Convergence of a Class of Interpolatory Splines

Abstract

The object of the present paper is to investigate the convergence properties of periodic cubic splines which interpolate a given function at one or more inner points of the given mesh intervals. If the number of interpolatory conditions in each mesh interval is more than one (as in our Theorem 2), the deficiency of the spline increases. On the other hand the increase of deficiency of the spline (which is equivalent to the decrease of differentiability at the joints) is compensated by the increase in smoothness at the points of interpolation (in fact we have three non-trivial continuous derivatives at the points of interpolation). Here we obtain the error bounds for cubic splines which interpolate at one or two points in each mesh interval. For the sake of simplicity we consider only equidistant points in Theorem 1, but in Theorem 2 this restriction is not needed.