Building blocks for designing arbitrarily smooth subdivision schemes with conic precision

• Non-stationary extension of Lane–Riesenfeld algorithm.
• New family of alternating primal/dual subdivision schemes reproducing conics.
• New family of non-stationary interpolatory 2n2n-point schemes reproducing conics.
• Explicit formulation and recurrence relations.
• Analysis of the main properties of the above families.

Since subdivision schemes featured by high smoothness and conic precision are strongly required in many application contexts, in this work we define the building blocks to obtain new families of non-stationary subdivision schemes enjoying such properties. To this purpose, we firstly derive a non-stationary extension of the Lane–Riesenfeld algorithm, and we exploit the resulting class of schemes to design a non-stationary family of alternating primal/dual subdivision schemes, all featured by reproduction of {1,x,etx,e−tx},t∈[0,π)∪iR+. Then, we focus our attention on interpolatory subdivision schemes with conic precision, that can be obtained as a byproduct of the above classes. In particular, we present a novel construction of a family of non-stationary interpolatory 2n2n-point schemes which generalizes the well-known Dubuc–Deslauriers family in such a way the nnth (n≥2n≥2) family member reproduces Π2n−3∪{etx,e−tx},t∈[0,π)∪iR+, and keeps the original smoothness of its stationary counterpart unchanged.