## Abstract

For any interpolatory ternary subdivision scheme with two-ring stencils for a regular triangular or quadrilateral mesh, in this paper we show that the critical H\"older smoothness exponent of its basis function cannot exceed $\log_3 11 (\approx 2.18266)$, where the critical H\"older smoothness exponent of a function $f : \RR^2\mapsto \RR$ is defined to be $$\nu_\infty(f):=\sup\{ \nu\; : \; f\in \hbox{Lip}\,\nu\}.$$ On the other hand, for both regular triangular and quadrilateral meshes, in this paper we present several examples of interpolatory ternary subdivision schemes with two-ring stencils such that the critical H\"older 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.

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