Explicit Error Bounds for Periodic Splines of Odd Order on a Uniform Mesh

Abstract

The dependence relationships connecting equal interval splines and their derivatives are analysed to obtain the form of the error term when the spline is replaced by a general function. The defining equations for periodic splines of odd order on a uniform mesh are then expressed in terms of a positive definite circulant matrix A and attainable bounds determined for the condition number of A and for the norm of A-1. In conjunction with the error term associated with the dependence relationships, this enables explicit error bounds to be established for the derivatives at the knots of the spline function. Some subsidiary results in the paper also relate to B-splines on a uniform mesh.