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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

abstract and applied analysis

Approximation by<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math>-Bernstein Polynomials in the Case<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>q</mml:mi><mml:mo>→</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo></mml:math>

Approximation byq-Bernstein Polynomials in the Caseq→1+