International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 61, Pages 3841-3871
doi:10.1155/S0161171203301206
Positive operators and approximation in function spaces on completely regular spaces
1Department of Mathematics, University of Bari, Via Orabona 4, Bari 70125, Italy
2Department of Economic Sciences, University of Bari, Via C. Rosalba 53, Bari 70124, Italy
Received 14 January 2003
Copyright © 2003 Francesco Altomare and Sabrina Diomede. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We discuss the approximation properties of nets of positive
linear operators acting on function spaces defined on Hausdorff
completely regular spaces. A particular attention is devoted to
positive operators which are defined in terms of integrals with
respect to a given family of Borel measures. We present several
applications which, in particular, show the advantages of such a
general approach. Among other things, some new Korovkin-type
theorems on function spaces on arbitrary topological spaces are
obtained. Finally, a natural extension of the so-called
Bernstein-Schnabl operators for convex (not necessarily compact)
subsets of a locally convex space is presented as well.