Asymptotic behaviour of fixed-order error constants of modified quadratureformulae for Cauchy principal value integrals

We consider quadrature formulae for Cauchy principal value integrals I-w,I-xi[f] = integral(a)(b) f(x)/x-xi w(x) dx, a < xi < b. The quadrature formulae considered here are so-called modified formulae, wh ich are obtained by first subtracting the singularity, and then applying so me standard quadrature formula Q(n). The aim of this paper is to determine the asymptotic behaviour of the constants kappa(i,n) in error estimates of the form \R-n(mod) [f; xi]\ < kappa(i,n) (xi) \\f((i)) \\(infinity) for fix ed i and n --> infinity, where R-n(mod) [f; xi] is the quadrature error. Th is is done for quadrature formulae Q(n) for which the Peano kernels K-i,K-n of fixed order i behave in a certain regular way, including, e.g., many in terpolatory quadrature formulae as Gauss-Legendre and Clenshaw-Curtis formu lae, as well as compound quadrature formulae. It turns out that essentially all the interpolatory formulae behave in a very similar way.