The remainder in the approximation by a generalized Bernstein operator: a representation by a convex combination of second-order divided differences

Abstract

We investigate the remainder of the approximation formula of a function f∈C[0,1] by means of a generalized Bernstein operator (1.3), depending on two nonnegative integer parameters, introduced by the author in 1984 in the paper [11]. The remainder is expressed in (2.1)–(2.2) by a formula generalizing the author's earlier representation (1.2) of the remainder in Bernstein's classical approximation formula. In (2.13)–(2.14) an expression is given for the remainder involving a linear functional (2.14) which is a convex combination of second-order divided differences.