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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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


Authors: Bin Han and Rong-Qing Jia
Journal: Math. Comp. 75 (2006), 1287-1308
MSC (2000): Primary 42C20, 41A05, 41A63, 65D05, 65D17
Published electronically: May 3, 2006
MathSciNet review: 2219029
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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

$\displaystyle \nu_\infty(f):=\sup\{ \nu\; : \; f\in \hbox{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.


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Additional Information

Bin Han
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: bhan@math.ualberta.ca

Rong-Qing Jia
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: rjia@ualberta.ca

DOI: http://dx.doi.org/10.1090/S0025-5718-06-01821-7
PII: S 0025-5718(06)01821-7
Keywords: Ternary subdivision schemes, interpolatory subdivision schemes, H\"older smoothness, projection method, joint spectral radius
Received by editor(s): March 18, 2004
Received by editor(s) in revised form: January 21, 2005
Published electronically: May 3, 2006
Additional Notes: Research supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under Grant RGPIN 228051 and Grant OGP 121336
Article copyright: © Copyright 2006 American Mathematical Society