Finite-part integrals and modified splines (Q1770937)

The authors recall a uniform convergence theorem for finite-part integrals, which are derivatives of weighted Cauchy principal value integrals. An example of a sequence of functions satisfying such a theorem is the sequence of complete cubic interpolatory splines. Approximating splines and optimal nodal splines based on locally uniform partitions do not satisfy the boundary conditions of the theorem. Therefore, after an introduction to these two classes of functions, a suitable two-stage process by modifying the splines and their derivatives at the end points of the integration interval is provided in such a way that all hypothesis of the uniform convergence theorem are satisfied. The splines so modified can now be used in the numerical evaluation of finite-part integrals.