On rational approximations of a singular integral on a segment by Abel — Poisson sums

Rational approximations on the segment $[-1,~1]$ are studied of singular integrals of the form
$$ \hat{f}(x) = \int\limits_{-1}^{+1} \frac{f(t)}{t-x}\sqrt{1-t^2} dt, \ x \in [-1,~1]. $$
The approximation apparatus is the Abel – Poisson sums of rational integral Fourier operators – Chebyshev associated with a system of rational Chebyshev – Markov functions, with an arbitrary fixed number of geometrically different poles. An integral representation of the approximations is established. In the case when the density of the singular integral has a power-law singularity, estimates are found pointwise approximations, uniform approximations with a certain majorant, its asymptotic expression and optimal values of parameters at which the majorant has the highest rate of decrease.
A consequence of the results obtained are estimates of approximations of singular integrals with density having a power-law singularity by Abel – Poisson sums of the polynomial Fourier – Chebyshev series.
Estimates of approximations of singular integrals with a density satisfying the Lipschitz condition on the segment $[-1,~1]$ by Abel – Poisson sums of the polynomial Fourier series – Chebyshev are established. The peculiarity of the estimates found is their dependence on the position of the point on the segment. Moreover, at the ends of the segment, the speed is higher than in the whole segment.
It is established that the classes of studied singular integrals with a density having a power-law singularity in some cases reflect the features of rational approximation in the sense that with a special choice of the velocity parameters of uniform rational approximations they turn out to be higher than the corresponding polynomial analogues.