Abstract
Rational approximations of the conjugate function on the segment \([ - 1,\,\,1]\) by Abel–Poisson sums of conjugate rational integral Fourier–Chebyshev operators with restrctions on the number of geometrically different poles are investigated. An integral representation of the corresponding approximations is established. Rational approximations on the segment \([ - 1,\,\,1]\) of the conjugate function with density \({{(1 - x)}^{\gamma }},\) \(\gamma \in ({\text{1/2}}, 1),\) by Abel–Poisson sums are studied. An integral representation of approximations and estimates of approximations taking into account the position of a point on the segment \([ - 1,\,\,1]\) are obtained. An asymptotic expression as \(r \to 1\) for the majorant of approximations, depending on the parameters of the approximating function, is established. In the final part, the optimal values of parameters which provide the highest rate of decrease of this majorant are found. As a corollary, we give some asymptotic estimates of approximations on the segment \([ - 1,\,\,1]\) of the conjugate function by Abel–Poisson sums of conjugate polynomial Fourier–Chebyshev series.
