A recursive construction of Hermite spline interpolants and applications

Let fk be the Hermite spline interpolant of class Ck and degree 2k+1 to a real function f which is defined by its values and derivatives up to order k at some knots of an interval [a,b]. We present a quite simple recursive method for the construction of fk. We show that if at the step k, the values of the kth derivative of f are known, then fk can be obtained as a sum of fk-1 and of a particular spline gk-1 of class Ck-1 and degree 2k+1. Beyond the simplicity of the evaluation of gk-1, we prove that it has other interesting properties. We also give some applications of this method in numerical approximation.