Since their discovery, the classical theorems of Korovkin on approximation of continuous functions on a compact interval have impressed many mathematicians for their simplicity. Several authors in fact have undertaken the program of extending these theorems to other settings and, in the last forty years, many interesting results have been determined; as a matter of fact, mathematicians refer to this research field as Korovkin-type approximation theory. A quite complete picture of what has been achieved in this field up to 1996 can be found in the monographs by F. Altomare and M. Campiti, 1994. More recent results that would be compared with those of this paper can be found in [3], [4], [5] and [25]. With the aim to extend and to treat in a more systematic way some old and more recent results, in this paper we develop the main aspects of the Korovkin-type approximation theory in the framework of a class of locally convex vector lattices of continuous functions on a locally compact Hausdorff space $X$, that we call regular l.c.v. lattices on X. Although several results obtained in the setting of abstract locally convex vector lattices are available in the literature, our approach here is more direct and simple and gives, in addition, new results. Examples of regular l.c.v. lattice include weighted function spaces (in particular, the space C_0(X) of all continuous real valued functions on X vanishing at infinity) and every sublattice of C(X) containing the continuous functions with compact support endowed with the topology of the pointwise convergence on X or with the topology of the uniform convergence on compact subsets of X. Therefore, as a matter of fact, our results generalize and/or add some new aspects to those of F. Altomare and M. Campiti ([2]) and H. Bauer and K. Donner ([9]) for C_0(X) and others. Due to its length, the paper is split up into two parts. In this first part, we present some preliminary results for regular l.c.v. lattices. In particular, we introduce some suitable enveloping functions related to a continuous positive linear operator and we study the corresponding space of generalized affine functions. Moreover, we present a natural notion of Choquet boundary and we characterize it by means of these enveloping functions. The case of the identity operator is also considered and, as a consequence, a Stone-Weierstrass type theorem is obtained. In the second part of the paper, by using the results stated in the first part, given a regular l.c.v. lattice E and a continuous positive linear operator T:E \rightarrow E, we characterize those subspaces H of E which are Korovkin subspaces in E for T , in the sense that every equicontinuous net of positive linear operators from E ito E converging T on H in E, automatically converges to T in E. By using this characterization, we present simple methods to construct Korovkin subspaces for particular positive linear operators, called finitely defined operators as well as for the identity operator. We also exhibit examples of finite dimensional Korovkin subspaces for these operators. Finally, we present some Korovkin-type theorems for continuous positive projections. This last part of the paper is a generalization of a previous work of the authors in the setting of adapted spaces (see also [2, Sect. 3.3]).

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`http://hdl.handle.net/11586/52869`

Titolo: | Regular vector lattices of continuous functions and Korovkin-type theorems — Part I, Studia Math. 171 (3) (2005), 239 – 260 |

Autori: | |

Data di pubblicazione: | 2005 |

Rivista: | |

Abstract: | Since their discovery, the classical theorems of Korovkin on approximation of continuous functions on a compact interval have impressed many mathematicians for their simplicity. Several authors in fact have undertaken the program of extending these theorems to other settings and, in the last forty years, many interesting results have been determined; as a matter of fact, mathematicians refer to this research field as Korovkin-type approximation theory. A quite complete picture of what has been achieved in this field up to 1996 can be found in the monographs by F. Altomare and M. Campiti, 1994. More recent results that would be compared with those of this paper can be found in [3], [4], [5] and [25]. With the aim to extend and to treat in a more systematic way some old and more recent results, in this paper we develop the main aspects of the Korovkin-type approximation theory in the framework of a class of locally convex vector lattices of continuous functions on a locally compact Hausdorff space $X$, that we call regular l.c.v. lattices on X. Although several results obtained in the setting of abstract locally convex vector lattices are available in the literature, our approach here is more direct and simple and gives, in addition, new results. Examples of regular l.c.v. lattice include weighted function spaces (in particular, the space C_0(X) of all continuous real valued functions on X vanishing at infinity) and every sublattice of C(X) containing the continuous functions with compact support endowed with the topology of the pointwise convergence on X or with the topology of the uniform convergence on compact subsets of X. Therefore, as a matter of fact, our results generalize and/or add some new aspects to those of F. Altomare and M. Campiti ([2]) and H. Bauer and K. Donner ([9]) for C_0(X) and others. Due to its length, the paper is split up into two parts. In this first part, we present some preliminary results for regular l.c.v. lattices. In particular, we introduce some suitable enveloping functions related to a continuous positive linear operator and we study the corresponding space of generalized affine functions. Moreover, we present a natural notion of Choquet boundary and we characterize it by means of these enveloping functions. The case of the identity operator is also considered and, as a consequence, a Stone-Weierstrass type theorem is obtained. In the second part of the paper, by using the results stated in the first part, given a regular l.c.v. lattice E and a continuous positive linear operator T:E \rightarrow E, we characterize those subspaces H of E which are Korovkin subspaces in E for T , in the sense that every equicontinuous net of positive linear operators from E ito E converging T on H in E, automatically converges to T in E. By using this characterization, we present simple methods to construct Korovkin subspaces for particular positive linear operators, called finitely defined operators as well as for the identity operator. We also exhibit examples of finite dimensional Korovkin subspaces for these operators. Finally, we present some Korovkin-type theorems for continuous positive projections. This last part of the paper is a generalization of a previous work of the authors in the setting of adapted spaces (see also [2, Sect. 3.3]). |

Handle: | http://hdl.handle.net/11586/52869 |

Appare nelle tipologie: | 1.1 Articolo in rivista |