Ognyan Kounchev

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.

Ognyan Kounchev, Damyan Kalaglarsky, Milcho Tsvetkov

We consider the application of the polyharmonic subdivision wavelets (of Daubechies type) to Image Processing, in particular to Astronomical Images. The results show an essential advantage over some standard multivariate wavelets and a potential for better compression.

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.

Bozhidar Srebrov, Ognyan Kounchev, Georgi Simeonov

We provide a Wavelet analysis of Big Data in Solar Terrestrial Physics. In order to explain and predict the dynamics of the geomagnetic phenomena we analyze high frequency time series data from different sources: 1. The Interplanetary Magnetic Field (from the ACE satellite). 2. The Ionospheric parameters - TEC (from ionospheric sounding stations). 3. The ground Geomagnetic data (from ground geomagnetic observatories, located in middle geographic latitudes). We seek for correlations in the wavelet coefficients which explain the dynamics of different magnetic phenomena in the Solar Terrestrial Physics. The large variety of data used in our research from both Solar Astronomy and Earth Observations makes it a contribution to the newly developing area of AstroGeoInformatics.

Ognyan Kounchev, Damyan Kalaglarsky

We continue the study of a new family of multivariate wavelets which are obtained by "polyharmonic subdivision". We provide the results of experiments considering the distribution of the wavelet coefficients for the Lena image and for astronomical images. The main purpose of this investigation is to find a clue for proper quantization algorithms.

Ognyan Kounchev

In the present paper we introduce the notion of harmonicity modulus and harmonicity K-functional and apply these notions to prove a Jackson type theorem for approximation of continuous functions by polyharmonic functions. For corresponding results on approximation by polynomials see [3, 7].

Werner Haussmann, Ognyan Kounchev

In the present paper we will introduce a new approach to multivariate interpolation by employing polyharmonic functions as interpolants, i.e. by solutions of higher order elliptic equations. We assume that the data arise from $C^{\infty}$ or analytic functions in the ball $B_{R}.$ We prove two main results on the interpolation of $C^{\infty}$ or analytic functions $f$ in the ball $B_{R}$ by polyharmonic functions $h$ of a given order of polyharmonicity $p.$

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, Damyan Kalaglarsky

We apply successfully the Compressive Sensing approach for Image Analysis using the new family of Polyharmonic Subdivision wavelets. We show that this approach provides a very efficient recovery of the images based on fewer samples than the traditional Shannon-Nyquist paradigm. We provide the results of experiments with PHSD wavelets and Daubechies wavelets, for the Lena image and astronomical images.

Ognyan Kounchev

We provide a definition of Multidimensional Chebyshev Systems of order N which is satisfied by the solutions of a wide class of elliptic equations of order 2N. This definition generalizes a very large class of Extended Complete Chebyshev systems in the one-dimensional case. This is the first of a series of papers in this area, which solves the longstanding problem of finding a satisfactory multidimensional generalization of the classical Chebyshev systems introduced already by A. Markov more than hundered years ago, and studied later by S. Bernstein and M. Krein.

Ognyan Kounchev

Recently the theory of widths of Kolmogorov-Gelfand has received a great deal of interest due to its close relationship with the newly born area of Compressive Sensing in Signal Processing. However fundamental problems of the theory of widths in multidimensional Theory of Functions remain untouched, as well as analogous problems in the theory of multidimensional Signal Analysis. In the present paper we provide a multidimensional generalization of the original result of Kolmogorov about the widths of an "ellipsoidal sets" consisting of functions defined on an interval.

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

Recently the theory of widths of Kolmogorov-Gelfand has received a great deal of interest due to its close relationship with the newly born area of Compressed Sensing. It has been realized that widths reflect properly the sparsity of the data in Signal Processing. However fundamental problems of the theory of widths in multidimensional Theory of Functions remain untouched, as well as analogous problems in the theory of multidimensional Signal Analysis. In the present paper we provide a multidimensional generalization of the original result of Kolmogorov by introducing a new hierarchy of infinite-dimensional spaces based on solutions of higher order elliptic equation.

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.