Asymptotic properties of powers of Bernstein operators

The order of approximation by positive linear operators

“…When r = 1 and j n = n for all ^ ^ 1, this result reduces to that of Kelisky and Rivlin [6] (cf. [5], [9], [10]). For extensive approximation properties by iterations of positive linear operators, we refer to [18].…”

“…In [12] we showed that there exists a unique strongly continuous semigroup {S(t); t^O} of Markov operators on C(X) such that for every feC(X) and every sequence {k n } of positive integers with whenever t 2: 0. Now take W(ί) = S(ί) (ί ^ 0) and let {C ί>λ ; ς > 0, λ ^ 0} and {2ί efi ; f, λ ^ 0} be the families of operators defined by (10) and (11), respectively. Then we have the following quantitative ergodic type theorem for iterations of continuous Cesaro and Abel means of the semigroup {S(t)}.…”

Geometric series of positive linear operators and the inverse Voronovskaya theorem on a compact interval

“…There exists a rich bibliography concerning the convergence of iterates of positive linear operators, starting with the paper of Kelisky and Rivlin [19] about the iterates of Bernstein operators. A non-exhaustive list of contributions in this direction is given in our references: [2], [5], [8], [9], [10], [11], [13], [14], [15], [21], [22], [27], [28].…”

Kantorovi? operators of second order

“…In particular we shall state theorems of Voronovskaja type for operator sequences (Q~,")#~, provided lim (kn/n) = 0 or lim (k,/n) = ~. Section 4 is essentially based on /~"* G(3 n---~ GO results of our earlier works [7], [8].…”