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Abstract: Optimal C2 two-dimensional interpolatory ternary
subdivision schemes with two-ring stencils

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Bin Han and Rong-Qing Jia

## 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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