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

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

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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• http://mi.mathnet.ru/eng/izv221
• https://doi.org/10.4213/im221
• http://mi.mathnet.ru/eng/izv/v62/i6/p59

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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. 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
2. P. A. Borodin, “The Banach–Mazur Theorem for Spaces with Asymmetric Norm”, Math. Notes, 69:3 (2001), 298–305
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
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
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
6. A. V. Pokrovskii, “The best asymmetric approximation in spaces of continuous functions”, Izv. Math., 70:4 (2006), 809–839
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
8. A.-R. K. Ramazanov, “Otsenka polinomialnykh priblizhenii ogranichennykh funktsii s vesom”, Dagestanskie elektronnye matematicheskie izvestiya, 2014, no. 2, 38–44
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