Snakes as an apparatus for approximating

E. A. Sevast'yanov ; E. Kh. Sadekova

Sbornik. Mathematics, Tome 199 (2008) no. 1, pp. 99-130

Résumé

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. Bibliography: 19 titles.