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

This article is cited in 13 scientific papers (total in 13 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 $d>1$.

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
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
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\pages 99--130
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  • 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  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  adsnasa  isi  elib
    11. I. G. Tsar'kov, “Continuous selections in asymmetric spaces”, Sb. Math., 209:4 (2018), 560–579  mathnet  crossref  crossref  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  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  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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