On Better Approximation of the Squared Bernstein Polynomials

Abstract

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.