BERNSTEIN-TYPE APPROXIMATION PROCESSES FOR VECTOR-VALUED FUNCTIONS (Nonlinear Analysis and Convex Analysis)
Toshihiko Nishishiraho · Institutional Repositories DataBase (IRDB) · 1998
E$ is equal to $\mathbb{R}$ , we $\mathrm{s}\mathrm{i}\mathrm{m}_{\mathrm{P}}1\mathrm{y}$ write $B(X)$ and $C(X)$ instead of $B(X, E)$ and $C(X, E),$ respectively.$\mathrm{T}\mathrm{h}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}_{\mathrm{o}\mathrm{u}\mathrm{t}}$ this paper we suppose that $E$ always contains an element $e$ such that $e>0,$ $||e||=1$ and $|a|\leq||a||e$ for all $a\in E$ .We call $e$ the normal order unit of $E$ .We define $\rho(x)=e$ and $1_{X}(x)=1$ for all $x\in X$ .Notice that $\rho$ and $1_{X}$ are the normal order units of $C(x_{\text{ノ}}.E)$ and $C(X),$ respectively.For any $a$ $\in E$ and $v\in B(X)$ , the function $v\otimes a$ is defined by $(v\otimes a)(x)=v(x)a$ for all $x\in X.$ Also, for any $v\in B(X)$ and $f\in B(X, E)$ , we define $(vf)(x)=v(x)f(x)$ for all $x\in X.$ Clearly, $v\otimes a$ and $vf$ belong to $B(X, E)$ , and $||v\otimes a||=$ $||v||||a||,$ $||vf||\leq||v||||f||$ and $\rho=1_{X}\otimes e$ .We shall denote by $C(X)\otimes E$ the linear subspace of $C(X, E)$ consisting of all finite sums of functions of the form $v\otimes a$ , where $v\in C(X)$ and $a$ $\in E$ .A $\mathrm{b}_{\mathrm{o}\mathrm{u}\mathrm{n}}\mathrm{d}\mathrm{e}\mathrm{d}$ linear operator $L$ of $C(X, E)$ into $B(X, E)$ is said to be quasi-positive if $v,$ $w\in C(X)$ and $|v|\leq w$ , then $||L(v\otimes a)(x)||\leq||L(w\otimes a)(x)||$ for all $a\in E_{+}$ and all $x\in X$ .(cf. [8], [9]).A typical example of such an