On Better Approximation of the Squared Bernstein Polynomials

The present paper is defined a new better approximation of the squared Bernstein polynomials. This better approximation has been built on a positive function  defined on the interval [0,1] which has some properties. First, the moderate uniform convergence theorem for a sequence of linear positive operators (the generalization of the Korovkin theorem) of these polynomials is improved. Then, the rate of convergence of these polynomials corresponding to the first and second modulus of continuity and Ditzian- Totik modulus of smoothness is given. Also, the quantitative Voronovskaja and the Grüss- Voronovskaja theorems are discussed. Finally, some numerically applied for these polynomials are given by choosing a test function  and two different  functions show the effect of the different chosen functions .  It turns the new better approximation of the squared Bernstein polynomials gives us a better numerical result than the numerical results of both the classical Bernstein polynomials and the squared Bernstein polynomials. MSC 2010. 41A10, 41A25, 41A36

On Sequences of J. P. King-Type Operators

This survey is devoted to a series of investigations developed in the last fifteen years, starting from the introduction of a sequence of positive linear operators which modify the classical Bernstein operators in order to reproduce constant functions and x2 on [0,1]. Nowadays, these operators are known as King operators, in honor of J. P. King who defined them, and they have been a source of inspiration for many scholars. In this paper we try to take stock of the situation and highlight the state of the art, hoping that this will be a useful tool for all people who intend to extend King's approach to some new contents within Approximation Theory. In particular, we recall the main results concerning certain King-type modifications of two well known sequences of positive linear operators, the Bernstein operators and the Szász-Mirakyan operators

Convergence properties of generalized Lupaş-Kantorovich operators

In the present paper, we consider the Kantorovich modification of generalized Lupaş operators, whose construction depends on a continuously differentiable, increasing and unbounded function $\rho$. For these new operators we give weighted approximation, Voronovskaya type theorem, quantitative estimates for the local approximation