Hermann Render

REAL BARGMANN SPACES, FISCHER DECOMPOSITIONS AND SETS OF UNIQUENESS FOR POLYHARMONIC FUNCTIONS

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 R n 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 decom- positions (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 suciently

Bernstein Operators for Exponential Polynomials

AbstractLet L be a linear differential operator with constant coefficients of order n and complex eigenvalues λ0,…,λn. Assume that the set Un of all solutions of the equation Lf=0 is closed under complex conjugation. If the length of the interval [a,b] is smaller than π/Mn, where Mn:=max {|Im λj|:j=0,…,n}, then there exists a basis pn,k , k=0,…,n, of the space Un with the property that each pn,k has a zero of order k at a and a zero of order n−k at b, and each pn,k is positive on the open interval (a,b). Under the additional assumption that λ0 and λ1 are real and distinct, our first main result states that there exist points a=t0<t1<⋅⋅⋅<tn=b and positive numbers α0,…,αn, such that the operator

Optimality of generalized Bernstein operators

B_{n}f:=\sum_{k=0}^{n}\alpha _{k}f(t_{k})p_{n,k}(x)

Invertibility of Holomorphic Functions with Respect to the Hadamard Product

satisfies

Nonstandard methods of completing quasi-uniform spaces

B_{n}e^{\lambda _{j}x}=e^{\lambda _{j}x}

Nonstandard topology on function spaces with applications to hyperspaces

, for j=0,1. The second main result gives a sufficient condition guaranteeing the uniform convergence of Bnf to f for each f∈C[a,b].

A moment problem for pseudo-positive definite functionals

Cardinal polysplines of order p on annuli are functions in C2p-2(Rn\{0}) which are piecewise polyharmonic of order p such that Δp-1 S may have discontinuities on spheres in Rn, centered at the origin and having radii of the form ej, j ∈ Z. The main result is an interpolation theorem for cardinal polysplines where the data are given by sufficiently smooth functions on the spheres of radius ej and center 0 obeying a certain growth condition in |j|. This result can be considered as an analogue of the famous interpolation theorem of Schoenberg for cardinal splines.

The Puritz Order and Its Relationship to the Rudin-Keisler Order

We show that a certain optimality property of the classical Bernstein operator also holds, when suitably reinterpreted, for generalized Bernstein operators on extended Chebyshev systems.