An iterative algorithm for improved approximation by Bernstein's operator using statistical perspective. (Q1427867)

The well-known Bernstein operator is defined as \[ B_n(f)(x) = \sum_{k=0}^n f\left(\frac{k}{n}\right) {n \choose k} x^k (1-x)^{n-k} \] for \(f \in C[0,1]\). The author suggests an iterative improvement of approximation by \(B_n\), which leads to the operator \[ I(k)B_n = 1 - (1-B_n)^{k+1}, \quad k = 0,1, \ldots \] Some numerical expreriments are presented.