Snakes as an apparatus for approximating functions in the Hausdorff metric

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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