Geodesic
Haar problem for sign-sensitive approximations
Sbornik. Mathematics, Tome 188 (1997) no. 2, pp. 265-297 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {E. A. Sevast'yanov},
     title = {Haar problem for sign-sensitive approximations},
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     pages = {265--297},
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     volume = {188},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1997_188_2_a4/}
}
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E. A. Sevast'yanov. Haar problem for sign-sensitive approximations. Sbornik. Mathematics, Tome 188 (1997) no. 2, pp. 265-297. http://geodesic.mathdoc.fr/item/SM_1997_188_2_a4/

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