## Electronically Available Papers and Preprints

Please be advised that according to the international law all papers may be downloaded for private use only!
Please note that sometimes the downloadable stuff is a preprint and not the paper which has appeared, and there might be some minor differences or small errors.

Kounchev, O.; Render, H. Cardinal interpolation with polysplines on annuli. J. Approx. Theory 137 (2005), no. 1, 89--107.

Preprint:   pdf (231K)

Resume:  We provide a complete analog to the one-dimensional interpolation theorem of Schoenberg where the data are on spheres of radii  e^j . The interesting feature of the cardinal polysplines on annuli is that they enrich our understanding of cardinality.

Alexiev, Kiril; Kounchev, Ognyan; Render, Hermann Stable recurrence relations for a class of $L$-splines and for polysplines. Geometric modeling and computing: Seattle 2003, 1--12, Mod. Methods Math., Nashboro Press, Brentwood, TN, 2004.

Preprint: Postscript   pdf (139K)

Resume: For a special class of exponential splines we prove the complete analog to the recurrence relations for the classical splines - the most remarkable is that they are stable since they have positive coefficients. The result has implications for  recurrence relations between polysplines.

Kounchev, Ognyan; Render, Hermann:  Polyharmonic splines on grids $\Bbb Z\times a\Bbb Z\sp n$ and their limits. Math. Comp. 74 (2005), no. 252, 1831—1841.

Preprint: Postscript  pdf (177K)

Resume: Limits of polyharmonic splines are polysplines. A result which is intuitively clear since the representation of the polyharmonic functions of order $p$ by means of the Green formula (for polyharmonic functions)  contains precisely the fundamental solution of the polyharmonic functions of order $p$. The Green formula may be considered as a continual analog to the polyharmonic splines.

Bejancu, Aurelian; Kounchev, Ognyan; Render, Hermann:            Cardinal interpolation with biharmonic polysplines on strips

appeared in: Curve and surface fitting (Saint-Malo, 2002), 41--58, Mod. Methods Math., Nashboro Press, Brentwood, TN, 2003.
Preprint: Postscript  pdf (191K)

Resume: The result is an analog to the cardinal polysplines interpolation on annuli which is partially available in the book "Multivariate Polysplines".

Kounchev, Ognyan; Render, Hermann:            The approximation order of polysplines.

appeared in:  Proc. Am. Math. Soc. 132, No.2, 455-461 (2004).
Preprint: Postscript  pdf (234K)

Resume:  The result shows a complete analog to the univariate case of approximation order by polynomial splines.

Kounchev, Ognyan; Wilson, Michael:

Application of PDE methods to Visualization of Heart Data.

appeared in: Michael J. Wilson, Ralph R. Martin (Eds.): Mathematics of Surfaces, 10th IMA International Conference, Leeds, UK, September 15-17, 2003, Proceedings. Lecture Notes in Computer Science 2768, Springer-Verlag,  2003;   pp. 377-391..

Preprint:    pdf (890K)

Resume: The polysplines on cylinder (or polysplines on strips with periodic conditions) are used to model the Left ventricle of the heart.

Kounchev, Ognyan; Render, Hermann:

Wavelet analysis of cardinal L-splines and construction of multivariate prewavelets.
Preprint University of Duisburg, SM-DU-516;

appeared in: Chui, Charles K. (ed.) et al., Approximation theory X. Wavelets, splines, and applications. Papers from the 10th international symposium, St. Louis, Mo, USA, March 26-29, 2001. Nashville, TN: Vanderbilt University Press. Innovations in Applied Mathematics. 333-353 (2002).

Preprint:    pdf (235K)

D. Dryanov and O. Kounchev:

Multivariate Bernoulli functions and polyharmonically exact cubature formula of Euler--Maclaurin,

appeared in: Mathematische Nachrichten, 226 (2001), 65--83. Preprint:    pdf (197K)

O. Kounchev, H. Render:

Symmetry properties of cardinal interpolation $L-$splines and polysplines,
Preprint University of Duisburg, SM-DU-483, 2000;

appeared in: "Trends in Approximation Theory", K. Kopotun, T. Lyche, and M. Neamtu (eds.), Vanderbilt University Press, Nashville, TN, 2001.

