On approximations of a singular integral on a segment by Fourier–Chebyshevʼs rational integral operators

Rational approximations on a segment –1, 1 of singular integrals of the form , by integral operators, in a sense related to each other are studied. The first of them is Fourier–Chebyshev's rational integral operator associated with the system of Chebyshev–Markov's rational functions. It is a natural generalization of partial sums of Fourier–Chebyshev's polynomial series. The second operator is the image of the first one when transformed by a singular integral under study. An integral representation of approximations is established for each of the operators. Approximations on the segment –1, 1 of a singular integral with a density having a power-law singularity are studied. For each of the operators, we consider the case of an arbitrary fixed number of geometrically different poles and the case when the poles represent some modifications of the "Newman" parameters. It is established that the classes of the studied singular integrals reflect the rational approximation features by the considered integral operators in the sense that with a special choice of parameters of approximating functions, the orders of their approximations turn out to be higher than the corresponding polynomial analogues.