On the convergence of interpolatory-type quadrature rules for evaluating Cauchy integrals

The aim of this work is to analyse the stability and the convergence for the quadrature rule of interpolatory-type, based on the trigonometric approximation, for the discretization of the Cauchy principal value integrals f_-1¹ f(τ)/(τ-t)dτ. We prove that the quadrature rule has almost optimal stability property behaving in the form O((logN+1)/sin²x), x=cost. Using this result, we show that the rule has an exponential convergence rate when the function f is differentiable enough. When f possesses continuous derivatives up to order p≥0 and the derivative f^(p)(t) satisfies Hölder continuity of order ρ, we can also prove that the rule has the convergence rate of the form O((A+BlogN +N^2v)/ N^p+p), where v is as small as we like, A and B are constants depending only on x.