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## Studia Mathematica

2003 | 155 | 2 | 131-143
Tytuł artykułu

### Some theorems of Korovkin type

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EN
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EN
We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, $C_{ℝ}(X)$) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., $C_{ℝ}(Y)$), and $ϕ_{∞}$ a linear isometry from M into C(Y) (resp., $C_{ℝ}(Y)$). We show, under the assumption that $Π_{N} ⊂ Π_{T}$, where $Π_{N}$ is the Choquet boundary for $N = Span(⋃_{1≤n≤∞}Nₙ)$, Nₙ = ϕₙ(M) (n = 1,2,..., ∞), and $Π_{T}$ the Choquet boundary for $T = ϕ_{∞}(S)$, that {ϕₙ(f)} converges pointwise to $ϕ_{∞}(f)$ for any f ∈ M provided {ϕₙ(f)} converges pointwise to ${ϕ_{∞}(f)}$ for any f ∈ S; that {ϕₙ(f)} converges uniformly on any compact subset of $Π_{N}$ to $ϕ_{∞}(f)$ for any f ∈ M provided {ϕₙ(f)} converges uniformly to $ϕ_{∞}(f)$ for any f ∈ S; and that, in the case where S is a function algebra, {ϕₙ} norm converges to $ϕ_{∞}$ on M provided {ϕₙ(f)} norm converges to $ϕ_{∞}$ on S. The proofs are in the spirit of the original one for the theorem of Korovkin.
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Kategorie tematyczne
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Rocznik
Tom
Numer
Strony
131-143
Opis fizyczny
Daty
wydano
2003
Twórcy
autor
• Mathematics Division, Sendai Shirayuri Gakuen High School, Murasaki-Yama 1-2-1, Izumi-ku, Sendai 981-3205, Japan
autor
• Department of Mathematical Sciences, Faculty of Science, Yamagata University, Yamagata 990-8560, Japan
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