International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 61, Pages 3841-3871

Positive operators and approximation in function spaces on completely regular spaces

Francesco Altomare1 and Sabrina Diomede2

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.


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.