Interpolation by weak Chebyshev spaces (Q1971919)

Let \(K\) be a totally ordered set; \( F(K) \) be the space of real functions defined on \( K \) and let \(U\) be an \(n\)-dimensional subspace of \( F(K). \) The subset \( T=\{t_1,\dots{},t_n\}\subseteq K \) is called an interpolation set with respect to \(U\) if for any \( \{y_1,\dots{},y_n\}\subseteq\mathbb R, \) there exists a unique function \( u\in U \) such that \( u(t_i)=y_i, i=1,\dots{},n. \) The space \(U\) is said to be weak Chebyshev if every \( u\in U \) has at most \( n-1 \) sign changes. Let \( Z(U):=\cap_{f\in U}\{t\in K: f(t)=0\} \) be the set of zeroes of \(U.\) The authors prove that if \( T\subseteq K\setminus Z(U) \) and \( t_1<\dots{}<t_n; t_{n+1}:=t_1 \) then \(T\) is an interpolation set with respect to the weak Chebyshev space \(U\) if and only if for all \( P\subseteq \{1,\dots{},n\}, \text{card} (T\cap\cup_{i\in P} [t_i,t_{i+1}])\leq \dim U|_{\cup_{i\in P}[t_i,t_{i+1}]}.\) Here \( [t_n,t_1]:=\{t\in K, t\geq t_n \) or \( t\leq t_1\}. \) The second characterization of interpolation sets with respect to the weak Chebyshev space \(U\) is given in the case when \(U\) posseses a locally linearly independent basis.