Simultaneous Spline Approximation and Interpolation Preserving Norms

Abstract

Abstract.In this paper, it is proved that splines of order k (k > 2) have property SAIN. The proof of this result is based on the important properties of B-splines. Introduction.In a recent manuscript [5], Lambert proved that the twice continuously differentiable cubic splines possess property SAIN (simultaneous approximation and interpolation which is norm preserving) on C [a, b] where the interpolatory constraints are point evaluations. In this paper we establish the more general result for splines of any order greater than 1 while at the same time supplying a simple proof. More precisely, we will show