O. Kounchev
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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.
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.
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.
appeared in: Curve and surface fitting (
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".
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.
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,
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.
Wavelet analysis
of cardinal L-splines and construction of multivariate prewavelets.
Preprint
appeared in: Chui, Charles K. (ed.) et al., Approximation theory X. Wavelets,
splines, and applications. Papers from the 10th international symposium,
Preprint: pdf
(235K)
Multivariate Bernoulli functions and polyharmonically exact cubature formula of
Euler--Maclaurin,
appeared in: Mathematische Nachrichten, 226 (2001), 65--83.
Preprint: pdf (197K)
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,
Preprint: pdf
(173K)
Polyharmonically
exact formula of Euler-Maclaurin, multivariate Bernoulli functions, and Poisson
type formula,
appeared in: C. R. de l'acad. des sci.
Preprint: pdf (110K)
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.
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)
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.
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
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.
Splines
constructed by pieces of polyharmonic functions,
appeared in: Wavelets, images, and surface fitting, Eds. P. Laurent et al., A.
K. Peters,
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.
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”.
On the approximation through polyharmonic operators.
Appeared in Approximation theory
(Memphis, TN, 1991), 377--384, Lecture
Notes in Pure and Appl. Math., 138,
Preprint: pdf
(260K)
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
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.
