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

This article is cited in 1 scientific paper (total in 1 paper)

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
Received: 16.01.2007 and 06.09.2007

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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\by E.~A.~Sevast'yanov, E.~Kh.~Sadekova
\paper Snakes as an apparatus for approximating
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\pages 101--132
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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. Sevastyanov E.A., Sadekova E.Kh., “Asimptoticheskie svoistva uzhei [Asymptotic properties of “snakes”]”, Anal. Math., 34:4 (2008), 277–305  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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