Optimal $C^2$ two-dimensional interpolatory ternary subdivision schemes with two-ring stencils

Abstract:

For any interpolatory ternary subdivision scheme with two-ring stencils for a regular triangular or quadrilateral mesh, we show that the critical Hölder smoothness exponent of its basis function cannot exceed $\log _3 11 (\approx 2.18266)$, where the critical Hölder smoothness exponent of a function $f : \mathbb {R}^2\mapsto \mathbb {R}$ is defined to be \[ \nu _\infty (f) \coloneq \sup \{ \nu : f\in \operatorname {Lip} \nu \}. \] On the other hand, for both regular triangular and quadrilateral meshes, we present several examples of interpolatory ternary subdivision schemes with two-ring stencils such that the critical Hölder smoothness exponents of their basis functions do achieve the optimal smoothness upper bound $\log _3 11$. Consequently, we obtain optimal smoothest $C^2$ interpolatory ternary subdivision schemes with two-ring stencils for the regular triangular and quadrilateral meshes. Our computation and analysis of optimal multidimensional subdivision schemes are based on the projection method and the $\ell _p$-norm joint spectral radius.