We consider Markov operators L on C[0, 1] such that for a certain $$c \in [0,1)$$ c ∈ [ 0 , 1 ) , $$\Vert (Lf)' \Vert \le c \Vert f' \Vert $$ ‖ ( L f ) ′ ‖ ≤ c ‖ f ′ ‖ for all $$ f \in C^1[0,1]$$ f ∈ C 1 [ 0 , 1 ] . It is shown that L has a unique invariant probability measure $$\nu $$ ν , and then $$\nu $$ ν is used in order to characterize the limit of the iterates $$L^m$$ L m of L. When L is a Kantorovich modification of a certain classical operator from approximation theory, the eigenstructure of this operator is used to give a precise description of the limit of $$L^m$$ L m . This way we extend some known results; in particular, we extend the domain of convergence of the dual functionals associated with the classical Bernstein operator, which gives a partial answer to a problem raised in 2000 by Cooper and Waldron (JAT 105:133–165, 2000, Remark after Theorem 4.20).
Positivity – Springer Journals
Published: Aug 23, 2016
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