On the Durrmeyer-Type Variant and Generalizations of Lototsky–Bernstein Operators

The starting points of the paper are the classic Lototsky–Bernstein operators. We present an integral Durrmeyer-type extension and investigate some approximation properties of this new class. The evaluation of the convergence speed is performed both with moduli of smoothness and with K-functionals of the Peetre-type. In a distinct section we indicate a generalization of these operators that is useful in approximating vector functions with real values defined on the hypercube [0,1]q, q>1. The study involves achieving a parallelism between different classes of linear and positive operators, which will highlight a symmetry between these approximation processes.