On optimal error bounds for derivatives of interpolating splines on a uniform partition

Based on Peano kernel technique, explicit error bounds (optimal For the hig hest order derivative) are proved for the derivatives of cardinal spline in terpolation (interpolating at the knots for odd degree splines and at the m idpoints between two knots for even degree splines). The results are based on a new representation of the Peano kernels and on a thorough investigatio n of their zero distributions. The bounds are given in terms of Euler-Frobe nius polynomials and their zeros. (C) 1999 Academic Press.