Geodesic
Approximation with limitations. The problem of Bernstein
Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 3 (2017), pp. 93-106

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An inverse theorem of approximation theory is obtained in an arbitrary Banach space in the case of approximations by elements of closed locally compact cones. The theorem is an analogue of the well-known theorem of S.N. Bernshtein, which he proved in the Banach space of continuous functions on an interval when approximating by finite-dimensional subspaces of algebraic polynomials.
Keywords: Banach space, best approximation, closed convex locally compact set, cone, finite-dimensional subspace, complete system in a normed space. 
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     author = {I. A. Drozhzhin},
     title = {Approximation with limitations. {The} problem of {Bernstein}},
     journal = {Vestnik Tverskogo gosudarstvennogo universiteta. Seri\^a Prikladna\^a matematika},
     pages = {93--106},
     publisher = {mathdoc},
     number = {3},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTPMK_2017_3_a6/}
}
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I. A. Drozhzhin. Approximation with limitations. The problem of Bernstein. Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 3 (2017), pp. 93-106. http://geodesic.mathdoc.fr/item/VTPMK_2017_3_a6/