Local expansions of periodic spline interpolation with some applications

In this paper we prove some asymptotic expansions of the error of interpola tion on equally spaced nodes with periodic smoothest splines of arbitrary d egree on a uniform partition. We obtain a local expansion in terms of deriv atives of the interpolate. Afterwards we apply this result to the asymptoti c study of the numerical solution of periodic integral equations of the sec ond kind by means of epsilon -collocation methods. We show some new superco nvergence results and give particular Forms of these expansions depending o n the choices of the parameter epsilon. We finally give some numerical expe riments, which corroborate the theory.