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

This article is cited in 16 scientific papers (total in 16 papers)

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 $d>1$.


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

Bibliographic databases:

UDC: 517.518.8
MSC: 41A65, 41A50, 46B20, 65J15
Received: 12.04.2001
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
\by E.~D.~Livshits
\paper Stability of the operator of $\varepsilon$-projection to the set of splines in~$C[0,1]$
\jour Izv. RAN. Ser. Mat.
\yr 2003
\vol 67
\issue 1
\pages 99--130
\jour Izv. Math.
\yr 2003
\vol 67
\issue 1
\pages 91--119

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    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  mathscinet  zmath  isi  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  mathnet  crossref  mathscinet
    4. I. G. Tsar'kov, “Continuous $\varepsilon$-selection”, Sb. Math., 207:2 (2016), 267–285  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. I. G. Tsar'kov, “Local and global continuous $\varepsilon$-selection”, Izv. Math., 80:2 (2016), 442–461  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. I. G. Tsar'kov, “Continuous selection for set-valued mappings”, Izv. Math., 81:3 (2017), 645–669  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    8. I. G. Tsar'kov, “Continuous $\varepsilon$-Selection and Monotone Path-Connected Sets”, Math. Notes, 101:6 (2017), 1040–1049  mathnet  crossref  crossref  mathscinet  isi  elib
    9. Tsar'kov I.G., “Continuous Selection From the Sets of Best and Near-Best Approximation”, Dokl. Math., 96:1 (2017), 362–364  crossref  mathscinet  zmath  isi  scopus
    10. I. G. Tsar'kov, “Continuous selections for metric projection operators and for their generalizations”, Izv. Math., 82:4 (2018), 837–859  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    11. I. G. Tsar'kov, “Continuous selections in asymmetric spaces”, Sb. Math., 209:4 (2018), 560–579  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    12. I. G. Tsar'kov, “New Criteria for the Existence of a Continuous $\varepsilon$-Selection”, Math. Notes, 104:5 (2018), 727–734  mathnet  crossref  crossref  mathscinet  isi  elib
    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  mathnet  crossref  crossref  mathscinet  isi  elib
    14. I. G. Tsar'kov, “Local Approximation Properties of Sets and Continuous Selections on Them”, Math. Notes, 106:6 (2019), 995–1008  mathnet  crossref  crossref  mathscinet  isi  elib
    15. I. G. Tsar'kov, “Weakly monotone sets and continuous selection in asymmetric spaces”, Sb. Math., 210:9 (2019), 1326–1347  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    16. I. G. Tsar'kov, “Approximative properties of sets and continuous selections”, Sb. Math., 211:8 (2020), 1190–1211  mathnet  crossref  crossref  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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