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 Tr. Mat. Inst. Steklova, 2018, Volume 303, Pages 17–25 (Mi tm3939)

Selections of the best and near-best approximation operators and solarity

A. R. Alimovab

a Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia
b Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

Abstract: In a finite-dimensional Banach space, a closed set with lower semicontinuous metric projection is shown to have a continuous selection of the near-best approximation operator. Such a set is known to be a sun. In the converse question of the stability of best approximation by suns, it is proved that a strict sun in a finite-dimensional Banach space of dimension at most $3$ is a $P$-sun, has a contractible set of nearest points, and admits a continuous $\varepsilon$-selection from the operator of near-best approximation for any $\varepsilon >0$. A number of approximative and geometric properties of sets with lower semicontinuous metric projection are obtained.

 Funding Agency Grant Number Russian Foundation for Basic Research 18-01-00333_à19-01-00332_à Ministry of Education and Science of the Russian Federation ÍØ-6222.2018.1 The work was supported by the Russian Foundation for Basic Research (project nos. 18-01-00333 and 19-01-00332-a) and by a grant of the President of the Russian Federation (project no. NSh-6222.2018.1).

DOI: https://doi.org/10.1134/S0371968518040027

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English version:
Proceedings of the Steklov Institute of Mathematics, 2018, 303, 10–17

Bibliographic databases:

UDC: 517.982.256+517.982.252

Citation: A. R. Alimov, “Selections of the best and near-best approximation operators and solarity”, Harmonic analysis, approximation theory, and number theory, Collected papers. Dedicated to Academician Sergei Vladimirovich Konyagin on the occasion of his 60th birthday, Tr. Mat. Inst. Steklova, 303, MAIK Nauka/Interperiodica, Moscow, 2018, 17–25; Proc. Steklov Inst. Math., 303 (2018), 10–17

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3939
• https://doi.org/10.1134/S0371968518040027
• http://mi.mathnet.ru/eng/tm/v303/p17

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This publication is cited in the following articles:
1. Alimov A.R., “Solarity of Sets in Max-Approximation Problems”, J. Fixed Point Theory Appl., 21:3 (2019), UNSP 76
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