Preprint:    pdf (173K)

D. Dryanov and O. Kounchev:

Polyharmonically exact formula of Euler-Maclaurin, multivariate Bernoulli functions, and Poisson type formula,

appeared in: C. R. de l'acad. des sci. Paris, t. 327, No. 5 (September), 1998.

Preprint:    pdf (110K)

W. Haussmann, O. Kounchev:

Peano kernel associated with the polyharmonic mean value property in the annulus.

appeared in: Numer. Funct. Anal. Optim. 21 (2000), no. 5-6, 683--692.

Preprint:  pdf (122K)

Resume: This is an analog to the paper W. Haussmann, O. Kounchev,  Definiteness of the Peano kernel associated with the polyharmonic mean value property. (see below) but for the case of the annulus.

W. Haussmann, O. Kounchev:

Definiteness of the Peano kernel associated with the polyharmonic mean value property.

appeared in: J. London Math. Soc. (2)  62 (2000), no. 1, 149--160.

Preprint  pdf (160K)

Resume: In previous papers (see below) we have demonstrated that  the mean value property of Picone and  Bramble-Payne for polyharmonic functions is analog to the difference operator for polynomials. Then it follows that the Peano kernel for that mean value property will be the analog to the B-splines. In W. Haussmann, O. Kounchev,  Preprint: "Peano kernel for harmonicity differences of order $p$" (see below), we have proved  that it is a polyspline. In the present paper the positivity of this kernel is proved which is the analog to the positivity of the B-splines.

O. Kounchev:

Extremizers for the multivariate Landau-Kolmogorov inequality

appeared in: Multivariate Approximation, W. Haussmann et al. (eds.), Akademie Verlag, 1997, p. 123-132.

Preprint:    pdf (126K)

W. Haussmann, O. Kounchev:

Preprint: "Peano kernel for harmonicity differences of order $p$"

containing  the paper "Variational property of the Peano kernel for harmonicity differences of order $p$" appeared in: Clifford Algebras and Their Applications in Mathematical Physics, V. Dietrich et al. (eds.),
Kluwer Acad. Publishers, 1998, p. 185-199.

Preprint:    pdf (178K)

Resume: The main idea is that the mean value property of Picone and Bramble-Payne for polyharmonic functions is a genuine generalization of the one-dimensional finite difference operator for polynomials. Since this property generates a functional which annihilates all polyharmonic functions of order  p, the functional has a Peano kernel which has a compact support, and is a polyspline of order p.

W. Haussmann, O. Kounchev:

Peano theorem for linear functionals vanishing on polyharmonic functions

appeared in: Approximation Theory VIII, Vol. 1, Ch. Chui and L. Schumaker (eds.), 1995, World
Scientific, p. 233-240. contained in the above preprint,
Preprint containing the above paper:   pdf (178K)

Resume: In the one-dimensional theory there is a natural class of functionals for which the so-called Peano kernel exists which is a  of the Taylor formula. We introduce multivariate functionals for which we apply the Green formula for polyharmonic functions (which is a multivariate analog to the Taylor formula) and we prove an analog to the Peano kernel.

O. Kounchev:

Optimal recovery of linear functionals of Peano type through data on manifolds.

appeared in: Comput. Math. Appl.  30 (1995), no. 3-6, p. 335--351.

Preprint:    pdf (188K)

Resume: An analog to the classical theorem of Schoenberg for optimal recovery with splines is proved; recall that it shows that the splines of odd degree are naturally generated by an optimal recovery problem which is an extremal problem. We prove that if we take data living on manifolds then the polysplines are most naturally generated by an optimal recovery problem. This may be considered as a basic result showing a natural appearance of polysplines.

O. Kounchev:

Splines constructed by pieces of polyharmonic functions,

appeared in: Wavelets, images, and surface fitting, Eds. P. Laurent et al., A. K. Peters, Wellesley, MA, 1994, p. 319-326.

