Some theorems of Korovkin type

Tomoko Hachiro¹ ; Takateru Okayasu²

¹ Mathematics Division Sendai Shirayuri Gakuen High School Murasaki-Yama 1-2-1, Izumi-ku Sendai 981-3205, Japan
² Department of Mathematical Sciences Faculty of Science Yamagata University Yamagata 990-8560, Japan

Studia Mathematica, Tome 155 (2003) no. 2, pp. 131-143

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, $C_{{\mathbb R}}(X)$) of all complex-valued (resp., real-valued) continuous functions on $X$, $S\subset M$ a complex (resp., real) function space on $X$, ${\{\phi_{n}\}}$ a sequence of unital linear contractions from $M$ into $C(Y)$ (resp., $C_{{\mathbb R}}(Y)$), and $\phi_{\infty}$ a linear isometry from $M$ into $C(Y)$ (resp., $C_{{\mathbb R}}(Y)$). We show, under the assumption that ${\mit\Pi}_{N} \subset {\mit\Pi}_{T}$, where ${\mit\Pi}_{N}$ is the Choquet boundary for $N=\mathop{\rm Span} (\bigcup_{1\le n\le \infty}N_n)$, $N_n=\phi_{n}(M)\ (n=1,2,\ldots , \infty)$, and ${\mit\Pi}_{T}$ the Choquet boundary for ${T=\phi_{\infty}(S)}$, that ${\{\phi_{n}(f)\}}$ converges pointwise to $\phi_{\infty}(f)$ for any $f\in M$ provided $\{\phi_{n}(f)\}$ converges pointwise to ${\phi_{\infty}(f)}$ for any $f\in S$; that ${\{\phi_{n}(f)\}} $ converges uniformly on any compact subset of ${{\mit\Pi}_N} $ to $\phi_{\infty}(f)$ for any $f\in M$ provided ${\{\phi_{n}(f)\}} $ converges uniformly to ${\phi_{\infty}(f)}$ for any $f\in S$; and that, in the case where $S$ is a function algebra, $\{\phi_n\}$ norm converges to $\phi_{\infty}$ on $M$ provided ${\{\phi_{n}(f)\}}$ norm converges to $\phi_{\infty}$ on $S$. The proofs are in the spirit of the original one for the theorem of Korovkin.

Détail

DOI : 10.4064/sm155-2-3

Keywords: another approach known theorem korovkin following situation compact hausdorff spaces unital subspace banach space respectively mathbb complex valued resp real valued continuous functions subset complex resp real function space phi sequence unital linear contractions resp mathbb phi infty linear isometry resp mathbb under assumption mit subset mit where mit choquet boundary mathop span bigcup infty phi ldots infty mit choquet boundary phi infty phi converges pointwise phi infty provided phi converges pointwise phi infty phi converges uniformly compact subset mit phi infty provided phi converges uniformly phi infty where function algebra phi norm converges phi infty provided phi norm converges phi infty proofs spirit original theorem korovkin