Chebyshev sets composed of subspaces in asymmetric normed spaces

A. R. Alimov ; I. G. Tsar'kov

Izvestiya. Mathematics , Tome 88 (2024) no. 6, pp. 1032-1049.

Résumé

By definition, a Chebyshev set is a set of existence and uniqueness, that is, any point has a unique best approximant from this set. We study properties of Chebyshev sets composed of finitely or infinitely many planes (closed affine subspaces, possibly degenerated to points). We show that a finite union of planes is a Chebyshev set if and only if is a Chebyshev plane. Under some conditions on a space or a set, we show that a countable union of planes is never a Chebyshev set (unless this union is a Chebyshev plane itself). As a corollary, we give the following partial answer to the famous Efimov–Stechkin–Klee problem on convexity of Chebyshev sets: in Hilbert spaces (and, more generally, in reflexive (CLUR)-spaces), an at most countable union of planes is a Chebyshev set if and only if this set is a Chebyshev plane. Results of this kind are obtained both in usual normed linear spaces and in spaces with asymmetric norm.

Détail

Keywords: Chebyshev set, best approximation, union of subspaces, asymmetric normed space, sun, ridge function.

Bibliographie

[1] A. R. Alimov and I. G. Tsar'kov, “Chebyshev unions of planes, and their approximative and geometric properties”, J. Approx. Theory, 298 (2024), 106009 | DOI | MR | Zbl

[2] I. G. Tsar'kov, “Chebyshev sets with piecewise continuous metric projection”, Math. Notes, 113:6 (2023), 840–849 | DOI

[3] Kiryung Lee and Y. Bresler, “Efficient and guaranteed rank minimization by atomic decomposition”, 2009 IEEE international symposium on information theory (Seoul 2009), IEEE, 2009, 314–318 | DOI

[4] C. Hegde, P. Indyk, and L. Schmidt, “Approximation algorithms for model-based compressive sensing”, IEEE Trans. Inform. Theory, 61:9 (2015), 5129–5147 | DOI | MR | Zbl

[5] V. E. Ismailov, Ridge functions and applications in neural networks, Math. Surveys Monogr., 263, Amer. Math. Soc., Providence, RI, 2021 | DOI | MR | Zbl

[6] R. A. DeVore, “Nonlinear approximation”, Acta Numer., 7, Cambridge Univ. Press, Cambridge, 1998, 51–150 | DOI | MR | Zbl

[7] A. R. Alimov and H. Berens, “Examples of Chebyshev sets in matrix spaces”, J. Approx. Theory, 99:1 (1999), 44–53 | DOI | MR | Zbl

[8] A. G. Robertson, “Strongly Chebyshev subspaces of matrices”, J. Approx. Theory, 55:3 (1988), 264–269 | DOI | MR | Zbl

[9] A. R. Alimov and I. G. Tsar'kov, “Connectedness and solarity in problems of best and near-best approximation”, Russian Math. Surveys, 71:1 (2016), 1–77 | DOI

[10] A. R. Alimov and I. G. Tsar'kov, Geometric approximation theory, Springer Monogr. Math., Springer, Cham, 2021 | DOI | MR | Zbl

[11] M. G. Kreĭn, “The $L$-problem in an abstract linear normed space”, Some questions in the theory of moments, Transl. Math. Monogr., 2, Amer. Math. Soc., Providence, RI, 1962, 175–204 | MR | Zbl

[12] V. Donjuán and N. Jonard-Pérez, “Separation axioms and covering dimension of asymmetric normed spaces”, Quaest. Math., 43:4 (2020), 467–491 | DOI | MR | Zbl

[13] Ş. Cobzaş, Functional analysis in asymmetric normed spaces, Front. Math., Birkhäuser/Springer Basel AG, Basel, 2013 | DOI | MR | Zbl

[14] A. R. Alimov, K. S. Ryutin, and I. G. Tsar'kov, “Existence, uniqueness, and stability of best and near-best approximations”, Russian Math. Surveys, 78:3 (2023), 399–442 | DOI