Preprint:    pdf (173K)

Resume: This paper contains the computation of the polysplines on strips. It proves that the convergence rate of the polysplines is the same as the convergence rate of the one-dimensional splines, in particular it is O(h^4) for the biharmonic polysplines which correspond to the cubic one-dimensional splines.

O. Kounchev:

Zeroes of non-negative sub-biharmonic functions and extremal problems in the inverse source problem for the biharmonic potential

appeared in: Inverse Problems: Principle and Applications in Geophysics, Technology, and Medicine, Eds. G. Anger et al., Akademie Verlag, 1993, p. 243-250.

Preprint:    pdf (92K)

Resume: As we know, Chebyshev discovered that the Gauss-Jacobi quadrature formula (and the respective measure) provide a solution to extremal problems for the Moment problem. For the case of Chebyshev systems an analogous phenomenon occurs and this is considered in detail in the monographs of Krein-Nudelman and Karlin-Studden mentioned below.  In the theory of the Moment problem for Chebyshev systems developed by A. Markov and M. Krein, the "canonical representations" play an important role since they provide solution to the extremal problems for the Moment problem, and also some very distinguished among them, called "chief representations" provide Gauss-Jacobi type quadrature formulas. There is a geometric approach to the theory of the canonical representations (see the book of M. Krein and A. Nudelman) where the canonical representations are alternatively defined as solutions of extremal problems. In the present paper we provide an attempt to find multivariate “canonical representations” by using extremal problems for the multivariate “distributed moment problem”.

O. Kounchev:

On the approximation through polyharmonic operators.

Appeared in Approximation theory (Memphis, TN, 1991), 377--384, Lecture Notes in Pure and Appl. Math., 138, Dekker, New York, 1992, a volume dedicated to the memory of Vassil Popov.

Preprint:    pdf (260K)

O. Kounchev:

Harmonicity modulus and application to the approximation by polyharmonic functions

appeared in: Approximation by Solutions of Partial Differential Equations, Eds. B. Fuglede et al., Kluwer, 1992, p.
111-125.

Preprint:    pdf (460K)

Resume: In the present paper a Jackson type theorem for the approximation by polyharmonic functions is proved. This paper should be considered as a couple to the “inverse theorems of Bernstein type” for the approximation by polyharmonic functions which has been proved in  Kounchev, O., Sharp estimate of the Laplacian of a polyharmonic function and applications. Trans. Amer. Math. Soc., 332 (1992), no. 1, 121--133. In order to prove the Jackson type theorem one needs an analog to the one-dimensional “modulus of continuity” – we introduce this analog which is called “harmonicity modulus”; it appears in a very natural way in the proof. This is a new fundamental notion which is further developed and generalized to “harmonicity modulus of order p”, see above the papers “Peano kernel for harmonicity differences of order $p$" and  “Peano theorem for linear functionals vanishing on polyharmonic functions”.

O. Kounchev:

Extremal problems for the distributed moment problem

appeared in: Potential Theory, Proc. conf. On Potential theory, Prague, 1987; eds. J. Kral, J. Lukes, I. Netuka, J. Vesely, p. 187-195.

Preprint:    pdf (2.172 K)

Resume: If a truncated Moment problem of degree 2n is given for the classical polynomials, then the orthogonal polynomials provide us with a method to find knots  t_j  (zeros of the n-th orthogonal polynomial P_n(t) ) and weights  \lambda_j (Christoffel coefficients) for a Gauss-Jacobi quadrature formula – this is in fact the measure  d\mu (t)= \sum_j=1^n   \lambda_j  \delta(t-t_j)  which solves the Moment problem. In the case of the truncated Moment problem for a Chebyshev system no such method exists. The replacement which has been developed by M. Krein is the method of maximal mass and construction of canonical representations (they have been studied by Markov) which I have mentioned below by the paper  “Distributed Moment problem and some related questions on approximation of functions of many variables”. This method provides a measure which has “a minimal support” and is an analog to the Gauss-Jacobi measure, and there is also a quadrature formula, see Theorem 4.1 in chapter 4, section 1, in the book M. Krein and A. Nudelman “The Markov Moment Problem and Extremal Problems”, AMS, 1976. So far there is also a so-called geometric approach for the construction of such Gauss-Jacobi type measures which is developed in chapter 4, section 6, and it is based on extremal problems.

