Hermann Render

Ognyan Kounchev, Hermann Render

A moment problem is presented for a class of signed measures which are termed pseudo-positive. Our main result says that for every pseudo-positive definite functional (subject to some reasonable restrictions) there exists a representing pseudo-positive measure. The second main result is a characterization of determinacy in the class of equivalent pseudo-positive representation measures. Finally the corresponding truncated moment problem is discussed.

Ognyan Kounchev, Hermann Render

The present paper has a twofold contribution: first, we introduce a new concept of Hardy spaces on a multidimensional complexified annular domain which is closely related to the annulus of the Klein-Dirac quadric important in Conformal Quantum Field Theory. Secondly, for functions in these Hardy spaces, we provide error estimate for the polyharmonic Gauß-Jacobi cubature formulas, which have been introduced in previous papers.

Hermann Render

In this paper we discuss convergence properties and error estimates of rational Bernstein operators introduced by P. Piţul and P. Sablonnière. It is shown that the rational Bernstein operators R_n converge to the identity operator if and only if Δ_n, the maximal difference between two consecutive nodes of R_n, is converging to zero. Error estimates in terms of Δ_n are provided. Moreover a Voronovskaja theorem is presented which is based on the explicit computation of higher order moments for the rational Bernstein operator.

Ognyan Kounchev, Hermann Render

We introduce a multivariate Markov transform which generalizes the well-known one-dimensional Stieltjes transform from the Moment problem and Spectral theory. Our main result states that two measures μ and ν with bounded support contained in the zero set of a polynomial P(x) are equal if they coincide on the subspace of all polynomials of polyharmonic degree N_{P} where the natural number N_{P} is explictly computed by the properties of the polynomial P(x). The method of proof depends on a definition of a multivariate Markov transform which 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 as in the one-dimensional case.

Ognyan Kounchev, Hermann Render

Methods of Padé approximation are used to analyse a multivariate Markov transform which has been recently introduced by the authors, and which is generalizing the well-known in Spectral theory Stieltjes transform (Markov function) of one-dimensional measure. 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 measures.

Ognyan Kounchev, Hermann Render

In the present paper we introduce a new concept of Hardy type space naturally defined on the Klein-Dirac quadric. We study different properties of the functions belonging to these spaces, in particular boundary value problems. We apply these new spaces to polyharmonic interpolation and to interpolatory cubature formulas.

Ognyan Kounchev, Hermann Render

In the monograph Kounchev, O. I., Multivariate Polysplines. Applications to Numerical and Wavelet Analysis, Academic Press, San Diego-London, 2001, and in the paper Kounchev O., Render, H., Cardinal interpolation with polysplines on annuli, Journal of Approximation Theory 137 (2005) 89--107, we have introduced and studied a new paradigm for cardinal interpolation which is related to the theory of multivariate polysplines. In the present paper we show that this is related to a new sampling paradigm in the multivariate case, whereas we obtain a Shannon type function $S(x) $ and the following Shannon type formula: $f(rθ) =\sum_{j=-\infty}^{\infty}\int_{\QTR{Bbb}{S}^{n-1}}S(e^{-j}rθ) f(e^{j}θ) dθ.$ This formula relies upon infinitely many Shannon type formulas for the exponential splines arising from the radial part of the polyharmonic operator $Δ^{p}$ for fixed $p\geq 1$. Acknowledgement. The first and the second author have been partially supported by the Institutes partnership project with the Alexander von Humboldt Foundation. The first has been partially sponsored by the Greek-Bulgarian bilateral project BGr-17, and the second author by Grant MTM2006-13000-C03-03 of the D.G.I. of Spain.

Ognyan Kounchev, Hermann Render

Due to its intimate relation to Spectral Theory and Schrödinger operators, the multivariate moment problem has been a subject of many researches, so far without essential success (if one compares with the one--dimensional case). In the present paper we reconsider a basic axiom of the standard approach - the positivity of the measure. We introduce the so--called pseudopositive measures instead. One of our main achievements is the solution of the moment problem in the class of the pseudopositive measures. A measure \ $μ$ is called pseudopositive if its Laplace-Fourier coefficients $μ_{k,l}(r) ,$ $r\geq0,$ in the expansion in spherical harmonics are non--negative. Another main profit of our approach is that for pseudopositive measures we may develop efficient ''cubature formulas'' by generalizing the classical procedure of Gauss--Jacobi: for every integer \ $p\geq1$ we construct a new pseudopositive measure $ν_{p}$ having ''minimal support'' and such that $μ(h) =ν_{p}(h) $ for every polynomial $h$ with $Δ^{2p}h=0.$ The proof of this result requires application of the famous theory of Chebyshev, Markov, Stieltjes, Krein for extremal properties of the Gauss-Jacobi measure, by employing the classical orthogonal polynomials $p_{k,l;j},$ $j\geq0,$ with respect to every measure $μ_{k,l}.$ As a byproduct we obtain a notion of multivariate orthogonality defined by the polynomials $p_{k,l;j}$. A major motivation for our investigation has been the further development of new models for the multivariate Schrödinger operators, which generalize the classical result of M. Stone saying that the one--dimensional orthogonal polynomials represent a model for the self--adjoint operators with simple spectrum.