations 1 - PublicationReproducing kernels for polyharmonic polynomialsThe reproducing kernel of the space of all homogeneous polynomi- als of degree k and polyharmonic order m is computed explicitly, solving a question of A. Fryant and M.K. Vemuri.
344ScopusÂ© Citations 5 - 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 ] .
306ScopusÂ© Citations 40 - PublicationHarmonic divisors and rationality of zeros of Jacobi polynomialsLet Pn (Î±,Î² ) ( x ) be the Jacobi polynomial of degree n with parameters Î±Î² The main result of the paper states the following: If bâ‰ 1 ; 3 and c are non-zero rel- atively prime natural numbers then P ( k +( d 3) = 2 ;k +( d 3) = 2) n p b=c 6 â‰ 0 for all natural numbers d;n and k 2 N 0 : Moreover, under the above assumption, the polynomial Q ( x ) = b c x 2 1 + ::: + x 2 d 1 + b c 1 x 2 d is not a harmonic divisor, and the Dirichlet problem for the cone f Q ( x ) < 0 g has polynomial harmonic solutions for polynomial data functions.
287 - 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.
247ScopusÂ© Citations 7 - 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
249ScopusÂ© Citations 20 - PublicationThe Goursat problem for a generalized Helmholtz operator in the planeWe consider the Goursat problem in the plane for partial differential operators whose principal part is the pth power of the standard Laplace operator. The data is posed on a union of 2p distinct lines through the origin. We show that the solvability of this Goursat problem depends on Diophantine properties of the geometry of lines on which the data is posed.
256ScopusÂ© Citations 5 - PublicationRegularity of generalized Daubechies wavelets reproducing exponential polynomials with real-valued parametersWe investigate non-stationary orthogonal wavelets based on a non-stationary interpolatory subdivision scheme reproducing a given set of exponentials with real-valued parameters. The construction is analogous to the construction of Daubechies wavelets using the subdivision scheme of Deslauriers-Dubuc. The main result is the existence and smoothness of these Daubechies type wavelets.
409ScopusÂ© Citations 16 - PublicationThe Khavinson-Shapiro conjecture and polynomial decompositionsThe main result of the paper states the following: Let Ïˆ be a polynomial in n variables of degree t: Suppose that there exists a constant C > 0 such that any polynomial f has a polynomial decomposition f = Ïˆ qf + hf with khf = 0 and deg qf deg f + C: Then deg Ïˆ 2k. Here âˆ†k is the kth iterate of the Laplace operator âˆ† : As an application, new classes of domains in Rn are identi ed for which the Khavinson-Shapiro conjecture holds.
304ScopusÂ© Citations 7