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Approximation order/smoothness tradeoff in Hermite subdivision schemes

It is well-known to waveleticians that refinable functions exhibit subtle relationships between their approximation order and smoothness properties. We show how one can exploit this phenomenon to construct Hermite subdivision schemes with optimal smoothness but suboptimal approximation order for a given support size of the subdivision mask. The construction method considered here is based on a blend of the theory of subdivision schemes and computational techniques in non-smooth optimization. Our construction method produces schemes which are much smoother than those constructed based on optimizing approximation orders. We discuss also several interesting bivariate Hermite schemes, with appealing symmetry property, and illustrate how they can be applied to build interpolating subdivision surfaces.