de Rham Transform of a Hermite Subdivision Scheme

For a Hermite subdivision scheme H of degree d, we define a spectral condition. We have already proved that if it is fulfilled, then there exists an associated affine subdivision scheme S. Moreover, if T is the subdivision matrix of the first difference process ∆S, if the spectral radius ρ(T ) is less than 1, then S is C0 and H is Cd. Generalizing the de Rham corner cutting, from every Hermite subdivision scheme H, we build a new Hermite subdivision scheme, the de Rham transform H̃. If H satisfies the spectral condition, then its de Rham transform fulfils it. If S̃ is the associated subdivision scheme to H̃ and if T and T̃ are the respective subdivision matrices corresponding to ∆S and ∆S̃, then the spectral radii ρ(T ) and ρ(T̃ ) allow the comparison between the indices of smoothness of the limit functions of the schemes H and H̃. We apply these results to the Merrien class of Hermite subdivision schemes of degree 1.