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 Holder smoothness exponent of its basis function cannot exceed log 3 11(≈ 2.18266), where the critical Holder smoothness exponent of a function f: R 2 R is defined to be v ∞ (f):= sup{v: f ∈ Lipv}. 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 Holder 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 l p -norm joint spectral radius.