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We present some direct and inverse results for the approximation by the genuine Bernstein–Durrmeyer operators Un. We also consider iterates of Un and discuss some convergence results towards the corresponding semigroup. Using the eigenstructure of the operators Un we give new proofs of some known qualitative results and obtain new quantitative estimates concerning the convergence of iterates...
. We continue the studies on the so–called genuine Bernstein–Durrmeyer operators Un by establishing a recurrence formula for the moments and by investigating the semigroup T(t) approximated by Un. Moreover, for sufficiently smooth functions the degree of this convergence is estimated. We also determine the eigenstructure of Un, compute the moments of T(t) and establish asymptotic formulas.
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