One has to mention that the construction of a Gauss-Jacobi type quadrature formula for the Chebyshev systems (or, their equivalent, called "chief representation measures" in  Krein-Nudelman) is the main driving force for the Krein-Nudelman monograph; it is the main focus also of the book of S. Karlin and W. Studden (Tchebycheff Systems: with Applications in Analysis and Statistics, Intersci. publishers, 1966). So far, the only case in which one has an explicit construction of the Gauss-Jacobi quadrature formula is the classical case - of algebraic or trigonometric polynomials, by means of the classical orthogonal polynomials. It seems to be a major challenge until today - to find the analog to the orthogonal polynomials for other Chebyshev systems.

In the present paper, I  tried to follow this geometric approach to construct analogs to the Gauss-Jacobi measures for the Distributed Moment Problem (which may be called "truncated polyharmonic Moment Problem") as a solution to an extremal problem  – see Theorem 1 and the conjecture thereafter;  even the biharmonic case is rather non-trivial, see above the paper “Zeroes of non-negative sub-biharmonic functions and extremal problems in the inverse source problem for the biharmonic potential”.

The truncated polyharmonic Moment Problem was understood and solved only recently in a joint research with H. Render and the solution brought a new concept into the game - the so-called pseudo-positive measures.

O. Kounchev:

Duality properties for the extreme values of integrals  in distributed moments

appeared in: Differential Equations and Applications, Proc. of the third conference, Rousse, 1985.

Preprint:    pdf (510 K)

Resume: This paper continues to establish the basic properties of the Distributed Moment Problem (truncated polyharmonic Moment Problem) – we prove an analog to Theorem 4.1, chapter 4, section 4 in the book M. Krein and A. Nudelman “The Markov Moment Problem and Extremal Problems”, AMS, 1976.

O. Kounchev:

Distributed Moment problem and some related questions  on approximation of functions of many variables

appeared in:  “Mathematics and Education in Mathematics”, Proceedings of 14th spring conference of the Union of Bulgarian Mathematicians, Sunny Beach, 1985, p. 454-458.

Preprint:    pdf (2.180 K)

Resume: The main point of this paper is to provide a proper setting for a new multivariate (truncated) Moment problem which is an analog to the one-dimensional truncated Moment problem, and which we called Distributed Moment Problem, following analogies with the Moment problem arising in the Optimal Control of PDEs (e.g., in the monograph A.G. Butkovskii, Theory of Optimal Control of Distributed Parameter Systems, Moscow, 1965 (English Translation, Elsevier, Amsterdam, 1969). This comes from the understanding that the polyharmonic functions of order p are an analog to the one-dimensional polynomials of degree p+1, and are a multivariate Chebyshev system (the Polyharmonic Paradigm). In this paper I tried to generalize the method of maximal mass of M. Krein (see Theorem 3.1 and 3.2 in chapter 3, section 3 in the book cited below) -  in Theorem 2 (there is a flaw in the proof). In Theorem 3 the basic duality theorem is proved which is an analog to the one-dimensional such considered in the book of  M. Krein and A. Nudelman “The Markov Moment Problem and Extremal Problems”, AMS, 1976 – see  Theorem 1.1 in chapter 9, section 1.

This kind of setting but from a different point of view - of the Inverse Problems in Potential Theory - has appeared for the first time in B.-W. Schulze; G. Wildenhain: Methoden der Potentialtheorie für elliptische Differentialgleichungen beliebiger Ordnung. Akademie-Verlag Berlin, Birkhäuser-Verlag Basel, Stuttgart (1977). In this book I saw cited the name of Dimiter Zidarov – a Bulgarian physicist and geophysicist working in the area of Inverse source problems in Potential theory.

