Approximation by q-Bernstein Polynomials in the Case q→1+ - Xuezhi Wu (Abstract and Applied Analysis)
Let Bn,q(f;x), q∈(0,∞) be the q-Bernstein polynomials of a function f∈C[0,1]. It has been known that, in general, the sequence Bn,qn(f) with qn→1+ is not an approximating sequence for f∈C[0,1], in contrast to the standard case qn→1-. In this paper, we give the sufficient and necessary condition under which the sequence Bn,qn(f) approximates f for any f∈C[0,1] in the case qn>1. Based on this condition, we get that if 1<qn<1+ln⁡2/n for sufficiently large n, then Bn,qn(f) approximates f for any f∈C[0,1]. On the other hand, if Bn,qn(f) can approximate f for any f∈C[0,1] in the case qn>1, then the sequence (qn) satisfies lim¯n→∞n(qn-1)≤ln2.