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

Tom 171 / 2005

Francesco Altomare, Mirella Cappelletti Montano Studia Mathematica 171 (2005), 239-260 MSC: 46E05, 46E10, 46E99. DOI: 10.4064/sm171-3-3

Streszczenie

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.

Autorzy

  • Francesco AltomareDepartment of Mathematics
    University of Bari
    Campus Universitario
    Via E. Orabona, 4
    70125 Bari, Italy
    e-mail
  • Mirella Cappelletti MontanoDepartment of Mathematics
    University of Bari
    Campus Universitario
    Via E. Orabona, 4
    70125 Bari, Italy
    e-mail

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