Abstract
In this paper we construct a family of ternary interpolatory Hermite subdivision schemes of order 1 with small support and \({\mathscr{H}}\mathcal {C}^{2}\)-smoothness. Indeed, leaving the binary domain, it is possible to derive interpolatory Hermite subdivision schemes with higher regularity than the existing binary examples. The family of schemes we construct is a two-parameter family whose \({\mathscr{H}}\mathcal {C}^{2}\)-smoothness is guaranteed whenever the parameters are chosen from a certain polygonal region. The construction of this new family is inspired by the geometric insight into the ternary interpolatory scalar three-point subdivision scheme by Hassan and Dodgson. The smoothness of our new family of Hermite schemes is proven by means of joint spectral radius techniques.
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Acknowledgements
The authors thank ‘Vienna Scientific Cluster’ (VSC) for providing computational resources. The authors thank the referees for their carefully reading of the paper and for the constructive comments and suggestions.
Funding
Open access funding provided by Università degli Studi di Firenze within the CRUI-CARE Agreement. Costanza Conti received partial support from INdAM-GNCS. Maria Charina and Thomas Mejstrik were sponsored by the Austrian Science Foundation (FWF) grant P33352-N.
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Communicated by: Larry L. Schumaker
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Appendix
Appendix
For the interested reader, we provide the details of our MATLAB routines via MATLAB exchange, see t-toolboxes [30]. For completeness, we provide here the transition matrices \(\left \{T_{\varepsilon }^{{\boldsymbol {A}}}|_{V_{2}^{{\boldsymbol {A}}}} : \varepsilon \in \{0,1,2\}\right \}\)
Similarly we obtain \(\left \{T_{\varepsilon }^{{\boldsymbol {B}}}|_{V_{0}^{{\boldsymbol {B}}}} : \varepsilon \in \{0,1,2\}\right \}\) given by (to be scaled by 1/243)
The mask B derived using the Taylor operator approach is given by (to be scaled by 1/243)
Last, we provide the computation of the entries (see (??)) of \(\tilde {\boldsymbol {B}}^{*}_{+}(z)= \left (\begin {array}{rrr}\tilde b_{00}(z)& \tilde b_{01}(z)& \tilde b_{02}(z)\\ \tilde b_{10}(z)& \tilde b_{11}(z)& \tilde b_{12}(z)\\ \tilde b_{20}(z)& \tilde b_{21}(z)& \tilde b_{22}(z)\\ \end {array} \right ):\)
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Charina, M., Conti, C., Mejstrik, T. et al. Joint spectral radius and ternary hermite subdivision. Adv Comput Math 47, 25 (2021). https://doi.org/10.1007/s10444-021-09854-x
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DOI: https://doi.org/10.1007/s10444-021-09854-x