The degree of convergence of over-iterated positive linear operators. (English) Zbl 1099.41011

The paper presents methods to determine the degree of approximation for the iterates of certain positive linear operators towards \(B_1\), the first Bernstein operator. O. Agratini and I. A. Rus [Comment. Math. Univ. Carol. 44, 555–563 (2003; Zbl 1096.41015)] proved convergence for over-iteration of certain general discretely defined operators. (Over-iteration means that for a fixed operator its \(m\)-th powers are investigated when \(m\) goes to infinity). Their proof uses the contraction principle.
The authors of this paper generalize the result of Agratini and Rus for a whole class of summation-type operators. Then they give a general quantitative result, without contraction arguments, and derive a full quantitative version of Agratini’s and Rus’ result. The quantitative assertions are illustrated by considering the Bernstein, Bernstein-Sheffer-Popoviciu, and Bernstein-Stancu operators, among others. Finally, the iterates of two types of operators are investigated: Beta operators (which are not discretely-defined), and Schoenberg spline operators (for which the contraction argument fails).


41A15 Spline approximation
41A25 Rate of convergence, degree of approximation
41A36 Approximation by positive operators
47H10 Fixed-point theorems


Zbl 1096.41015