Martensen splines and finite-part integrals

Abstract

We state a uniform convergence theorem for finite-part integrals which are derivatives of weighted Cauchy principal value integrals. We prove that a sequence of Martensen splines, based on locally uniform meshes, satisfies the sufficient conditions required by the theorem. We construct the quadrature rules based on such splines and illustrate their behaviour by presenting some numerical results and comparisons with composite midpoint, Simpson and Newton-Cotes rules.

Smoothness and error bounds of Martensen splines

“…The obtained uniform convergence results (42) and (43), together with interpolation conditions 1. of Theorem 4 at t 0 and t Rn , allow the use of Martensen splines M R f for the numerical evaluation of certain finite-part integrals [7,8,12].…”