On the Korovkin approximation theorem and Volkov-type theorems

Nihan Uygun

doi:10.1186/1029-242x-2014-89

Abstract

In this short paper, we give a generalization of the classical Korovkin approximation theorem (Korovkin in Linear Operators and Approximation Theory, 1960), Volkov-type theorems (Volkov in Dokl. Akad. Nauk SSSR 115:17-19, 1957), and a recent result of

Highlights

In this paper, the classical Korovkin theorem and one of the key results (Theorem ) of [ ] will be generalized to arbitrary compact Hausdorff spaces
Let X be a compact Hausdorff space and E be a subspace of C(X)
Let (An)n∈N be a sequence of positive operators from Hw,h into C(X × Y ) satisfying: (i) An(f ) – f → ; (ii) An(f ) – f → ; (iii) An(f ) – f → ; (iv) An(f + f ) – (f + f ) →

Summary

Introduction

1 Introduction In this paper, the classical Korovkin theorem (see [ ]) and one of the key results (Theorem ) of [ ] will be generalized to arbitrary compact Hausdorff spaces. Let X be a compact Hausdorff space and E be a subspace of C(X). We define Hw,h as the set of all continuous functions f ∈ C(X ×Y ) such that for all (u, v), (x, y) ∈ X ×Y , one has f (u, v) – f (x, y) ≤ wh(f ) h (u, v), (x, y) .

Results

Conclusion