Let τ = (a = x 0 < x 1 < · · · < x n = b) be a partition of an interval [a, b] of R, and let f be a piecewise function of class C k on [a, b] except at knots x i where it is only of class C k i , k i k. We study in this paper a novel method which smooth the function f at x i , 0 i n. We first define a new basis of the space of polynomials of degree 2k + 1, and we describe algorithms for smoothing the function f . Then, as an application, we give a recursive computation of classical Hermite spline interpolants, and we present a method which allows us to compress Hermite data. The most part of these results are illustrated by some numerical examples.
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