On simultaneous approximation of the Bernstein Durrmeyer operators

The present paper deals with the study of the rate of convergence of the Bézier variant of certain Bernstein Durrmeyer type operators in simultaneous approximation.     

On the approximation of convex functions by Bernstein-type operators

As a consequence of Jensen's inequality, centered operators of probabilistic type (also called Bernstein-type operators) approximate convex functions from above. Starting from this fact, we consider several pairs of classical operators and determine, in each case, which one is better to approximate convex functions. In almost all the discussed examples, the conclusion follows from a simple argument concerning composition of operators. However, when comparing Szász-Mirakyan operators with Bernstein operators over the positive semi-axis, the result is derived from the convex ordering of the involved probability distributions. Analogous results for non-centered operators are also considered.     

On uniform approximation by some classical Bernstein-type operators

We investigate the functions for which certain classical families of operators of probabilistic type over noncompact intervals provide uniform approximation on the whole interval. The discussed examples include the Szász operators, the Szász-Durrmeyer operators, the gamma operators, the Baskakov operators, and the Meyer-König and Zeller operators. We show that some results of Totik remain valid for unbounded functions, at the same time that we give simple rates of convergence in terms of the usual modulus of continuity. We also show by a counterexample that the result for Meyer-König and Zeller operators does not extend to Cheney and Sharma operators.     

Studying Baskakov–Durrmeyer operators and quasi-interpolants via special functions

We prove that the kernels of the Baskakov–Durrmeyer and the Szász–Mirakjan–Durrmeyer operators are completely monotonic functions. We establish a Bernstein type inequality for these operators and apply the results to the quasi-interpolants recently introduced by Abel. For the Baskakov–Durrmeyer quasi-interpolants, we give a representation as linear combinations of the original Baskakov–Durrmeyer operators and prove an estimate of Jackson–Favard type and a direct theorem in terms of an appropriate K-functional.     

On converse approximation theorems

We establish sufficient conditions for obtaining a strong converse inequality of type B in terms of a unified K-functional for a sequence of linear positive operators (Ln)_n?1, . This K-functional, introduced by Guo et al. (see, e.g., [S. Guo, Q. Qi, Strong converse inequalities for Baskakov operators, J. Approx. Theory 124 (2003) 219-231]), will be considered for more general weight functions. As applications we investigate the situation for Baskakov type operators and Szász-Mirakjan type operators.     

Durrmeyer type modification of generalized Baskakov operators

In the present paper, Durrmeyer type modification of generalized Baskakov operators is introduced and some direct results are established for these operators.     

Szász-Mirakian-Baskakov型算子的同时逼近

研究了Baskakov和Szász-Mirakian型算子的线性组合的同时逼近问题,得到了Voronovskaja型的渐进展开公式以及误差估计．     

Approximation by a class of modified Szász-Durrmeyer operators

We modify Szász-Durrmeyer operators by means of three-diagonal generalized matrix which overcomes a difficulty in extending a Berens-Lorentz result to the Szász-Durrmeyer operators for second order of smoothness. The direct and converse theorems for these operators inL _p are also presented by Ditzian-Totik modulus of smoothness.This project is supported by Zhejiang Provincial Foundation of China.     

Convergence of iterates of genuine and ultraspherical Durrmeyer operators to the limiting semigroup: -estimates

In the present note a general inequality for the degree of approximation of semigroups by iterates of commuting bounded linear operators on Banach spaces is given. Combining this with a recent quantitative Voronovskaja-type result applications to Durrmeyer operators with ultraspherical weights are derived. Our considerations include the genuine Bernstein–Durrmeyer operators.     

Localization results for generalized Baskakov/Mastroianni and composite operators

In this paper we study localization results for classical sequences of linear positive operators that are particular cases of the generalized Baskakov/Mastroianni operators and also for certain class of composite operators that can be derived from them by means of a suitable transformation. Amongst these composite operators we can find classical sequences like the Meyer-König and Zeller operators and the Bleimann, Butzer and Hahn ones. We extend in different senses the traditional form of the localization results that we find in the classical literature and we show several examples of sequences with different behavior to this respect.     

Pointwise approximation by Bézier variant of integrated MKZ operators

In this paper the pointwise approximation of Bézier variant of integrated MKZ operators for general bounded functions is studied. Two estimate formulas of this type approximation are obtained. The approximation of functions of bounded variation becomes a special case of the main result of this paper. In the case of functions of bounded variation, Theorem B of the paper corrects the mistake of Theorem 1 of the article [V. Gupta, Degree of approximation to functions of bounded variation by Bézier variant of MKZ operators, J. Math. Anal. Appl. 289 (2004) 292-300].     

Uniform approximation by some Durrmeyer operators

The purpose of this paper is to give the inverse theorems for uniform approximation by Bernstein and Szász-Mirakjan Durrmeyer operators. A new K-functional is introduced which shows the instinct difference between Feller and non-Feller operators. A new approach to the inverse theorem for Bernstein operators is also presented. Supported by National Science Foundation of China     

Some eigenvalue results for maximal monotone operators

We study the eigenvalue problem of the form

0∈Tx−λCx,     

Generalized Bernstein–Durrmeyer Operators and the Associated Limit Semigroup

The aim of this paper is the study of a new sequence of positive linear approximation operators M_n, λ on C([0, 1]) which generalize the classical Bernstein–Durrmeyer operators. After proving a Voronovskaja-type result, we show that there exists a strongly continuous positive contraction semigroup on C([0, 1]) which may be expressed in terms of powers of these operators. As a direct consequence, we are able to represent explicitly the solutions of the Cauchy problems associated with a particular class of second order differential operators.     

Approximation by complex szász-durrmeyer operators in compact disks

In the present paper, we deal with the complex Szász-Durrmeyer operators and study Voronovskaja type results with quantitative estimates for these operators attached to analytic functions of exponential growth on compact disks. Also, the exact order of approximation is found.     

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.     

Szász-Mirakian-Baskakov型算子的同时逼近

研究了Baskakov和Szász-Mirakian型算子的线性组合的同时逼近问题,得到了Voronovskaja型的渐进展开公式以及误差估计.     

Log-concavity for Bernstein-type operators using stochastic orders

This paper aims to study the preservation of log-concavity for Bernstein-type operators. In particular, attention is focused on positive linear operators, defined on the positive semi-axis, admitting a probabilistic representation in terms of a process with independent increments. This class includes the classical Gamma, Szász, and Szász–Durrmeyer operators. With respect to the first and second operators, the results of this paper correct two erroneous counterexamples in [10]. As a main tool in our results we use stochastic order techniques. Our results include, as a particular case, the log-concavity of certain functions related to the incomplete Gamma function.     

Weighted statistical approximation by kantorovich type q-Szász-Mirakjan operators

In the present paper, we introduce a Kantorovich type modification of q-Szász-Mirakjan operators and obtain weighted statistical approximation properties of these operators. Also for introduced operators, we give a Voronovskaja type theorem related to q-derivatives.     

Best constants in preservation of global smoothness for Szász-Mirakyan operators

We obtain the best possible constants in preservation inequalities concerning the usual first modulus of continuity for the classical Szász-Mirakyan operator. The probabilistic representation of this operator in terms of the standard Poisson process is used.