Approximations with a sign-sensitive weight: existence and uniqueness theorems

Approximations with a sign-sensitive weight generally take into account both the absolute value of the error of approximation and its sign. We study the problems of existence, uniqueness and plurality for the element of best uniform approximation with a given sign-sensitive weight $p=(p_-,p_+)$ by functions of a given family $L$ on an interval $\Delta$. We also study these problems for approximations in normed linear spaces $\mathcal L$ by elements of a family $L\subset\mathcal L$, where the deviation of an element $x$ from another element $y$ is measured by the value $P(x-y)$ of some non-negative sublinear functional $P$. A very important role is played by the rigidity and freedom of the systems $(p,L)$ and $(P;L)$. These notions are also studied in the paper, with special attention being given to the case of Chebyshev subspaces $L$.