Smoothness and error bounds of Martensen splines

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