Asymptotic Expansions for Interpolating Periodic Splines

A detailed study is made of the structure of smooth periodic interpolating splines of odd degree over a uniform mesh. For splines of degree $2r - 1$, all terms of the asymptotic expansion of the error at the mesh points of the first $2r - 2$ derivatives are given. This result is used to derive a number of other high order spline approximations.