Some theorems of Korovkin type

Tomoko Hachiro, and Takateru Okayasu. "Some theorems of Korovkin type." Studia Mathematica 155.2 (2003): 131-143. <http://eudml.org/doc/284543>.

@article{TomokoHachiro2003,
abstract = {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.},
author = {Tomoko Hachiro, Takateru Okayasu},
journal = {Studia Mathematica},
keywords = {Korovkin type approximation; function space; linear contraction},
language = {eng},
number = {2},
pages = {131-143},
title = {Some theorems of Korovkin type},
url = {http://eudml.org/doc/284543},
volume = {155},
year = {2003},
}

TY - JOUR
AU - Tomoko Hachiro
AU - Takateru Okayasu
TI - Some theorems of Korovkin type
JO - Studia Mathematica
PY - 2003
VL - 155
IS - 2
SP - 131
EP - 143
AB - 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.
LA - eng
KW - Korovkin type approximation; function space; linear contraction
UR - http://eudml.org/doc/284543
ER -