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 Izv. RAN. Ser. Mat., 2003, Volume 67, Issue 1, Pages 99–130 (Mi izv420)

Stability of the operator of $\varepsilon$-projection to the set of splines in $C[0,1]$

E. D. Livshits

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 $\varepsilon$-selection to $L$ ($L\subset C[0,1]$) 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 $\varepsilon$-selection for a set of splines of degree $d>1$.

DOI: https://doi.org/10.4213/im420

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English version:
Izvestiya: Mathematics, 2003, 67:1, 91–119

Bibliographic databases:

UDC: 517.518.8
MSC: 41A65, 41A50, 46B20, 65J15
Revised: 28.08.2002

Citation: E. D. Livshits, “Stability of the operator of $\varepsilon$-projection to the set of splines in $C[0,1]$”, Izv. RAN. Ser. Mat., 67:1 (2003), 99–130; Izv. Math., 67:1 (2003), 91–119

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv420
• https://doi.org/10.4213/im420
• http://mi.mathnet.ru/eng/izv/v67/i1/p99

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Livshits E. D., “Continuous selections of operators of almost best approximation by splines in the space $L_p[0,1]$. I”, Russ. J. Math. Phys., 12:2 (2005), 215–218
2. E. D. Livshits, “On Almost-Best Approximation by Piecewise Polynomial Functions in the Space $C[0,1]$”, Math. Notes, 78:4 (2005), 586–591
3. 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
4. I. G. Tsar'kov, “Continuous $\varepsilon$-selection”, Sb. Math., 207:2 (2016), 267–285
5. 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
6. I. G. Tsar'kov, “Local and global continuous $\varepsilon$-selection”, Izv. Math., 80:2 (2016), 442–461
7. I. G. Tsar'kov, “Continuous selection for set-valued mappings”, Izv. Math., 81:3 (2017), 645–669
8. I. G. Tsar'kov, “Continuous $\varepsilon$-Selection and Monotone Path-Connected Sets”, Math. Notes, 101:6 (2017), 1040–1049
9. Tsar'kov I.G., “Continuous Selection From the Sets of Best and Near-Best Approximation”, Dokl. Math., 96:1 (2017), 362–364
10. I. G. Tsar'kov, “Continuous selections for metric projection operators and for their generalizations”, Izv. Math., 82:4 (2018), 837–859
11. I. G. Tsar'kov, “Continuous selections in asymmetric spaces”, Sb. Math., 209:4 (2018), 560–579
12. I. G. Tsar'kov, “New Criteria for the Existence of a Continuous $\varepsilon$-Selection”, Math. Notes, 104:5 (2018), 727–734
13. I. G. Tsar'kov, “Weakly monotone sets and continuous selection from a near-best approximation operator”, Proc. Steklov Inst. Math., 303 (2018), 227–238
14. I. G. Tsar'kov, “Local Approximation Properties of Sets and Continuous Selections on Them”, Math. Notes, 106:6 (2019), 995–1008
15. I. G. Tsar'kov, “Weakly monotone sets and continuous selection in asymmetric spaces”, Sb. Math., 210:9 (2019), 1326–1347
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