Estimates for the differences of positive linear operators and their derivatives

Abstract

The present paper deals with the estimate of the differences of certain positive linear operators and their derivatives. Our approach involves operators defined on bounded intervals, as Bernstein operators, Kantorovich operators, genuine Bernstein-Durrmeyer operators, Durrmeyer operators with Jacobi weights. The estimates in quantitative form are given in terms of first modulus of continuity. In order to analyze the theoretical results in the last section we consider some numerical examples.

Difference of Some Positive Linear Approximation Operators for Higher-Order Derivatives

“…Various new generalizations of these operators were considered in, for example, [17,18]. In [19], estimates of the differences of the Bernstein operators and their derivatives were obtained. Recently, some interesting results on this topic were published in [20][21][22][23][24][25].…”

Differences of Positive Linear Operators on Simplices

“…Differences of positive linear operators were intensively investigated in the last years; see [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and the references therein. The operators involved in these studies act usually on continuous functions defined on real intervals, and the differences are estimated in terms of moduli of smoothness and K-functionals.…”

Modified Operators Interpolating at Endpoints

“…We will give estimates forL n f − f andL n f − L n f . Using methods from [2][3][4][5][6] we will investigate differences between classical operators and modified operators. General results concerning the iterates and the invariant measures for operators S of the form (1) are given in [1].…”