A certain family of summation-integral type operators

In the present paper, the authors introduce and investigate a new sequence of linear positive operators G_n,c, which includes some well-known operators as its special cases. The results obtained here include an estimate on the rate of convergence of G_n,c (f, x) by means of the decomposition technique for functions of bounded variation.     

Direct and inverse estimates for a new family of linear positive operators

In the present paper we introduce a new family of linear positive operators and study some direct and inverse results in simultaneous approximation.     

Some results on modified Szász-Mirakjan operators

In this paper we study the mixed summation-integral type operators having Szász and Beta basis functions. We extend the study of Gupta and Noor [V. Gupta, M.A. Noor, Convergence of derivatives for certain mixed Szász-Beta operators, J. Math. Anal. Appl. 321 (1) (2006) 1-9] and obtain some direct results in local approximation without and with iterative combinations. In the last section are established direct global approximation theorems.     

Global smoothness and uniform convergence of smooth Poisson-Cauchy type singular operators

In this article we introduce the smooth Poisson-Cauchy type singular integral operators over the real line. Here we study their simultaneous global smoothness preservation property with respect to the L_p norm, 1?p?∞, by involving higher order moduli of smoothness. Also we study their simultaneous approximation to the unit operator with rates involving the modulus of continuity with respect to the uniform norm. The produced Jackson type inequalities are almost sharp containing elegant constants, and they reflect the high order of differentiability of the engaged function.     

Convergence of derivatives for certain mixed Szasz-Beta operators

In this paper we study the mixed summation-integral type operators having Szasz and Beta basis functions in summation and integration, respectively, we obtain the rate of point-wise convergence, a Voronovskaja type asymptotic formula and an error estimate in simultaneous approximation.     

Derivatives of multidimensional Bernstein operators and smoothness

We characterize the directional derivatives of multidimensional Bernstein operators by a new measure of smoothness. This task is carried out by means of establishing the relation between the asymptotic behavior of the derivatives and the smoothness of the functions they approximate. The obtained results generalize the corresponding ones for univariate Bernstein operators.     

在C（S）空间中多元的Bernstein—Kantorovich算子的正逆定理

在C（S）空间中，对定义在单形上的Bernstein－Kantorovich算子Kn（f），给出了一个积分型估计式及一个弱型逆定理，得到了当0＜α＜1时，其中为定义在C（S）空间中的Ditzian－Totick光滑模[1]．     

On an extremal relation of Bernstein operators

In this note we present a new characterization of Bernstein operators by showing that they are the only solution of a certain extremal relation.     

A New Characterization of Weighted Peetre K-Functionals

The purpose of this paper is to present a characterization of certain types of generalized weighted Peetre K-functionals by means of a modulus of smoothness. This new modulus is based on the classical one taken on a certain linear transform of the function. A new modulus of smoothness which describes the best algebraic approximation is introduced.     

Bernstein-Kantorovich 算子线性组合同时逼近的正逆定理

借助光滑模ωtφ三(f,t)给出了Bernstein-Kantorovich 算子线性组合同时逼近的正逆定理,其中φ是一般步权函数,对已有的结果进行了补充和完善.     

Properties of convergence for the q-Meyer-König and Zeller operators

In this paper, we discuss properties of convergence for the q-Meyer-König and Zeller operators Mn,q. Based on an explicit expression for Mn,q(t²,x) in terms of q-hypergeometric series, we show that for qn∈(0,1], the sequence (Mn,qn(f))_n?1 converges to f uniformly on [0,1] for each f∈C[0,1] if and only if limn→∞qn=1. For fixed q∈(0,1), we prove that the sequence (Mn,q(f)) converges for each f∈C[0,1] and obtain the estimates for the rate of convergence of (Mn,q(f)) by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions. We also give explicit formulas of Voronovskaya type for the q-Meyer-König and Zeller operators for fixed 0<q<1. If 0<q<1, f∈C¹[0,1], we show that the rate of convergence for the Meyer-König and Zeller operators is o(qn) if and only if

     

Strong type of Steckin inequality for the linear combination of Bernstein operators

In this paper we obtain a new strong type of Steckin inequality for the linear combinations of Bernstein operators, which gives the optimal approximation rate. Moreover, a method to prove lower estimates for linear operators is introduced. As a result the lower estimate for the linear combinations of Bernstein operators is obtained by using the Ditzian–Totik modulus of smoothness.     

Iterative solution of linear equations with unbounded operators

A convergent iterative process is constructed for solving any solvable linear equation in a Hilbert space, including equations with unbounded, closed, densely defined linear operators. The method is proved to be stable towards small perturbation of the data. Some abstract results are established and used in an analysis of variational regularization method for equations with unbounded linear operators. The dynamical systems method (DSM) is justified for unbounded, closed, densely defined linear operators. The stopping time is chosen by a discrepancy principle. Equations with selfadjoint operators are considered separately. Numerical examples, illustrating the efficiency of the proposed method, are given.     

Global smoothness and uniform convergence of smooth Gauss–Weierstrass singular operators

In this article we continue with the study of smooth Gauss–Weierstrass singular integral operators over the real line regarding their simultaneous global smoothness preservation property with respect to the L_p norm, 1≤p≤∞, by involving higher order moduli of smoothness. Also we study their simultaneous approximation to the unit operator with rates involving the modulus of continuity with respect to the uniform norm. The produced Jackson type inequalities are almost sharp containing elegant constants, and they reflect the high order of differentiability of the engaged function.     

Approximation by Durrmeyer–Bezier operators

In the present paper, we consider the Bezier variant M_n,α(f,x) of the generalized Durrmeyer type operators, and obtain an estimate on the rate of convergence of M_n,α(f,x) for the decomposition technique of functions of bounded variation. In the end we propose an open problem for the readers and give an asymptotic formula for these generalized Durrmeyer type operators.     

Best constants for tensor products of Bernstein type operators

For the tensor product of k copies of the same one-dimensional Bernstein-type operator L, we consider the problem of finding the best constant in preservation of the usual modulus of continuity for the lp-norm on Rk. Two main results are obtained: the first one gives both necessary and sufficient conditions in order that 1+k1−1/p is the best uniform constant for a single operator; the second one gives sufficient conditions in order that 1+k1−1/p is the best uniform constant for a family of operators. The general results are applied to several classical families of operators usually considered in approximation theory. Throughout the paper, probabilistic concepts and methods play an important role.     

Exponential rate of convergence for some Markov operators

The exponential rate of convergence for Markov operators is established. The operators correspond to continuous iterated function systems which are a very useful tool in some cell cycle models.     

Certain positive linear operators with better approximation properties

The present paper deals with a new positive linear operator which gives a connection between the Bernstein operators and their genuine Bernstein‐Durrmeyer variants. These new operators depend on a certain function τ defined on [0,1] and improve the classical results in some particular cases. Some approximation properties of the new operators in terms of first and second modulus of continuity and the Ditzian‐Totik modulus of smoothness are studied. Quantitative Voronovskaja–type theorems and Grüss‐Voronovskaja–type theorems constitute a great deal of interest of the present work. Some numerical results that compare the rate of convergence of these operators with the classical ones and illustrate the relevance of the theoretical results are given.     

Meyer-K(o)nig-Zeller 算子加权逼近的收敛阶

利用加权Ditzin-Totik 光滑模ω2φλ(f;t)w,借助Peetre K-泛函研究了Meyer-K(o)nig-Zeller算子,给出其特征刻画.