N.: Improved orders of approximation derived from interpolatory cubic splines

Let s be a cubic spline, with equally spaced knots on [a,b], interpolating a given function y at the knots. The parameters which determine s are used to construct a piecewise defined polynomial P of degree four. It is shown that P can be used to give better orders of approximation to y and its derivatives than those obtained from s. It is also shown that the known superconvergence properties of the derivatives of s, at specific points [a,b], are all special cases of the mai