A degree by degree recursive construction of Hermite spline interpolants

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