Multiplicative perturbations of the Laplacian and related approximation problems

Of concern are multiplicative perturbations of the Laplacian acting on weighted spaces of continuous functions on \mathbbR^N,  N 3 1{{\mathbb{R}}^{N},\; N\geq1} . It is proved that such differential operators, defined on their maximal domains, are pre-generators of positive quasicontractive C ₀-semigroups of operators that fulfill the Feller property. Accordingly, these semigroups are associated with a suitable probability transition function and hence with a Markov process on \mathbbR^N{{\mathbb{R}}^{N}} . An approximation formula for these semigroups is also stated in terms of iterates of integral operators that generalize the classical Gauss-Weierstrass operators. Some applications of such approximation formula are finally shown concerning both the semigroups and the associated Markov processes.     

A Korovkin-Type Approximation Theorem and Power Series Method

In this paper, using power series methods we give an abstract Korovkin type approximation theorem for a sequence of positive linear operators mapping \({C\left(X, \mathbb{R}\right)}\) into itself.     

一类Post-Widder算子线性组合的加权Lp逼近

本文研究了一类Post-Widder算子的线性组合加Jacobi型权的逼近问题.运用K-泛函和光滑模方法,建立了这类算子在Lp(1≤p≤∞)空间的逼近正、逆定理及特征刻划,所获得结果推广了经典的Post-Widder算子逼近的相关结论.     

On a certain seqence of positive linear operators and tow optimization inequalities

In this Paper the approximation of continuous functions by positive linear operators of Bernstein type is investigated. The consideered operators are constructed using a system of rational functions with prescribed matrix of real poles. A certain general problem of S Bernstein concerning a scheme of construction of a sequence of positive linear operators is discussed. The answer on the Bernstein's hypothesis is given. The optimal limiting relations for the norm of the second central moment of our sequence of operators are established.     

Galerkin method for Wiener-Hopf operators with piecewise continuous symbol

This paper is concerned with the applicability of maximum defect polynomial (Galerkin) spline approximation methods with graded meshes to Wiener-Hopf operators with matrix-valued piecewise continuous generating function defined on R. For this, an algebra of sequences is introduced, which contains the approximating sequences we are interested in. There is a direct relationship between the stability of the approximation method for a given operator and invertibility of the corresponding sequence in this algebra. Exploring this relationship, the methods of essentialization, localization and identification of the local algebras are used in order to derive stability criteria for the approximation sequences.Supported by grant Praxis XXI/BD/4501/94 from FCT.Partly supported by FCT/BMFT grant 423.     

On Approximation Properties of Generalized q-Bernstein Operators

In this study, we identify a generalization of q-Bernstein type operators and investigate approximation properties of a sequence of these operators . We estimate rate of approximation by modulus of continuity. We prove Voronovskaya type theorem for these operators.     

The complete asymptotic expansion for Bernstein operators

In this paper we study the asymptotic behavior of the classical Bernstein operators, applied to q-times continuously differentiable functions. Our main results extend the results of S.N. Bernstein and R.G. Mamedov for all q-odd natural numbers and thus generalize the theorem of E.V. Voronovskaja. The exact degree of approximation is also proved.     

The Durrmeyer type modification of the q-Baskakov type operators with two parameter α and β

In this paper, we are dealing with q analogue of Durrmeyer type modified the Baskakov operators with two parameter α and β, which introduces a new sequence of positive linear q-integral operators. We show that this sequence is an approximation process in the polynomial weighted space of continuous function defined on the interval [0, ∞). We study moments, weighted approximation properties, the rate of convergence using a weighted modulus of smoothness, asymptotic formula and better error estimation for these operators.     

APPROXIMATION ORDER AND INTERPOLATION SPACES

In this paper we introduce the two-parameter operators on Abelian group and establish their interpolation theorems of approximation, which are extensions of the interpolation theorems for nonlinear best approximation by R. Devore and are suitable for the approximation of oprators.     

