Eigenstructure and iterates for uniquely ergodic Kantorovich modifications of operators. (English) Zbl 1375.37016
Summary: We consider Markov operators \(L\) on \(C[0,1]\) such that for a certain \(c \in [0,1)\), \(\| (Lf)' \| \leq c \| f' \| \) for all \( f \in C^1[0,1]\). It is shown that \(L\) has a unique invariant probability measure \(\nu \), and then \(\nu \) is used in order to characterize the limit of the iterates \(L^m\) 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 \(L^m\). 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 by S. Cooper and S. Waldron [J. Approx. Theory 105, No. 1, 133–165 (2000; Zbl 0963.41006), Remark after Theorem 4.20].
MSC:
| 37A30 | Ergodic theorems, spectral theory, Markov operators |
| 41A36 | Approximation by positive operators |
Keywords:
uniquely ergodic operator; Kantorovich modification; iterates of operators; dual functionalsCitations:
Zbl 0963.41006References:
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