Quantitative estimates for the tensor product (p,q)-Balázs-Szabados operators and associated generalized Boolean sum operators

In this study, we give some approximation results for the tensor product of (p,q)-Bal?zs-Szabados operators associated generalized Boolean sum (GBS) operators. Firstly, we introduce tensor product (p,q)-Bal?zs-Szabados operators and give an uniform convergence theorem of these operators on compact rectangular regions with an illustrative example. Then we estimate the approximation for the tensor product (p,q)-Bal?zs-Szabados operators in terms of the complete modulus of continuity, the partial modulus of continuity, Lipschitz functions and Petree?s K-functional corresponding to the second modulus of continuity. After that, we introduce the GBS operators associated the tensor product (p,q)-Bal?zs-Szabados operators. Finally, we improve the rate of smoothness by the mixed modulus of smoothness and Lipschitz class of B?gel continuous functions for the GBS operators.

Better approximation of functions by genuine Bernstein-Durrmeyer type operators

The main object of this paper is to construct a new genuine Bernstein-Durrmeyer type operators which have better features than the classical one. Some direct estimates for the modified genuine Bernstein-Durrmeyer operator by means of the first and second modulus of continuity are given. An asymptotic formula for the new operator is proved. Finally, some numerical examples with illustrative graphics have been added to validate the theoretical results and also compare the rate of convergence.

INTERPOLATORY ESTIMATES IN MONOTONЕ PIECEWISE–POLYNOMIAL APPROXIMATION

Recently we obtained an interpolation estimate for a monotonе piecewise-polynomial approximation of a continuous on a segment function, involving the second modulus of continuity of the r-th derivative, and also have shown that this estimate is valid only for a sufficiently large number of segments of the partition, which essentially depends on the function. In paper [2] both results are widespread to the case of a convex approximation and obtained in somewhat more preciseform. We show that in the case of a monotone approximation, the corresponding results are also valid in an appropriate more precise form.

Stancu type generalization of Szász-Durrmeyer operators involving Brenke-type polynomials

In the present paper, we introduce a Stancu type generalization of Sz?sz- Durrmeyer operators including Brenke type polynomials. We give convergence properties of these operators via Korovkin?s theorem and the order of convergence by using a classical approach. As an example, we consider a Stancu type generalization of the Durrmeyer type integral operators including Hermite polynomials of variance v. Then, we obtain the rates of convergence by using the second modulus of continuity. Also, for these operators including Hermite polynomials of variance v, we present a Voronovskaja type theorem and r-th order generalization of these positive linear operators.

On Some Extensions of Szasz Operators Including Boas-Buck-Type Polynomials

This paper is concerned with a new sequence of linear positive operators which generalize Szasz operators including Boas-Buck-type polynomials. We establish a convergence theorem for these operators and give the quantitative estimation of the approximation process by using a classical approach and the second modulus of continuity. Some explicit examples of our operators involving Laguerre polynomials, Charlier polynomials, and Gould-Hopper polynomials are given. Moreover, a Voronovskaya-type result is obtained for the operators containing Gould-Hopper polynomials.