A saturation theorem for combinations of Bernstein-Durrmeyer polynomials

Abstract

We prove a local saturation theorem in ordinary approximation for combinations of Durrmeyer’s integral modification of Bernstein polynomials. Introduction. The Bernstein–Durrmeyer polynomial of order n is defined by Mn(f, x) = 1 ∫ 0 W (n, x, t)f(t) dt , f ∈ L1[0, 1] , where W (n, x, t) = (n+ 1) n ∑ ν=0 pnν(x)pnν(t) , pnν(x) being (nν )x ν(1−x)n−ν , x ∈ [0, 1]. These operators were introduced by Durrmeyer [5] by replacing f(ν/n) in Bn(f, x), the Bernstein polynomials, by (n+1) ∫ 1 0 pnν(t)f(t) dt. Several authors (see [1]–[4], [6], [8], [9]) have studied the operators Mn and obtained direct and inverse results both in supnorm and Lp-norm. In this paper we study the saturation behaviour of the linear combination Mn(f, k, x) [7]. It turns out that even though Bernstein– Durrmeyer polynomials are not exponential type operators [7] yet their saturation behaviour is similar to that of the operators of exponential type. The linear combination Mn(f, k, x) of Mdjn(f, x), j = 0, 1, . . . , k, is defined by Mn(f, k, x) = k ∑ j=0 C(j, k)Mdjn(f, x) , 1991 Mathematics Subject Classification: 41A30, 41A36.