Local Expansions of Periodic Spline Interpolation with Some Applications

In this paper we prove some asymptotic expansions of the error of interpolation on equally spaced nodes with periodic smoothest splines of arbitrary degree on a uniform partition. We obtain a local expansion in terms of derivatives of the interpolate. Afterwards we apply this result to the asymptotic study of the numerical solution of periodic integral equations of the second kind by means of ? – collocation methods. We show some new superconvergence results and give particular forms of these expansions depending on the choices of the parameter ?. We finally give some numerical experiments, which corroborate the theory.