Haar problem for sign-sensitive approximations

The Haar problem for sign-sensitive approximations consists in finding necessary and sufficient conditions for a finite-dimensional subspace $L$ of the space $C(E)$ of continuous functions on a compact subset $E$ of $\mathbb R$ and a sign-sensitive weight $p(x)=\bigl (p_-(x),p_+(x)\bigr )$, $x \in E$, ensuring that for each function $f$ in $L$ there exists a unique element of best approximation with weight $p$. Several conditions of this kind are established. These conditions are shown to be closely connected with the topological properties of the annihilators of the functions $p_-(x)$ and $p_+(x)$. In particular, the sign-sensitive weights $p=(p_-,p_+)$ are described such that the same condition as the one introduced by Haar for uniform approximations (that is, for $p(x) \equiv (1,1)$) serves the corresponding Haar problem.