O. Kounchev:

A nonlocal maximum principle about for the biharmonic equation, Almansi type formulae for operators which are squares of elliptic operators of second order and approximation by their solutions.

appeared in: Jubilee session devoted to the 100th anniversary of the birth of Acad. L. Chakalov, Samokov, 1986, p. 88-92.
111-125.

Preprint:    pdf (2.160 K)

Resume: This paper is a continuation of the paper below “About the harmonic function which deviates least from a given continuous in the circle (in Russian)”,  Doklady Belorussian Acad. of Sci.  29 (1985), no. 4, p. 293-295, but for the polyharmonic case. It is a desperate attempt to do what is possible to do on the best uniform approximation by polyharmonic functions. It is just a beautiful alternance sufficient condition. Could one expect more?

By that time it has become clear to me that there is not much to say about the zero sets of the polyharmonic functions and that even the biharmonic functions may show very monstrous properties compared with the nice harmonic functions – I have seen in a book by Atakhodzaev (which has been published in Russian in Tashkent in 1983 or so) an example of a biharmonic function in the plane which is zero on two convex ovals which are concentric. Not less weird is the behavior of the Green function of the clamped plate – it has been conjectured in 1908 by Hadamard that it is sign positive; it has been disproved by Duffin in 1948 and a series of counterexamples have appeared in 1953 by Garabedian and Szegoe.

O. Kounchev:

About the harmonic function which deviates least from a given continuous in the circle (in Russian)

appeared in:  Doklady Belorussian Acad. of Sci.  29 (1985), no. 4, p. 293-295.

Preprint:    pdf (1.800 K)

Resume: The title is reminding of the famous paper of  P. L. Chebyshev “About the polynomial which deviates least from a given continuous function” where the notion of alternance polynomial appeared. The context of the above paper is the following: In 1983 it has become clear to me that the polyharmonic functions provide a genuine multivariate generalization of the so-called Chebyshev systems (one needs a proper definition!) – later I called this “Polyharmonic Paradigm”.  Then I decided to prove some results which are available in the one-dimensional case about best approximation – in the book of  M. Krein and A. Nudelman “The Markov Moment Problem and Extremal Problems”, AMS, 1976,  one finds the characterization of best L_1 and best uniform approximations by (one-dimensional) Chebyshev systems. I decided that an analog to the Chebyshev’s best uniform approximation by polynomials would be the best proof that the polyharmonic functions (of order p) are really a genuine multidimensional Chebyshev system (of order p); for simplicity sake, I decided first to manage the case with the harmonic functions since I could not find any deeper results about the qualitative behavior of the polyharmonic functions; the harmonic functions in  n dimensions  correspond to the linear polynomials in 1 dimension, thus the above result settles the simplest possible case. In 1988 Werner Haussmann (from University of Duisburg, Germany) brought to my attention that this result has been proved and published in a proceedings paper in 1984 by W. Hayman, D. Kershaw and T. Lyons, and as is seen, my paper was submitted on March 30, 1984 (so I did not plagiarize from them …).  By the way my paper was submitted by academician Vladimir Krylov (known best by his famous monograph “Approximate Calculation of Integrals”, 1962; he should not be mixed with Alexei Nikolaevich Krylov after whom the Krylov subspace method is named)  whom I visited at his home. I explained the result, and he wrote a recommendation letter (report) about the publication in Doklady. He liked very much the fact that the main result is a generalization of Chebyshev’s theorem since Krylov came from Leningrad (St. Petersburg) where also Chebyshev lived 100 year ago.

Anyway, it has been discovered also by Werner Haussmann that the same result has been published already by H. Burchard (USA)  in a proceedings paper in 1976  J .

Today I think that this start, with trying to prove a generalization of Chebyshev’s alternance theorem, was one of the most difficult one could imagine, but I wanted to do a beautiful and deep mathematics.

Further papers will be added.