Approximation byq-Bernstein Polynomials in the Caseq→1+

Abstract

LetBn,q(f;x),q∈(0,∞)be theq-Bernstein polynomials of a functionf∈C[0,1]. It has been known that, in general, the sequenceBn,qn(f)withqn→1+is not an approximating sequence forf∈C[0,1], in contrast to the standard caseqn→1-. In this paper, we give the sufficient and necessary condition under which the sequenceBn,qn(f)approximatesffor anyf∈C[0,1]in the caseqn>1. Based on this condition, we get that if1<qn<1+ln⁡2/nfor sufficiently largen, thenBn,qn(f)approximatesffor anyf∈C[0,1]. On the other hand, ifBn,qn(f)can approximateffor anyf∈C[0,1]in the caseqn>1, then the sequence(qn)satisfieslim¯n→∞n(qn-1)≤ln2.