Regular vector lattices of continuous functions and Korovkin-type theorems-Part I

Francesco Altomare, and Mirella Cappelletti Montano. "Regular vector lattices of continuous functions and Korovkin-type theorems-Part I." Studia Mathematica 171.3 (2005): 239-260. <http://eudml.org/doc/284795>.

@article{FrancescoAltomare2005,
abstract = { We introduce and study a new class of locally convex vector lattices of continuous functions on a locally compact Hausdorff space, which we call regular vector lattices. We investigate some general properties of these spaces and of the subspaces of so-called generalized affine functions. Moreover, we present some Korovkin-type theorems for continuous positive linear operators; in particular, we study Korovkin subspaces for finitely defined operators, for the identity operator and for positive projections. Due to its length, the paper is split up into two parts of distinct character; in this first part, we introduce the class of regular vector lattices, we prove an integral representation theorem for continuous positive linear forms and we study some enveloping functions related to a continuous positive operator, together with the corresponding space of generalized affine functions. Finally, we obtain a Stone-Weierstrass type theorem. In the second part, which will appear in the same journal, we will present some Korovkin-type theorems, together with some applications. },
author = {Francesco Altomare, Mirella Cappelletti Montano},
journal = {Studia Mathematica},
keywords = {vector lattice of continuous functions; integral representation theorem; generalized affine function; Choquet boundary; Stone-Weierstrass theorem},
language = {eng},
number = {3},
pages = {239-260},
title = {Regular vector lattices of continuous functions and Korovkin-type theorems-Part I},
url = {http://eudml.org/doc/284795},
volume = {171},
year = {2005},
}

TY - JOUR
AU - Francesco Altomare
AU - Mirella Cappelletti Montano
TI - Regular vector lattices of continuous functions and Korovkin-type theorems-Part I
JO - Studia Mathematica
PY - 2005
VL - 171
IS - 3
SP - 239
EP - 260
AB - We introduce and study a new class of locally convex vector lattices of continuous functions on a locally compact Hausdorff space, which we call regular vector lattices. We investigate some general properties of these spaces and of the subspaces of so-called generalized affine functions. Moreover, we present some Korovkin-type theorems for continuous positive linear operators; in particular, we study Korovkin subspaces for finitely defined operators, for the identity operator and for positive projections. Due to its length, the paper is split up into two parts of distinct character; in this first part, we introduce the class of regular vector lattices, we prove an integral representation theorem for continuous positive linear forms and we study some enveloping functions related to a continuous positive operator, together with the corresponding space of generalized affine functions. Finally, we obtain a Stone-Weierstrass type theorem. In the second part, which will appear in the same journal, we will present some Korovkin-type theorems, together with some applications.
LA - eng
KW - vector lattice of continuous functions; integral representation theorem; generalized affine function; Choquet boundary; Stone-Weierstrass theorem
UR - http://eudml.org/doc/284795
ER -