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Izv. RAN. Ser. Mat., 1998, Volume 62, Issue 6, Pages 59–102 (Mi izv221)  

This article is cited in 7 scientific papers (total in 8 papers)

Approximations with a sign-sensitive weight: existence and uniqueness theorems

E. P. Dolzhenko, E. A. Sevast'yanova

a Moscow Institute of Municipal Economy and Construction

Abstract: Approximations with a sign-sensitive weight generally take into account both the absolute value of the error of approximation and its sign. We study the problems of existence, uniqueness and plurality for the element of best uniform approximation with a given sign-sensitive weight $p=(p_-,p_+)$ by functions of a given family $L$ on an interval $\Delta$. We also study these problems for approximations in normed linear spaces $\mathcal L$ by elements of a family $L\subset\mathcal L$, where the deviation of an element $x$ from another element $y$ is measured by the value $P(x-y)$ of some non-negative sublinear functional $P$. A very important role is played by the rigidity and freedom of the systems $(p,L)$ and $(P;L)$. These notions are also studied in the paper, with special attention being given to the case of Chebyshev subspaces $L$.

DOI: https://doi.org/10.4213/im221

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English version:
Izvestiya: Mathematics, 1998, 62:6, 1127–1168

Bibliographic databases:

MSC: 41A65, 41A50, 41A52
Received: 27.05.1997

Citation: E. P. Dolzhenko, E. A. Sevast'yanov, “Approximations with a sign-sensitive weight: existence and uniqueness theorems”, Izv. RAN. Ser. Mat., 62:6 (1998), 59–102; Izv. Math., 62:6 (1998), 1127–1168

Citation in format AMSBIB
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\paper Approximations with a~sign-sensitive weight: existence and uniqueness theorems
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. E. P. Dolzhenko, E. A. Sevast'yanov, “Approximations with a sign-sensitive weight. Stability, applications to the theory of snakes and Hausdorff approximations”, Izv. Math., 63:3 (1999), 495–534  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. P. A. Borodin, “The Banach–Mazur Theorem for Spaces with Asymmetric Norm”, Math. Notes, 69:3 (2001), 298–305  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. A. I. Kozko, “On the order of the best approximation in spaces with asymmetric norm and sign-sensitive weight on classes of differentiable functions”, Izv. Math., 66:1 (2002), 103–131  mathnet  crossref  crossref  mathscinet  zmath
    4. A. A. Chumak, “Construction of the polynomial of least deviation for approximations with a sign-sensitive weight”, Comput. Math. Math. Phys., 42:2 (2002), 135–147  mathnet  mathscinet  zmath
    5. A. K. Ramazanov, “Characterization of the best polynomial approximation with a sign-sensitive weight to a continuous function”, Sb. Math., 196:3 (2005), 395–422  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. A. V. Pokrovskii, “The best asymmetric approximation in spaces of continuous functions”, Izv. Math., 70:4 (2006), 809–839  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. A. I. Aptekarev, P. A. Borodin, B. S. Kashin, Yu. V. Nesterenko, P. V. Paramonov, A. V. Pokrovskii, A. G. Sergeev, A. T. Fomenko, “Evgenii Prokof'evich Dolzhenko (on his 80th birthday)”, Russian Math. Surveys, 69:6 (2014), 1143–1148  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    8. A.-R. K. Ramazanov, “Otsenka polinomialnykh priblizhenii ogranichennykh funktsii s vesom”, Dagestanskie elektronnye matematicheskie izvestiya, 2014, no. 2, 38–44  mathnet  crossref  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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