Approximation Properties For A Kind Of Modified Bernstein Operators And Bernstein-Durrmeyer Operators

The operator approximation is an important research direction in the approximation theory, which mainly discusses the convergence rate of operator series. In recent years, in order to improve the approximation rate of the operators, many scholars have modified some famous linearoperators ( such as Bernstein operators, Bernstein-Kantorovich operators, Szasz-Mirakjan operators and so on), and the better rates of convergence were obtained.In this paper, firstly, we study the pointwise approximation properties of the modified Bernstein operators. Secondly, a modification of Bernstein-Durrmeyer operators are introduced and the related approximation properties are also studied. The main results are as follows:1. For the modified Bernstein operatorswherewe obtain the following pointwise direct and equivalent theorems:Theorem A Let f∈C[0,1], 0≤λ≤1, thenTheorem B Let f∈C[0,1], 0≤λ≤1,0 <α< 1, then2. The modified Bernstein-Durrmeyer operators are defined bywhereFirstly, we prove that (?)(f,x) are bounded and linear operators from L_p[0,1] into the spaceπ_n of polynomial of degree n, and preserve linear functions.Secondly, we study some approximation properties of operators (?)(f,x) and obtain the following direct theorems, Steckin-Marchaud inequalities and equivalent theorems: Theorem C Let f∈L_p[0,1], 1≤p≤+∞, thenTheorem D Let f∈L_p[0,1], 1≤p≤+∞, r∈N, E_n(f) = ||(?) - f||_p, thenTheorem E Let f∈L_p[0,1], 1≤p≤+∞, 0＜α＜2, thenFrom quantitative estimates, the two kinds of modified operators have better rates of convergencein some cases.