On the Convergence of the q-Bernstein Polynomials for Power Functions

Abstract

The aim of this paper is to present new results related to the convergence of the sequence of the complex q-Bernstein polynomials $$ \{B_{n,q}(f_\alpha ;z)\},$$ where $$0<q\ne 1$$ and $$f_\alpha =x^\alpha ,\,\alpha \ge 0,$$ is a power function on [0, 1]. This study makes it possible to describe all feasible sets of convergence K for such polynomials. Specifically, if either $$0<q<1$$ or $$\alpha \in {\mathbb {N}}_0,$$ then $$K ={\mathbb {C}},$$ otherwise $$K=\{0\}\cup \{q^{-j}\}_{j=0}^\infty $$ . In the latter case, this identifies the sequence $$K=\{0\}\cup \{q^{-j}\}_{j=0}^\infty $$ as the ‘minimal’ set of convergence for polynomials $$B_{n,q}(f;z),\;f\in C[0,1]$$ in the case $$q>1.$$ In addition, the asymptotic behavior of the polynomials $$ \{B_{n,q}(f_\alpha ;z)\},$$ with $$q>1$$ has been investigated and the obtained results are illustrated by numerical examples.