Some theorems of Korovkin type

Tomoko Hachiro ; Takateru Okayasu

Studia Mathematica, Tome 157 (2003), p. 131-143 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2003-01-01

Zbl 1011.41011

EUDML-ID : urn:eudml:doc:284543