Approximations with a sign-sensitive weight. Stability, applications to the theory of snakes and Hausdorff approximations

Sign-sensitive approximations take into account not only the absolute value of the approximation error but also its sign. In the previous paper with the same title and the subtitle “existence and uniqueness theorems” we studied the problems of existence, uniqueness and plurality for the element of best uniform approximation with a sign-sensitive weight $p=(p_-,p_+)$ ($p_\pm(x)\geqslant 0$, $x\in E$) by some (in particular, Chebyshev) family $L$ of bounded functions on a set $E\subset\mathbb R$. An important role was played by the notions of rigidity and freedom of the system $(p,L)$. Here we consider the stability of this process of approximation, that is, whether the least deviations $E(p,L,f)$ and the best approximations $l(p,L,f)$ by elements $l\in L$ depend continuously on $p$ if the variation of $p$ is measured in the so-called $d$-metric. The results are applied to the theory of snakes and Hausdorff approximations of special multivalued functions.