A recursive construction of Hermite spline interpolants and applications

Let f k be the Hermite spline interpolant of class C k and degree 2 k + 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 f k . We show that if at the step k, the values of the k th derivative of f are known, then f k can be obtained as a sum of f k - 1 and of a particular spline g k - 1 of class C k - 1 and degree 2 k + 1 . Beyond the simplicity of the evaluation of g k - 1 , we prove that it has other interesting properties. We also give some applications of this method in numerical approximation.