Convergence of Iterated Boolean-type Sums and Their Iterates

Abstract

The convergence of iterated Boolean sums for different sequences of operators has been studied keeping fixed the number M of iterations or recently letting for a fixed operator. Here we consider some convergence properties of iterated Boolean-type sums of the classical Bernstein and Bernstein–Durrmeyer operators in the case where both the operators and their order of iterations are not fixed. In the case of Bernstein–Durrmeyer operators, we obtain a general solution of this problem. We can also state some alternative expressions of the semigroups and resolvent operators associated with the Voronovskaja’s formula for these sequences of operators.