Overconvergence properties of quintic interpolatory splines

摘要

Let Q be a quintic spline with equi-spaced knots on [a, b] interpolating a given function y at the knots. The parameters which determine Q are used to construct a piecewise defined polynomial P of degree six. It is shown that P can be used to give at any point of [a, b] better orders of approximation to y and its derivatives than those obtained from Q. It is also shown that the superconvergence properties of the derivatives of Q, at specific points of [a, b], are all simple consequences of the properties of P.