Parametric Families of Hermite Subdivision Schemes in Dimension 1

We introduce a class of stationary 1-D interpolating subdivision schemes, denoted by Hermite(m, L, k), which classifies all stationary Lagrange or Hermite interpolating subdivision schemes with prescribed multiplicity, support and polynomial reproduction property: Given m > 0, L > 0 and 0 ≤ k ≤ 2mL − 1, Hermite(m, L, k) is a family parametrized by m2L− m(k +1)/2 (in the symmetric case) or 2m2L−m(k +1) (in the asymmetric case) parameters which characterizes all Lagrange (when m = 1) or Hermite (when m > 1) interpolating subdivision schemes with multiplicity m, support width L and degree of polynomial reproduction k. We develop analytical and computational tools to determine upper and lower bounds for the critical Hölder exponents of Hermite schemes and report the so obtained bounds for various (m, L, k). Motivated by these computational results, we conjecture that a well-known property of Deslauriers-Dubuc schemes, which implies that the critical Hölder exponents of Deslauriers-Dubuc schemes are determined by the spectral radii of finite matrices, has a nature extension to all Hermite(m, L, 2mL − 1). We prove this conjecture in a few specific cases and, in particular, show that the Hermite(2, 2, 7) scheme is almost C3. We introduce the notion of smoothest scheme among these parametric families; and use our tools for regularity analysis in conjunction with optimization techniques to show that the smoothest scheme in S k<7 Hermite(2, 2, k) is at least C4, i.e. two differentiation orders higher than Hermite(2, 2, 7).