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 classical Gamma, Szász and Szász-Durrmeyer operators. As a main tool in our results we use stochastic orders techniques. Our results include, as a particular case, the log-concavity of certain functions related to the gamma incomplete function.     

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.     

Some approximation results for Durrmeyer operators

In the present paper, we obtain a sequence of positive linear operators which has a better rate of convergence than the Szász-Mirakian Durrmeyer and Baskakov Durrmeyer operators and their Voronovskaya type results.     

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.     

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.     

The best asymptotic constant of a class of approximation operators

The best asymptotic constant was established by Esseen for Bernstein operators. In this paper, we extend Esseen's result to a class of linear positive operators and as byproduct we obtain the best asymptotic constant for Szász, Baskakov, Gamma, and B-Spline operators.     

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.     

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.     

Convergence of the q analogue of Szász-Beta operators

In the present paper we introduce the q analogue of the well known Szász-Beta operators [11]. We also establish the approximation properties of these operators and estimate convergence results. In the end we propose an open problem.     

On the preservation of log convexity and log concavity under some classical Bernstein-type operators

This paper analyzes the preservation of both the log convexity and the log concavity under certain Bernstein-type operators. Some results are provided for the Bernstein, Szász, Baskakov, the gamma-type and the Weierstrass operators. Probabilistic methods support the proofs of these results.     

Rate of convergence in simultaneous approximation for Szász-Mirakyan-Durrmeyer operators

In the present paper, we study the rate of convergence in simultaneous approximation for the Szász-Mirakyan-Durrmeyer operators by using the decomposition technique of functions of bounded variation.     

Equivalent theorems on simultaneous approximation by combinations of Gamma operators

In this paper,some equivalent theorems on simultaneous approximation for combinations of Gamma operators by weighted moduli of smoothness ωr(4)λ(f,t)w(4)s(O≤λ≤1)are given.The relation between derivatives of combinations of Gamma operators and smoothness of derivatives of functions is also investigated.     

Simultaneous approximation for Bézier variant of Szász-Mirakyan-Durrmeyer operators

We study the rate of convergence in simultaneous approximation for the Bézier variant of Szász-Mirakyan-Durrmeyer operators by using the decomposition technique of functions of bounded variation.     

Upper Estimates in Direct Inequalities for Bernstein-Type Operators

We obtain explicit upper estimates in direct inequalities with respect to the usual sup-norm distance for Bernstein-type operators. Our approach combines analytical and probabilistic techniques based on representations of the operators in terms of stochastic processes. We illustrate our results by considering some classical families of operators, such as Weierstrass, Szàsz, and Bernstein operators.     

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.     

Approximation properties of modified Gamma operators

In this paper we present a survey of rates of pointwise approximation of modified Gamma operators Gn for locally bounded functions and absolutely continuous functions by using some inequalities and results of probability theory with the method of Bojanic-Cheng. In the paper a kind of locally bounded functions is introduced with different growth conditions in the fields of both ends of interval (0,+∞), and it is found out that the operators have different properties compared to the Gamma operators discussed in [X.M. Zeng, Approximation properties of Gamma operators, J. Math. Anal. Appl. 311 (2005) 389-401]. And we obtain two main theorems. Theorem 1 gives an estimate for locally bounded functions which subsumes the approximation of functions of bounded variation as a special case. Theorem 2 gives an estimate for absolutely continuous functions which is best possible in the asymptotical sense.     

Voronovskaya Type Theorems in Weighted Spaces

In this article, we introduce a generalization of Gamma operators based on a function ρ having some properties and prove quantitative Voronovskaya and quantitative Grüss type Voronovskaya theorems via weighted modulus of continuity.     

Rate of pointwise approximation for locally bounded functions by Szász operators

In this paper the asymptotic behavior of Szász operators for locally bounded functions f is studied at points x where f(x+) and f(x−) exist. An asymptotic estimate of this type approximation is obtained by using some techniques and results of probability theory. The estimate essentially is the best possible for continuous points of f.     

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.