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

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