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 Mat. Sb., 2008, Volume 199, Number 1, Pages 101–132 (Mi msb3831)

Snakes as an apparatus for approximating functions in the Hausdorff metric

E. A. Sevast'yanov, E. Kh. Sadekova

Moscow Engineering Physics Institute (State University)

Abstract: The Bulgarian mathematicians Sendov, Popov, and Boyanov have well-known results on the asymptotic behaviour of the least deviations of $2\pi$-periodic functions in the classes $H^\omega$ from trigonometric polynomials in the Hausdorff metric. However, the asymptotics they give are not adequate to detect a difference in, for example, the rate of approximation of functions $f$ whose moduli of continuity $\omega(f;\delta)$ differ by factors of the form $(\log(1/\delta))^\beta$. Furthermore, a more detailed determination of the asymptotic behaviour by traditional methods becomes very difficult. This paper develops an approach based on using trigonometric snakes as approximating polynomials. The snakes of order $n$ inscribed in the Minkowski $\delta$-neighbourhood of the graph of the approximated function $f$ provide, in a number of cases, the best approximation for $f$ (for the appropriate choice of $\delta$). The choice of $\delta$ depends on $n$ and $f$ and is based on constructing polynomial kernels adjusted to the Hausdorff metric and polynomials with special oscillatory properties.
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DOI: https://doi.org/10.4213/sm3831

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English version:
Sbornik: Mathematics, 2008, 199:1, 99–130

Bibliographic databases:

UDC: 517.518.83+517.518.845+517.518.863
MSC: Primary 42A10; Secondary 41A50, 41A60, 42A05

Citation: E. A. Sevast'yanov, E. Kh. Sadekova, “Snakes as an apparatus for approximating functions in the Hausdorff metric”, Mat. Sb., 199:1 (2008), 101–132; Sb. Math., 199:1 (2008), 99–130

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb3831
• https://doi.org/10.4213/sm3831
• http://mi.mathnet.ru/eng/msb/v199/i1/p101

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This publication is cited in the following articles:
1. Sevastyanov E.A., Sadekova E.Kh., “Asimptoticheskie svoistva uzhei [Asymptotic properties of “snakes”]”, Anal. Math., 34:4 (2008), 277–305
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