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Positive operators and approximation in function spaces on completely regular spaces. (English) Zbl 1042.41017

The paper is concerned with approximation properties of nets of positive linear operators acting on function spaces defined on Hausdorff completely regular spaces. Special attention is paid to positive operators which are defined in terms of integrals with respect to a given family of Borel measures. By using a very simple and direct approach the authors extend some well-known Korovkin-type theorems and establish new ones in the setting of completely regular spaces. The qualitative results are completed with pointwise and uniform estimates of the rate of convergence. The final section is devoted to the study of two approximating sequences of positive linear operators acting on spaces of weakly continuous functions defined on a convex subset of some locally convex Hausdorff space. They generalize the sequences of Bernstein-Schnabl operators which have been intensively investigated in the setting of compact convex sets.

MSC:

41A36 Approximation by positive operators
47A58 Linear operator approximation theory
47B65 Positive linear operators and order-bounded operators
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