The r-th Powers of the Rational Bernstein Polynomials

Abstract

This paper is defined and studied a new -th powers of rational Bernstein polynomials. The convergence theorem, the recurrence relation for the -th order moment, and the Voronovskaya-type asymptotic formula for these polynomials in ordinary approximation are given. Also, a numerical example for these polynomials is applied to approximate the test function . The results obtained from this example are shown that these polynomials are given better than the corresponding numerical results for the classical Bernstein polynomials and the square rational Bernstein polynomials. The comparison is done by plot the graphs of the function and its approximations as well as the evaluation of the average absolute errors for these approximations.