Title: On consistency relations for cubic splines-on-splines and asymptotic error estimates
Abstract: In the present paper we consider the spline-on-spline technique for calculating the derivative of a function from its values on a uniform mesh. There is computational evidence that this yields better results than the traditional process using a single spline [ 11. Dolezal and Tewarson [2] have recently obtained error bounds for spline-on-spline interpolation. The aim of this paper is to derive new consistency relations bletween a cubic spline and a cubic spline-on-spline interpolant of its first derivative, and to furnish asymptotic error estimates for the interpolation. For any integer n3 1, let A,,: O=.Y,,<.Y, < ..’ < .Y,, = 1 denote a uniform partition of I = [0, 1 ] with knots x, = ih. Given a sufficiently smooth functionf(x) defined on I, let s be an interpolatory cubic spline of f and p a cubic spline-on-spline interpolant of s’ defined by