Approximation by q-Bernstein Polynomials in the Case q → 1+

Let B-n,B-q(f; x), q is an element of (0, infinity) be the q-Bernstein polynomials of a function f is an element of C[0, 1]. It has been known that, in general, the sequence (B-n,B-qn(f)) with q(n) -> 1+ is not an approximating sequence for f is an element of C[0, 1], in contrast to the standard case q(n) -> 1-. In this paper, we give the sufficient and necessary condition under which the sequence (B-n,B-qn(f)) approximates f for any f is an element of C[0, 1] in the case q(n) > 1. Based on this condition, we get that if 1 < q(n) < 1 + ln 2/n for sufficiently large n, then (B-n,B-qn(f)) approximates f for any f is an element of C[0, 1]. On the other hand, if (B-n,B-qn(f)) can approximate f for any f is an element of C[0, 1] in the case q(n) > 1, then the sequence (q(n)) satisfies (lim) over bar (n ->infinity)n(q(n) - 1) <= ln2.