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 ⨍−11f(τ)/(τ−t) dτ. We prove that the quadrature rule has almost optimal stability property behaving in the form O((log N+1)/sin2 x), x=cos t. 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+B log N+N2ν)/Np+p), where ν is as small as we like, A and B are constants depending only on x.