On Some Classes of q-parametric Positive linear Operators

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Abstract

Recently, G.M. Phillips [6], [7] has introduced the q-parametric Bernstein polynomials denoted by B n (f, x, q), 0 < q ≤ 1. In [6] the investigation of the properties of these polynomials is based on some generalization of divided differences and on the Newton interpolation formula. A. Il’inskii and S. Ostrovska [4] have considered linear operators B (f, x, q) which were derived by means of B n (f,x, q) with the help of an informal passing to the limit for n → ∞. In this paper we give another more natural proof of the main results of [6], [4]. Moreover, for the Voronovskaya’s asymptotic formula we obtain the estimate of the remainder term. We also consider the modifi- cation of these operators in order to improve the degree of approximation of twice differentiable functions.