On rational approximations of the Markov function on the segment by the Fejer sums with a fixed number of poles

P. G. Patseika ; Y. A. Rovba

Trudy Instituta matematiki, Tome 30 (2022) no. 1, pp. 63-83

Résumé

Approximations of Markov functions on the segment $[-1,~1]$ by Fejer sums of the rational integral Fourier operator–Chebyshev with restrictions on the number of geometrically different poles are investigated. An integral representation of approximations and an estimate of uniform approximations are obtained. In the case when the measure $\mu$ satisfies the following conditions $\mathrm{supp} \mu = [1,a], a>1,$ $ d\mu(t)= \varphi(t) dt $ and $ \varphi(t)\asymp(t-1)^\alpha $ on $ [1,a], $ estimates of pointwise and uniform approximations are established, the asymptotic expression for $n\to \infty$ majorants of uniform approximations. The optimal values of the parameters providing the highest rate of decrease of this majorant are found. As a consequence, estimates of the corresponding uniform approximations of some elementary functions are established. It follows from the results obtained that rational approximations by Fejer sums of the Markov function with measures $\mu(t)$ with "low smoothness"  are better in terms of order than the corresponding polynomial ones.