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Best simultaneous approximation of an infinite set of functions. (English) Zbl 0974.41025

Let\(( X,\|\cdot\|_{X}) \) be a normed linear space and let \(( B,\|\cdot\|_{B}) \) be a normed linear space of sequences \(\{ a_{i}\} _{i=1}^{\infty }.\) Let \(F=( \varphi _{1},\varphi _{2},\dots ,\varphi _{i},\dots) \) be a fixed sequence in \(X\) and let \[ \|F\|=\max \Biggl\{\Biggl\|\sum_{i=1}^{\infty }a_{i}\varphi _{i}\Biggr\|_{X}:a=\{ a_{i}\} _{i=1}^{\infty }\in U\Biggr\} \] where \(U\) is the unit ball in \(B\). For \(S\subset X\) and \(f=( \varphi ,\varphi ,\dots ,\varphi ,\dots) \) with \(\varphi \in S\) one looks for \(f^{\ast }=( \varphi ^{\ast },\varphi ^{\ast },\dots) ,\) \(\varphi ^{\ast }\in S,\) such that \(d( F,S) =\|F-f^{\ast }\|\) (the problem of best simultaneous approximation (b.s.a)). Under the hypothesis \(\sup \{ \textstyle\sum\nolimits_{i=1}^{\infty }|a_{i}|:a\in U\} <\infty \) let \(\overline{U}\) denote the \(w^{\ast }\) closure of the set \(U\) in \(B\) and let \(W\) be the unit ball in the dual space of \(X\). For \(F=( \varphi _{1},\dots ,\varphi _{i},\dots) ,\) \( \varphi _{i}\in X\) define \(g_{F}( a,w) =\sum_{i=1}^{\infty }a_{i}\langle w,\varphi _{i}\rangle,\) for all \(( a,w) \in \overline{U}\times W\). The authors find characterization and uniqueness results for the elements \( f^{\ast }\) of b.s.a. For instance, if \(S\subset X\) is a sun for the b.s.a and \(\{ \varphi _{i}\} _{i\geq 1}\subset X\) is such that \(\varphi _{i}\rightarrow \varphi _{0}\) \(( i\rightarrow \infty) \) then \(f^{\ast }\) is a b.s.a element if and only if for every \(f=( \varphi ,\varphi ,\dots ,\varphi ,\dots) \) there exist \(a\in \text{ext }\overline{U}\) and \(w\in \text{ext }W\) such that \(g_{F-f^{\ast }}( a,w) =\|F-f^{\ast }\|\) and \(g_{f^{\ast }-f}( a,w) \geq 0\) (Theorem 2). If \(X\) is uniformly convex in any direction and \(S\) is a sun then for any \(F=( \varphi _{1},\dots ,\varphi _{i},\dots) \) such that \(d( F,X) <d( F,S) \) there exists at most one b.s.a. element \(f^{\ast }\) (Theorem 4). Sufficient conditions for the strong uniqueness of \(f^{\ast }\) are found also.

MSC:

41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
41A28 Simultaneous approximation
41A52 Uniqueness of best approximation
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