Rational interpolation of a function $|x|^{\alpha}$ with Chebyshev – Markov nodes of the first kind

This paper considers the approximations of the function $|x|^{\alpha}, ~\alpha > 0$, by interpolation rational Lagrange functions on the interval $[-1, 1]$. Zeros of the rational Chebyshev – Markov function of the first kind are chosen as interpolation nodes. An integral representation of the interpolation remainder and an upper estimation for the considered uniform approximations are obtained. The polynomial and general rational cases are studied in detail. In the polynomial case, an asymptotic estimate for uniform approximations is found. When approximating by interpolation rational Lagrange functions with Chebyshev – Markov nodes of the first kind, the upper and lower estimations are found. These estimations are close to that of the best uniform approximations of the function under consideration on the interval $[-1, 1]$.