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. Classification (2000). 41A36, 41A25, 41A17, 41A10, 20Mxx, 47A75. Mathematics SubjectKeywords. Genuine Bernstein-Durrmeyer operators, iterates of operators, limiting semigroup, degree of approximation, eigenstructure, moments of positive linear operators, asymptotic formulas.

In the center of our paper are two counterexamples showing the independence of the concepts of global smoothness preservation and variation diminution for sequences of approximation operators. Under certain additional assumptions it is shown that the variation-diminishing property is the stronger one. It is also demonstrated, however, that there are positive linear operators giving an optimal pointwise degree of approximation, and which preserve global smoothness, monotonicity and convexity, but are not variationdiminishing.

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We consider iterates of certain (general) positive linear operators preserving linear functions and derive quantitative upper estimates in terms of weighted and non-weighted moduli of smoothness and related K-functionals. We show that corresponding lower estimates in terms of the classical moduli are not possible, while for the Ditzian-Totik modulus the situation can be different. The results can be applied to several well-known operators; we present here the Bernstein and the genuine Bernstein-Durrmeyer operators. Classification (2000). 41A36, 41A25, 41A10. Mathematics SubjectKeywords. Iterates of operators, upper and lower inequalities, degree of approximation, K-functionals, moduli of smothness, Bernstein operators, genuine Bernstein-Durrmeyer operators. Main toolsLet f ∈ L ∞ [0, 1] be the space of essentially bounded measurable functions, f ∞ = vrai sup x∈ [0,1] |f (x)|, ϕ(x) = x(1 − x), x ∈ [0, 1] and r a natural number. Consider the K-functionalwhere the infimum is taken over all g such that g (r−1) ∈ AC loc (0, 1) (i.e., g (r−1) is absolutely continuous in every closed finite subinterval of (0, 1) and ϕ r g (r) ∞ < ∞).

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