On semi-cardinal interpolation with cubic splines

We prove the existence and uniqueness of a cubic spline that interpolates data of power growth at the set of all non-negative integer knots, subject to one end condition. Our approach models a hierarchy of automatic end conditions in terms of finite differences for B-spline coefficients. We establish the order of convergence of the associated Lagrange schemes for semi-cardinal interpolation, the maximal order 4 being obtained for the not-a-knot condition.