Approximation Properties For The Iterated Boolean Sums Of Bernstein-Kantorovich Operators And Szász-Durrmeyer Operators

The approximation of operator is one of the popular subject in approximation theory both at home and abroad all the time. The approximation of operator mainly studies the convergence property of linear operator series, its convergence rate and related question. It is well known that the optimal approximation orders of some well-known operators, for example, Bernstein operators, Szasz operators and their Modified form of Kantorovich, Durrmeyer operators, etc. are O(n^-1). In order to obtain faster convergence rate the linear combinations of these operatorsare considered by many Mathematician. Recently many scholars research the iterated Boolean sums of operators. In this paper, we discuss the approximation properties for the iterated Boolean sums of Bernstein-Kantorovich operators in L_∞[0,1] and the approximation properties for the iterated Boolean sums of Szasz-Durrmeyer operators in L_p[0, +∞). The main results are as follows.1. For the iterated Boolean sums of Bernstein-Kantorovich operators (?)K_n, firstly, we give the expression and the upper estimate of the moments of (?)K_n and K_n^r. Secondly, we study the approximation properties for the iterated Boolean sums of Bernstein-Kantorovich operators in L_∞[0,1]. We obtain approximation direct theorems and equivalent theorems as follows:Theorem A Let r∈N,f∈L_∞[0,1], thenTheorem B Let r∈N,f∈L_∞[0,1], 1≤p≤+∞, then for 0＜α＜2r we haveThese results improve the previous ones.2. For the iterated Boolean sums of Szasz-Durrmeyer operators (?)L_n, firstly, we give the expression and the upper estimate of the moments of (?)L_n and L_n^r. Secondly, we study the approximation properties for the iterated Boolean sums of Szasz-Durrmeyer operators in L_p[0, +∞). We obtain direct theorems, inverse inequalities and equivalent theorems as follows:Theorem C Let r∈N,f∈L_p[0,+∞), then Theorem D Let r∈N,f∈L_p[0,+∞),1≤p≤+∞, then for any m∈N, we haveTheorem E Let r∈N,f∈L_p[0,+∞),1≤p≤+∞, then for 0＜α＜2r, we haveFrom the above results we can see (?)L_n do accelerate the degree of convergence of the original operators.