Quadratic spline interpolation on uniform meshes

Abstract

Abstract.The interpolation problem at uniform mesh points of a quadratic spline s(x~) = f~, i = 0, 1 ..... N and s'(xo) = f~ is considered. It is known that IIs -f II ~ = O(h 3) and ]1 s' -f' rr ~ = O(h2), where h is the step size, and that these orders cannot be improved. Contrary to recently published results we prove that superconvergence cannot occur for any particular point independent of f other than mesh points where s = f by assumption. Best error bounds for some compound formulae approximating f~ and f~3~ are also derived.