A degree by degree recursive construction of Hermite spline interpolants

Based on the classical Hermite spline interpolant H2n−1H2n−1, which is the piecewise interpolation polynomial of class Cn−1Cn−1 and degree 2n−12n−1, a piecewise interpolation polynomial H2nH2n of degree 2n2n is given. The formulas for computing H2nH2n by H2n−1H2n−1 and computing H2n+1H2n+1 by H2nH2n are shown. Thus a simple recursive method for the construction of the piecewise interpolation polynomial set {Hj}{Hj} is presented. The piecewise interpolation polynomial H2nH2n satisfies the same interpolation conditions as the interpolant H2n−1H2n−1, and is an optimal approximation of the interpolant H2n+1H2n+1. Some interesting properties are also proved.