An application of Bernstein-Durrmeyer operators

In the present paper, we find that the Bemstein-Durrmeyer operators, besides their better applications in approximation theory and some other fields, are good tools in constructing translation network. With the help of the de la Vallee properties of the Bernstein-Durrmeyer operators a sequence of translation network operators is constructed and its degree of approximation is dealt.     

Simultaneous approximation by a class of Bernstein-Durrmeyer operators preserving linear functions

We introduce and study a one-parameter class of positive linear operators constituting a link between the well-known operators of S.N. Bernstein and their genuine Bernstein-Durrmeyer variants. Several limiting cases are considered including one relating our operators to mappings investigated earlier by Mache and Zhou. A recursion formula for the moments is proved and estimates for simultaneous approximation of derivatives are given.     

Rate of approximation for functions of bounded variation by integral operators

We estimate pointwise convergence rates of approximation for functions of bounded variation and for functions which are exponentially bounded and locally of bounded variation. The approximation is through the operation of a sequence of integral operators with not necessarily positive kernel functions. The general result is then applied to deduce estimates for particular operators, such as Beta operators, Fourier–Legendre operators, Picard operators, and Gauss–Weierstrass operators.     

Approximation by nonlinear integral operators via summability process

In this paper, we study the approximation properties of nonlinear integral operators of convolution-type by using summability process. In the approximation, we investigate the convergence with respect to both the variation semi-norm and the classical supremum norm. We also compute the rate of approximation on some appropriate function classes. At the end of the paper, we construct a specific sequence of nonlinear operators, which verifies the summability process. Some graphical illustrations and numerical computations are also provided.     

Rate of approximation for functions with derivatives of bounded variation

We estimate pointwise convergence rates of approximation for functions with derivatives of bounded variation and for functions which are exponentially bounded and have derivatives locally of bounded variation. The approximation is made through the operation of a sequence of integral operators with not necessarily positive kernel functions. The general result is then applied to deduce estimates for Beta operators, Hermite-Fejér operators, Picard operators, Gauss-Weierstrass operators, Baskakov operators, Mirakjan-Szász operators, Bleimann-Butzer-Hahn operators, Phillips operators, and Post-Widder operators.     

W_2~m(R~n)中连续泛函及连续算子的神经网络逼近

本文主要讨论定义在Sobolev空间Wm2 (Rn)中紧集上连续泛函及连续算子的逼近 .     

Approximation theorems by positive linear operators in weighted spaces

In this study, we obtain some Korovkin type approximation theorems by positive linear operators on the weighted space of all real valued functions defined on the real two-dimensional Euclidean space \mathbbR²{\mathbb{R}^2}. This paper is mainly consisted of two parts: a Korovkin type approximation theorem via the concept of A-statistical convergence and a Korovkin type approximation theorem via A{\mathcal {A}}-summability.     

Complete Asymptotic Expansions for Altomare Operators

In the present paper, we study an approximation process by a sequence of operators with a prescribed asymptotic behavior. The method of construction of these operators is due to Altomare and Amiar. The operators have several applications in semigroup theory. Our investigation provides an insight in their asymptotic behaviour. Finally, we apply our results to concrete examples of approximation operators recently defined and used by Altomare and Milella for the study of semigroups.     

Vallée-Poussin算子逼近连续函数的最优估计

通过将Vallée-Poussin算子逼近连续函数的能力转化为对辅助数列{g(n)}的上确界的计算,首先利用数列单调有界定理证明辅助数列极限的存在性,之后借助夹逼准则求得辅助数列{g(n)}的极限,即数列{g(n)}的上确界,进而得到Vallée-Poussin算子逼近连续函数的最优估计常数.     

The Simultaneous Approximation Average Errors for Bernstein Operators on the $r$-Fold Integrated Wiener Space

For weighted approximation in $L_p$ -norm, we determine strongly asymptotic orders for the average errors of both function approximation and derivative approximation by the Bernstein operators sequence on the $r$-fold integrated Wiener space.     

On a new sequence of positive linear operators related to squared Bernstein polynomials

We define a new sequence of positive linear approximation operators by means of the squared Bernstein polynomials and estimate the rate of approximation.