Order-Preserving Approximations to Successive Derivatives of Periodic Functions by Iterated Splines

A method is presented for the iterative use of periodic splines of order $2r$ to accurately approximate successive derivatives of a periodic function on a uniform mesh. Asymptotic expansions are derived for the error in this method of approximating the mth derivative at the meshpoints. It is found that the leading order term in this error expansion is O(h²ʳ ), independently of m. A numerical example is given, showing excellent agreement with theoretical predictions.