The approximation by q-Bernstein polynomials in the case q ↓ 1

Abstract.

Let B _n (f, q; x), n=1, 2, ... , 0 < q < ∞, be the q-Bernstein polynomials of a function f, B _n (f, 1; x) being the classical Bernstein polynomials. It is proved that, in general, {B _n (f, q _n ; x)} with q _n ↓ 1 is not an approximating sequence for f ∈C[0, 1], in contrast to the standard case q _n ↓ 1. At the same time, there exists a sequence 0 < δ _n ↓ 0 such that the condition $1 \leqq q_{n} \leqq \delta _{n}$ implies the approximation of f by {B _n (f, q _n ; x)} for all f ∈C[0, 1].