The Combination of Bernstein Polynomials with Positive Functions Based on a Positive Parameter \(s\)

Abstract

This paper deals with a sequence of the combination of Bernstein polynomials with a positive function \(\tau\) and based on a parameter \(s>-\frac{1}{2}\). These polynomials have preserved the functions \(1\) and \(\tau\). First, the convergence theorem for this sequence is studied for a function \(f\in C[0,1]\). Next, the rate of convergence theorem for these polynomials is descript by using the first, second modulus of continuous and Ditzian-Totik modulus of smoothness. Also, the Quantitative Voronovskaja and Gruss-Voronovskaja are obtained. Finally, two numerical examples are given for these polynomials by chosen a test function \(f\in C[0,1]\) and two functions for \(\tau\) to show that the effect of the different values of \(s\) and the different chosen functions \(\tau\).