Rational interpolation of the function ${\left| x \right|}^{\alpha}$by the system of Chebyshev–Markov of the second kind

The article discusses the approximation of the function $ {\left| x \right|}^{\alpha}, \ \alpha> 0 $, by rational Lagrange interpolation functions on the segment $ [-1, 1] $. The zeros of the rational Chebyshev – Markov function of the second kind are chosen as interpolation nodes. An integral representation of the interpolation remainder and an upper bound for the considered uniform approximations are obtained. On their basis, various cases of the arrangement of the poles of the approximating rational function are studied in detail: polynomial, a fixed number of geometrically different poles, and general rational.