Existence, uniqueness, and stability of best and near-best approximations

Abstract

Existence and stability of $\varepsilon$-selections (selections of operators of near-best approximation) are studied. Results relating the existence of continuous $\varepsilon$-selections with other approximative and structural properties of approximating sets are given. Both abstract and concrete sets are considered - the latter include $n$-link piecewise linear functions, $n$-link $r$-polynomial functions and their generalizations, $k$-monotone functions, and generalized rational functions. Classical problems of the existence, uniqueness, and stability of best and near-best generalized rational approximations are considered. Bibliography: 70 titles.