The method of mechanical quadratures for integral equations with fixed singularity

We investigate the method of mechanical quadratures for integral equations with fixed singularity. We establish estimates of the error of this method based on a quadrature process, which is the best in the class of differentiable functions. We prove the convergence of the method in finite-dimensional and uniform metrics. We find that the investigated quadrature method is optimal by order on the Hölder class of functions.