Shape Constraints and optimal bases for C^1 Hermite interpolatory subdivision schemes

We address the problem of constructing C^1 subdivision interpolating curves preserving salient shape properties of the data. We use the Hermite subdivision scheme introduced by J.-L. Merrien in [19], which depends on two parameters to be chosen in an appropriate region of convergence. With shape preservation in view, we try and nd normalised totally positive bases in the associated four-dimensional limit spaces. We actually exhibit two sub-regions: when choosing the pair of parameters in the rst one, no totally positive basis exists in the limit space; on the opposite, when choosing it in the second one, the limit space does possess normalised totally positive bases, and we explicitly describe the optimal one. As special cases, we recover the bases presented in [23], [12] and [22], of which we thus obtain the optimality.