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Render, Hermann
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Render, Hermann
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Render, Hermann
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Now showing 1 - 10 of 25
- 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.
311 - 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].
219 - 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.
661 - 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, Δp. 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).
236Scopus© Citations 5 - PublicationConvergence of rational Bernstein operatorsIn this paper we discuss convergence properties and error estimates of rational Bernstein operators introduced by P. Pit¸ul and P. Sablonni`ere. It is shown that the rational Bernstein operators converge to the identity operator if and only if the maximal difference between two consecutive nodes is converging to zero. Further a Voronovskaja theorem is given based on the explicit computation of higher order moments for the rational Bernstein operator
350Scopus© Citations 5 - PublicationHarmonic divisors and rationality of zeros of Jacobi polynomialsLet Pn (α,β ) ( x ) be the Jacobi polynomial of degree n with parameters αβ The main result of the paper states the following: If b≠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 ≠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.
395 - PublicationThe Dirichlet problem for the slab with entire data and a difference equation for harmonic functionsIt is shown that the Dirichlet problem for the slab (a,b)×Rd with entire boundary data has an entire solution. The proof is based on a generalized Schwarz reflection principle. Moreover, it is shown that for a given entire harmonic function g the inhomogeneous difference equation h(t+1,y)−h(t,y)=g(t,y) has an entire harmonic solution h.
301Scopus© Citations 4 - PublicationPolyharmonic Hardy spaces on the complexified annulus and error estimates of cubature formulasThe present paper has a twofold contribution: first, we intro- duce a new concept of Hardy spaces on a multidimensional complexified annular domain which is closely related to the annulus of the Klein-Di rac quadric important in Conformal Quantum Field Theory. Secondly, for functions in these Hardy spaces, we provide error estimate for the p oly- harmonic Gauß-Jacobi cubature formulas, which have been introduced in previous papers.
387Scopus© Citations 5 - 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
611Scopus© Citations 33 - 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 λ 1, ..., λ N+1, let E( N+1) be the space of all C∞-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 εn to εn+1 involving real-analytic functions, and with εN+1 = E(Λ N+1) if and only if contiguous eigenvalues are equally spaced.
280Scopus© Citations 2