[15] A. R. Alimov and I. G. Tsar'kov, “Connectedness and approximative properties of sets in asymmetric spaces”, Filomat, 38:9 (2024), 3243–3259 | MR

[16] A. R. Alimov and I. G. Tsar'kov, “Smoothness of subspace sections of the unit balls of $C(Q)$ and $L^1$”, J. Approx. Theory, 265 (2021), 105552 | DOI | MR | Zbl

[17] A. R. Alimov and I. G. Tsar'kov, “Suns, moons, and $\mathring{B}$-complete sets in asymmetric spaces”, Set-Valued Var. Anal., 30:3 (2022), 1233–1245 | DOI | MR | Zbl

[18] Th. Jahn and Ch. Richter, “Coproximinality of linear subspaces in generalized Minkowski spaces”, J. Math. Anal. Appl., 504:1 (2021), 125351 | DOI | MR | Zbl

[19] I. G. Tsar'kov, “Uniform convexity in nonsymmetric spaces”, Math. Notes, 110:5 (2021), 773–783 | DOI

[20] I. G. Tsar'kov, “Continuous selections of set-valued mappings and approximation in asymmetric and semilinear spaces”, Izv. Math., 87:4 (2023), 835–851 | DOI

[21] I. G. Tsarkov, “Uniformly and locally convex asymmetric spaces”, Russ. J. Math. Phys., 29:1 (2022), 141–148 | DOI | MR | Zbl

[22] A. R. Alimov and I. G. Tsar'kov, “Ball-complete sets and solar properties of sets in asymmetric spaces”, Results Math., 77:2 (2022), 86 | DOI | MR | Zbl

[23] A. R. Alimov and I. G. Tsar'kov, “Solarity and proximinality in generalized rational approximation in spaces $C(Q)$ and $L^p$”, Russ. J. Math. Phys., 29:3 (2022), 291–305 | DOI | MR | Zbl

[24] A. R. Alimov and I. G. Tsarkov, “$\mathring{B}$-complete sets: approximative and structural properties”, Siberian Math. J., 63:3 (2022), 412–420 | DOI

[25] S. V. Konyagin and I. G. Tsar'kov, “Efimov–Stechkin spaces”, Moscow Univ. Math. Bull., 41:5 (1986), 20–28

[26] I. G. Tsar'kov, “Relations between certain classes of sets in Banach spaces”, Math. Notes, 40:2 (1986), 597–610 | DOI

[27] M. G. Kreĭn and A. A. Nudel'man, The Markov moment problem and extremal problems, Ideas and problems of P. L. Čebyšev and A. A. Markov and their further development, Transl. Math. Monogr., 50, Amer. Math. Soc., Providence, RI, 1977 | MR | Zbl

[28] R. J. Duffin and L. A. Karlovitz, “Formulation of linear programs in analysis. I. Approximation theory”, SIAM J. Appl. Math., 16:4 (1968), 662–675 | DOI | MR | Zbl

[29] E. A. Sevast'yanov, “Haar problem for sign-sensitive approximations”, Sb. Math., 188:2 (1997), 265–297 | DOI

[30] E. P. Dolzhenko and E. A. Sevast'yanov, “Approximations with a sign-sensitive weight: existence and uniqueness theorems”, Izv. Math., 62:6 (1998), 1127–1168 | DOI

[31] A. V. Pokrovskii, “The best asymmetric approximation in spaces of continuous functions”, Izv. Math., 70:4 (2006), 809–839 | DOI

[32] A. R. Alimov, “Strict protosuns in asymmetric spaces of continuous functions”, Results Math., 78:3 (2023), 95 | DOI | MR | Zbl

[33] I. G. Tsar'kov, “Properties of suns in the spaces $L^1$ and $C(Q)$”, Russ. J. Math. Phys., 28:3 (2021), 398–405 | DOI | MR | Zbl