Geodesic
Existence, uniqueness, and stability of best and near-best approximations
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 78 (2023) no. 3, pp. 399-442

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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.
Keywords: generalized rational functions, $\varepsilon$-selection, near-best approximant, sun, monotone path-connected set, stability of approximation, piecewise-polynomial function. 
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     title = {Existence, uniqueness, and stability of best and near-best approximations},
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A. R. Alimov; K. S. Ryutin; I. G. Tsar'kov. Existence, uniqueness, and stability of best and near-best approximations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 78 (2023) no. 3, pp. 399-442. http://geodesic.mathdoc.fr/item/RM_2023_78_3_a0/