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Strict Protosuns in Asymmetric Spaces of Continuous Functions

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Abstract

Several new results of geometric approximation theory in asymmetric normed spaces of continuous functions (spaces \(C_{\psi }(Q)\) and \(C_{0;\!\psi }(Q)\) with asymmetric weight) are put forward. A number of properties characterizing the strict protosuns (also called Kolmogorov sets) in spaces \(C_{\psi }(Q)\) and \(C_{0;\!\psi }(Q)\) are obtained. As characterization conditions we consider the Brosowski–Wegmann connectedness, ORL-continuity of the metric projection operator, \(\mathring{B}\)-completeness, unimodality, and lunarity.

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Data Availibility Statement

The author declares that the data supporting the findings of this study are available within the article and the references cited therein.

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No (program) code was used in the research.

Notes

  1. Kolmogorov sets were introduced by Brosowski and Wegmann [12, 13] as the sets satisfying the Kolmogorov criterion for an element of best approximation; in modern literature on approximation such sets are called strict protosuns. This name was introduced in [2] to avoid confusion in the definitions of a “strict sun of existence” (a strict sun) and a “strict sun which is not necessarily an existence set” (a strict protosun)—all such sets satisfy (in some or other form) the Kolmogorov criterion; cf. also [5, §5].

  2. The (algebraic) kernel of a set A in a linear space X is defined as the set of all points \(x \in A\) such that for every \(v \in X\) there exists a number \(\varepsilon = \varepsilon (v) > 0\) for which \(x + tv \in A\) whenever \(|t| < \varepsilon \). If X is a normed space, then every inner point of A belongs to the algebraic kernel, but the algebraic kernel can be larger than the interior of A.

  3. Here (MS) comes from the phrase “every moon in X is a sun” (see [9]).

  4. In this notation, “\(\mathring{B}\)” stands for an open ball.

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Acknowledgements

The author would like to thank the referees for their for constructive criticism and detailed comments and suggestions. The author is especially grateful to T. Jahn or reading the paper and drawing my attention to several imperfections.

Funding

This research was carried out at Lomonosov Moscow State University with the financial support of the Russian Science Foundation (Grant No. 22-11-00129).

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Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia, 119991

    Alexey R. Alimov

  2. Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia, 119991

    Alexey R. Alimov

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Alimov, A.R. Strict Protosuns in Asymmetric Spaces of Continuous Functions. Results Math 78, 95 (2023). https://doi.org/10.1007/s00025-023-01876-9

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