A discrete systems approach to cardinal spline Hermite interpolation

Abstract

A cardinal spline Hermite interpolation problem is posed by specifying values, and m − 1 derivatives, m ⩾ 1, at uniformly spaced knots t k ; it may be solved by means of a generalized spline function w( t) (a standard spline function when m = 1), piecewise a polynomial of degree n − 1 = 2 m + p − 1, p ⩾ 0, with w ( j) ( t) continuous across the knots for j = 0, 1, 2, … , m + p − 1. The problem is studied here for p > 0 in the context of an ( m + p)-dimensional system of linear recursion equations satisfied by the values of the m-th through m + p − 1-st derivatives of w( t) at the knots, whose homogeneous term involves a p × p matrix A . In the case m = 1 we relate the characteristic polynomial of A and certain controllability notions to the standard B-spline and we proceed to show how systems-theoretic ideas can be used to generate systems of basis splines for higher values of m.