On the Riesz summation of rational Fourier-Chebyshev integral operators and approximations of functions with a power singularity

In the present paper Riesz sums of Fourier-Chebyshev rational integral operators with restrictions on the number of geometrically distinct poles are introduced. Approximation of the function \((1-x)^\gamma\), \(\gamma \in (0,1)\), by this method is considered. Estimates of pointwise and uniform approximation are established, as well as asymptotic expressions for the uniform approximation majorant. Additionally, the optimal values of the parameters of the approximating function, at which the rate of decrease of the majorant is the greatest are found. In the case of Riesz sums of a polynomial Fourier-Chebyshev series, approximation of functions satisfying the Lipschitz condition of order \(\gamma\) on the segment \([-1,1]\) is investigated.