Abstract:
We study the problem of the existence of a continuous selection for the metric projection to the set of n-link piecewise-linear functions in the space C[0,1]. We show that there is a continuous selection if and only if n=1 or n=2. We establish that there is a continuous
ε-selection to L (L⊂C[0,1]) if L belongs to a certain class of sets that contains, in particular, the set of algebraic rational fractions and the set of piecewise-linear functions. We construct an example showing that sometimes there is no ε-selection
for a set of splines of degree d>1.
\Bibitem{Liv03}
\by E.~D.~Livshits
\paper Stability of the operator of $\varepsilon$-projection to the set of splines in~$C[0,1]$
\jour Izv. Math.
\yr 2003
\vol 67
\issue 1
\pages 91--119
\mathnet{http://mi.mathnet.ru/eng/im420}
\crossref{https://doi.org/10.1070/IM2003v067n01ABEH000420}
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https://www.mathnet.ru/eng/im420
https://doi.org/10.1070/IM2003v067n01ABEH000420
https://www.mathnet.ru/eng/im/v67/i1/p99
This publication is cited in the following 19 articles:
A. R. Alimov, K. S. Ryutin, I. G. Tsar'kov, “Existence, uniqueness, and stability of best and near-best approximations”, Russian Math. Surveys, 78:3 (2023), 399–442
I. G. Tsar'kov, “Approximative properties of sets and continuous selections”, Sb. Math., 211:8 (2020), 1190–1211
I. G. Tsar'kov, “Local Approximation Properties of Sets and Continuous Selections on Them”, Math. Notes, 106:6 (2019), 995–1008
I. G. Tsar'kov, “Weakly monotone sets and continuous selection in asymmetric spaces”, Sb. Math., 210:9 (2019), 1326–1347
I. G. Tsar'kov, “Continuous selections for metric projection operators and for their generalizations”, Izv. Math., 82:4 (2018), 837–859
I. G. Tsar'kov, “Continuous selections in asymmetric spaces”, Sb. Math., 209:4 (2018), 560–579
I. G. Tsar'kov, “New Criteria for the Existence of a Continuous ε-Selection”, Math. Notes, 104:5 (2018), 727–734
I. G. Tsar'kov, “Weakly monotone sets and continuous selection from a near-best approximation operator”, Proc. Steklov Inst. Math., 303 (2018), 227–238
I. G. Tsar'kov, “Continuous selection for set-valued mappings”, Izv. Math., 81:3 (2017), 645–669
I. G. Tsar'kov, “Continuous ε-Selection and Monotone Path-Connected Sets”, Math. Notes, 101:6 (2017), 1040–1049
Tsar'kov I.G., “Continuous Selection From the Sets of Best and Near-Best Approximation”, Dokl. Math., 96:1 (2017), 362–364
I. G. Tsarkov, “NEPRERYVNYE VYBORKI IZ MNOZhESTVA BLIZhAIShIKh I POChTI BLIZhAIShIKh TOChEK, “Doklady Akademii nauk””, Doklady Akademii Nauk, 2017, no. 4, 373
I. G. Tsar'kov, “Continuous ε-selection”, Sb. Math., 207:2 (2016), 267–285
A. R. Alimov, I. G. Tsar'kov, “Connectedness and solarity in problems of best and near-best approximation”, Russian Math. Surveys, 71:1 (2016), 1–77
I. G. Tsar'kov, “Local and global continuous ε-selection”, Izv. Math., 80:2 (2016), 442–461
A. R. Alimov, I. G. Tsar'kov, “Connectedness and other geometric properties of suns and Chebyshev sets”, J. Math. Sci., 217:6 (2016), 683–730
Dušan Repovš, Pavel V. Semenov, Recent Progress in General Topology III, 2014, 711
Livshits E. D., “Continuous selections of operators of almost best approximation by splines in the space Lp[0,1]. I”, Russ. J. Math. Phys., 12:2 (2005), 215–218
E. D. Livshits, “On Almost-Best Approximation by Piecewise Polynomial Functions in the Space C[0,1]”, Math. Notes, 78:4 (2005), 586–591
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