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Multivariate weighted Kantorovich operators. (English) Zbl 1503.41014

Herein, the authors introduce a class of multidimensional weighted Kantorovich operators \(K_n\), \(n\geq 1\), whose definition is given on the space of continuous functions \(C(Q_{d})\) (where \(Q_d\) is the \(d\)-dimensional hypercube \([0,1]^{d}\), \(d\geq 1\)), and it involves the well-known Bernstein polynomials.
In this setting, they prove the existence of a unique probability measure on \(Q_d\) which is invariant with respect to \(K_n\), and they determine such a measure.
Furthermore, the authors give a convergence result of the iterates \(K_n f\) to \(f\), uniformly on \(C(Q_d)\).
Finally, they point out that the above class \(K_n\) is a generalization of some Kantotovich type operators, consequently their approach is unifying for the study of approximation results for \(K_n\).

MSC:

41A36 Approximation by positive operators
26A16 Lipschitz (Hölder) classes
26D10 Inequalities involving derivatives and differential and integral operators
37L40 Invariant measures for infinite-dimensional dissipative dynamical systems
46A32 Spaces of linear operators; topological tensor products; approximation properties
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References:

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