On integral bernstein operators in some classes of measurable bivariate functions

The two main theorems are concerned with the approximations of (complex-valued) functions on the real plane by sums of Bernstein pseudoentire functions. They are formulated and proved in Section 4, after prior determination of the suitable integral operators. Analogous results for pseudopolynomial approximations were obtained by Brudnyî, Gonska, and Jetter ([2],[3]).Research supported by KBN grant 2 1079 91 01.     

Convergence rate of a new Bezier variant of Chlodowsky operators to bounded variation functions

In the present paper, we estimate the rate of pointwise convergence of the Bézier Variant of Chlodowsky operators _Cn,α$C_{n,\alpha }$ for functions, defined on the interval extending infinity, of bounded variation. To prove our main result, we have used some methods and techniques of probability theory.     

Convergence in Variation and Rate of Approximation for Nonlinear Integral Operators of Convolution Type

In this paper we obtain estimates, convergence results and rate of approximation for functions belonging to BV–spaces (spaces of functions with bounded variation) by means of nonlinear convolution integral operators. We treat both the periodic and the non-periodic case using, respectively, the classical Jordan variation and the multidimensional variation in the sense of Tonelli.     

On the convergence properties of basic series representing special monogenic functions

In this paper, it is shown that certain classes of special monogenic functions cannot be represented by the basic series in the whole space. New definitions for the order of basis of special monogenic polynomials are given together with theorems on representation of classes of special monogenic functions in certain balls and at a point. Received: 8 January 2002     

Approximation by complex genuine Durrmeyer type polynomials in compact disks

In this paper, the order of simultaneous approximation and Voronovskaja kind results with quantitative estimate for the complex genuine Durrmeyer polynomials attached to analytic functions on compact disks are obtained. In this way, we put in evidence the overconvergence phenomenon for the genuine Durrmeyer polynomials, namely the extensions of the approximation properties (with quantitative estimates) from real intervals to compact disks in the complex plane.     

The square root base of polynomials in Clifford analysis

In this paper the problem of taking the square root of bases of special monogenic polynomials is studied, thus leading to a number of results under some additional conditions of associated infinite matrices, related essentially to the so-called algebraicness of these matrices.     

Quasi-interpolation for linear functional data

Quasi-interpolation has been studied extensively in the literature. However, most studies of quasi-interpolation are usually only for discrete function values (or a finite linear combination of discrete function values). Note that in practical applications, more commonly, we can sample the linear functional data (the discrete values of the right-hand side of some differential equations) rather than the discrete function values (e.g., remote sensing, seismic data, etc). Therefore, it is more meaningful to study quasi-interpolation for the linear functional data. The main result of this paper is to propose such a quasi-interpolation scheme. Error estimate of the scheme is also given in the paper. Based on the error estimate, one can find a quasi-interpolant that provides an optimal approximation order with respect to the smoothness of the right-hand side of the differential equation. The scheme can be applied in many situations such as the numerical solution of the differential equation, construction of the Lyapunov function and so on. Respective examples are presented in the end of this paper.     

WEIERSTRASS' THEOREM IN THE TWENTIETH CENTURY: A SELECTION

Abstract

We survey some of the extensions and ramifications of Weierstrass' theorem in the twentieth century, discussing first qualitative theory (can we approximate?) and then quantitative theory (how fast can we approximate?).     

Some properties of the Cauchy-type integral for the time-harmonic Maxwell equations

We study the analog of the Cauchy-type integral for the theory of time-harmonic electromagnetic fields in case of a piece-wise Liapunov surface of integration and we prove the Sokhotski-Plemelj theorem for it as well as the necessary and sufficient condition for the possibility to extend a given pair of vector fields from such a surface up to a solution of the time-harmonic Maxwell equations in a domain. Formula for the square of the singular Cauchy-type integral is given. The proofs of all these facts are based on intimate relations between time-harmonic solutions of the Maxwell equations and some versions of quaternionic analysis.     

On best error bounds for approximation by piecewise polynomial functions

Summary An analog of the well-known Jackson-Bernstein-Zygmund theory on best approximation by trigonometric polynomials is developed for approximation methods which use piecewise polynomial functions. Interpolation and best approximation by polynomial splines, Hermite and finite element functions are examples of such methods. A direct theorem is proven for methods which are stable, quasi-linear and optimally accurate for sufficiently smooth functions. These assumptions are known to be satisfied in many cases of practical interest. Under a certain additional assumption, on the family of meshes, an inverse theorem is proven which shows that the direct theorem is sharp.The work presented in this paper was supported by the ERDA Mathematics and Computing Laboratory, Courant Institute of Mathematical Sciences, New York University, under Contract E(11-1)-3077 with the Energy Research and Development Administration.     

Robust exponential attractors for singularly perturbed Hodgkin-Huxley equations

Here we consider a singular perturbation of the Hodgkin-Huxley system which is derived from the Lieberstein's model. We study the associated dynamical system on a suitable bounded phase space, when the perturbation parameter ε (i.e., the axon specific inductance) is sufficiently small. We prove the existence of bounded absorbing sets as well as of smooth attracting sets. We deduce the existence of a smooth global attractor Aε. Finally we prove the main result, that is, the existence of a family of exponential attractors {Eε} which is Hölder continuous with respect to ε.     

Limit problems for interpolation by analytic radial basis functions

Interpolation problems for analytic radial basis functions like the Gaussian and inverse multiquadrics can degenerate in two ways: the radial basis functions can be scaled to become increasingly flat, or the data points coalesce in the limit while the radial basis functions stay fixed. Both cases call for a careful regularization, which, if carried out explicitly, yields a preconditioning technique for the degenerating linear systems behind these interpolation problems. This paper deals with both cases. For the increasingly flat limit, we recover results by Larsson and Fornberg together with Lee, Yoon, and Yoon concerning convergence of interpolants towards polynomials. With slight modifications, the same technique can also handle scenarios with coalescing data points for fixed radial basis functions. The results show that the degenerating local Lagrange interpolation problems converge towards certain Hermite–Birkhoff problems. This is an important prerequisite for dealing with approximation by radial basis functions adaptively, using freely varying data sites.     

On the Notion of the Bochner–Martinelli Integral for Domains with Rectifiable Boundary

In this paper we discuss the notion of the Bochner–Martinelli kernel for domains with rectifiable boundary in , by expressing the kernel in terms of the exterior normal due to Federer (see [17,18]). We shall use the above mentioned kernel in order to prove both Sokhotski–Plemelj and Plemelj–Privalov theorems for the corresponding Bochner–Martinelli integral, as well as a criterion of the holomorphic extendibility in terms of the representation with Bochner–Martinelli kernel of a continuous function of two complex variables. Explicit formula for the square of the Bochner–Martinelli integral is rediscovered for more general surfaces of integration extending the formula established first by Vasilevski and Shapiro in 1989. The proofs of all these facts are based on an intimate relation between holomorphic function theory of two complex variables and some version of quaternionic analysis. Submitted: September 6, 2006. Accepted: November 1, 2006.     

Pseudospectral method for quadrilaterals

In this paper, we investigate the pseudospectral method on quadrilaterals. Some results on Legendre-Gauss-type interpolation are established, which play important roles in the pseudospectral method for partial differential equations defined on quadrilaterals. As examples of applications, we propose pseudospectral methods for two model problems and prove their spectral accuracy in space. Numerical results demonstrate the efficiency of the suggested algorithms. The approximation results and techniques developed in this paper are also applicable to other problems defined on quadrilaterals.