Approximation by \(q\)-Bernstein polynomials in the case \(q \rightarrow 1 +\) (Q1722474)

Summary: Let \(B_{n, q}(f; x)\), \(q \in(0, \infty)\) be the \(q\)-Bernstein polynomials of a function \(f \in C [0,1]\). It has been known that, in general, the sequence \(\left(B_{n, q_n}(f)\right)\) with \(q_n \rightarrow 1 +\) is not an approximating sequence for \(f \in C [0,1]\), in contrast to the standard case \(q_n \rightarrow 1 -\). In this paper, we give the sufficient and necessary condition under which the sequence \(\left(B_{n, q_n}(f)\right)\) approximates \(f\) for any \(f \in C [0,1]\) 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 \(\left(B_{n, q_n}(f)\right)\) approximates \(f\) for any \(f \in C [0,1]\). On the other hand, if \(\left(B_{n, q_n}(f)\right)\) can approximate \(f\) for any \(f \in C [0,1]\) in the case \(q_n > 1\), then the sequence \((q_n)\) satisfies \(\overline{\text{lim}}_{n \rightarrow \infty} n(q_n - 1) \leq \text{ln} 2\).