Estimates related to the iterates of positive linear operators and their multidimensional analogues

Abstract

The starting point of this paper is the construction of a general family \( (L_{n})_{n\ge 1}\) of positive linear operators of discrete type. Considering \((L_{n}^{k})_{k\ge 1}\) the sequence of iterates of one of such operators, \(L_{n}\), our goal is to find an expression of the upper edge of the error \(\Vert L_{n}^{k}f-f^{*}\Vert \), \(f\in C[0,1]\), where \(f^{*} \) is the fixed point of \(L_{n}.\) The estimate makes use of the error formula for the sequence of successive approximations in Banach’s fixed point theorem and the error of approximation of the operator \(L_{n}.\) Examples of special operators are inserted. Some extensions to multidimensional approximation operators are also given.