Evaluation of the remainder in an approximation formula of D. D. Stancu

Abstract

Abstract:In a paper published in 1983 in this journal [6] D. D. Stancu introduced and investigated a positive linear operator , of Bernstein type, depending on a non-negative integer parameter r and two real parameters α and β, such that 0 ≤α≤β. Special attention was given to the case of the operator . This operator enjoys the variation diminishing property in the sense of I. J. Schoenberg [3]. He evaluated the orders of approximation, the remainder and the point spectrum of this operator. In the present paper we evaluate the remainder of the approximation formula of a function by means of the general operator of D.–D. Stancu, by making use of first and second order divided differences.