The loss of Holder regularity of four-point interpolatory subdivision on irregularly spaced points

Daubechies, Guskov, and Sweldens studied four-point, cubic-based interpolatory subdivision on irregularly spaced grid points and showed that if a ‘dyadic’ mesh-ratio, λ, where 1/2 ≤ λ ≤ 1, satisfies the bound λ ≤ 2/3, the limit function has Holder regularity C2− for any small > 0. They also conjectured that C2− regularity is maintained for all λ < 1. We show, on the contrary, that for certain grids, with λ > λ1, where λ1 ≈ 0.8847, regularity is lost. In fact, the regularity of the scheme can approach C1 as λ approaches 1. Math Subject Classification: 65D05, 65D10