Uniform convergence of optimal order quadrature rules for Cauchy principal value integrals

AbstractFor the numerical evaluation of Cauchy principal value integrals of the form , λε(−1, 1), f εCs[− 1, 1], we consider a quadrature method based on spline interpolation of odd degree 2k + 1,k ∈N0. We show that these rules converge uniformly for λ ∈ (− 1, 1). In particular, we calculate the exact order of magnitude of the error and show that it is equal to the order of the optimal remainder in the class of functions with bounded sth derivative if s ε s;;2k + 1, 2k + 2};. Finally, we compare the rule to the well-known quadrature rule of Elliott and Paget which only converges pointwise