Smoothness and error bounds of Martensen splines

Martensen splines MfMf of degree nn interpolate ff and its derivatives up to the order n−1n−1 at a subset of the knots of the spline space, have local support and exactly reproduce both polynomials and splines of degree ≤n≤n. An approximation error estimate has been provided for f∈Cn+1f∈Cn+1.This paper aims to clarify how well the Martensen splines MfMf approximate smooth functions on compact intervals. Assuming that f∈Cn−1f∈Cn−1, approximation error estimates are provided for Djf,j=0,1,…,n−1Djf,j=0,1,…,n−1, where DjDj is the jth derivative operator. Moreover, a set of sufficient conditions on the sequence of meshes are derived for the uniform convergence of DjMfDjMf to DjfDjf, for j=0,1,…,n−1j=0,1,…,n−1.