<p>The paper is mainly concerned with those linear function subspaces on topological Hausdorff spaces, which guarantee that arbitrary sequences of positive linear operators locally converge, on bounded locally continuous functions, toward a given positive linear operator or, in particular, toward the identity operator, provided the same convergence property holds for them. For short, here such linear subspaces are called local Korovkin subspaces with respect to a positive linear operator or the identity operator. The main results are established by using order-theoretical methods based on upper and lower enveloping functions. In particular, a special class of subsets <i>S</i> of continuous functions on a completely regular Hausdorff space is recognized such that, setting <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$S^{2}:=\{u^{2}|u \in S \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>S</mi> <mn>2</mn> </msup> <mo>:</mo> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <msup> <mi>u</mi> <mn>2</mn> </msup> <mo stretchy="false">|</mo> <mi>u</mi> <mo>∈</mo> <mi>S</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, then the linear subspace generated by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\{{\textbf {1}}\}\cup S \cup S^{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">{</mo> <mn mathvariant="bold">1</mn> <mo stretchy="false">}</mo> </mrow> <mo>∪</mo> <mi>S</mi> <mo>∪</mo> <msup> <mi>S</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> is a local Korovkin subspace for every weighted composition operator associated with a homeomorphic mapping and, in particular, for the identity operator. In the final section, some applications are presented concerning the sequences of Bernstein-Schnabl operators acting on convex Borel subsets of locally convex spaces together with some recent generalizations of them. In some particular cases, by means of such operators, a constructive Weierstrass-type density result for the above mentioned class of functions in terms of polynomials is obtained as well.</p>

Local convergence problems for sequences of positive linear operators

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发表日期：

2025年11月13日

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Francesco Altomare

摘要

The paper is mainly concerned with those linear function subspaces on topological Hausdorff spaces, which guarantee that arbitrary sequences of positive linear operators locally converge, on bounded locally continuous functions, toward a given positive linear operator or, in particular, toward the identity operator, provided the same convergence property holds for them. For short, here such linear subspaces are called local Korovkin subspaces with respect to a positive linear operator or the identity operator. The main results are established by using order-theoretical methods based on upper and lower enveloping functions. In particular, a special class of subsets S of continuous functions on a completely regular Hausdorff space is recognized such that, setting $S^{2}:=\{u^{2}|u \in S \}$ $S^{2} : = {u^{2} | u \in S}$ , then the linear subspace generated by $\{{\textbf {1}}\}\cup S \cup S^{2}$ ${1} \cup S \cup S^{2}$ is a local Korovkin subspace for every weighted composition operator associated with a homeomorphic mapping and, in particular, for the identity operator. In the final section, some applications are presented concerning the sequences of Bernstein-Schnabl operators acting on convex Borel subsets of locally convex spaces together with some recent generalizations of them. In some particular cases, by means of such operators, a constructive Weierstrass-type density result for the above mentioned class of functions in terms of polynomials is obtained as well.