"Iterates of multidimensional approximation operators via Perov theorem"

Abstract

"The starting point is an approximation process consisting of linear and positive operators. The purpose of this note is to establish the limit of the iterates of some multidimensional approximation operators. The main tool is a Perov’s result which represents a generalization of Banach fixed point theorem. In order to support the theoretical aspects, we present three applications targeting respectively the operators Bernstein, Cheney-Sharma and those of binomial type. The last class involves an incursion into umbral calculus"

On the iterates of uni-and multidimensional operators

“…Consequently, it has a unique fixed point x * , its iterates L k (x) converge to x * and estimate (7) holds. In particular, if λ ij < 1 for all i and j, matrix Γ is convergent to zero as shown in [1]. Indeed, letting λ := max {λ ij : 1 ≤ i, j ≤ p} , one has λ < 1 and Γ ≤ λM, where M := [σ ij ] 1≤i,j≤p and the powers of M are dominated by the matrix U having all entries equal to 1 (this is proved by induction).…”

“…It remains to be clarified the contraction property of L. This aspect in the context of the theory of linear approximation operators was first highlighted by Rus [15] for Bernstein's operators. The contraction property was later highlighted for other well-known classes of operators (see, e.g., [1], [2], [3], [6], [7]).…”