New error bounds for modified quadrature formulas for Cauchy principal value integrals

For the numerical approximation of Cauchy principal value integrals, we consider the so-called modified quadrature formulas, i.e. formulas obtained by first subtracting out the singularity and then applying a classical quadrature formula. We are interested in error bounds holding uniformly for all possible positions of the singular point. The standard error bounds are based on suprema of derivatives, but they often overestimate the true errors by a factor that grows with the number of nodes of the quadrature formula. We give new bounds involving the total variation Var -(s) and LP-normst|-(s)t|p of some derivative of the integrand function. These bounds give additional possibilities for sharper estimations of the error.