Dolomites Research Notes on Approximation

Hermite subdivision schemes act on vector valued sequences that are not only considered as functions values of a vector valued function from R to Rr , but as evaluations of a function and its consecutive derivatives. Starting with data on `r (Z), r = d+1, interpreted as function value and d = r-1 consecutive derivatives, we compute successive iterations to define values on `r (2-nZ) and an r-vector valued limit function for whose first component Cd–smoothness is generally expected. In this paper, we construct univariate Hermite subdivision schemes such that, for any given initial data, it is possible to reach a limit function with smoothness d + p for any p > 0. The result is obtained with a generalized Taylor factorization and a smoothness condition for vector subdivision schemes.