Ognyan Kounchev

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

Minimizing the Laplacian of a function squared with prescribed values on interior boundaries- Theory of polysplines

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

Application of PDE Methods to Visualization of Heart Data

satisfies

Extremal Problems for the Distributed Moment Problem

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

A moment problem for pseudo-positive definite functionals

, 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 Binomial-Tree Model for Convertible Bond Pricing

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.

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

In this paper we consider the minimization of the integral of the Laplacian of a real-valued function squared (and more general functionals) with prescribed values on some interior boundaries r, with the integral taken over the domain D. We prove that the solution is a biharmonic function in D except on the interior boundaries r, and satisfies some matching conditions on r. There is a close analogy with the one-dimensional cubic splines, which is the reason for calling the solution a polyspline of order 2, or biharmonic polyspline. Similarly, when the quadratic functional is the integral Of (<qf)2, q a positive integer, then the solution is a polyharmonic function of order 2q, /\2qf(x) = O, for x E D \ r, satisfying matching conditions on r, and is called a polyspline of order 2q. Uniqueness and existence for polysplines of order 2q, provided that the interior boundaries r are sufficiently smooth surfaces arld AD C r, is proved. Three examples of data sets r possessing symmetry are considered, in which the computation of polysplines is reduced to computation of onedimensional L-splines.

Padé approximation for a multivariate Markov transform

We show that the scaling spaces defined by the polysplines of order p provide approximation order 2p. For that purpose we refine the results on one-dimensional approximation order by L-splines obtained by de Boor, DeVore, and Ron (1994).