The Combination of Bernstein Polynomials with Positive Functions Based on a Positive Parameter \(s\) - Mohammad, Ali J.; Kathem, Rafah F. (Communications in Mathematics and Applications, 2022)
This paper deals with a sequence of the combination of Bernstein polynomials with a positive function \(τ\) and based on a parameter \(s>-12\). These polynomials have preserved the functions \(1\) and \(τ\). First, the convergence theorem for this sequence is studied for a function \(fC[0,1]\). 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 \(fC[0,1]\) and two functions for \(τ\) to show that the effect of the different values of \(s\) and the different chosen functions \(τ\).