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Interpolating refinable functions and \(n_s\)-step interpolatory subdivision schemes

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Abstract

Standard interpolatory subdivision schemes and their underlying interpolating refinable functions are of interest in CAGD, numerical PDEs, and approximation theory. Generalizing these notions, we introduce and study \(n_s\)-step interpolatory \(\textsf{M}\)-subdivision schemes and their interpolating \(\textsf{M}\)-refinable functions with \(n_s\in \mathbb {N}\cup \{\infty \}\) and a dilation factor \(\textsf{M}\in \mathbb {N}\backslash \{1\}\). We completely characterize \(\mathscr {C}^m\)-convergence and smoothness of \(n_s\)-step interpolatory subdivision schemes and their interpolating \(\textsf{M}\)-refinable functions in terms of their masks. Inspired by \(n_s\)-step interpolatory stationary subdivision schemes, we further introduce the notion of r-mask quasi-stationary subdivision schemes, and then we characterize their \(\mathscr {C}^m\)-convergence and smoothness properties using only their masks. Moreover, combining \(n_s\)-step interpolatory subdivision schemes with r-mask quasi-stationary subdivision schemes, we can obtain \(r n_s\)-step interpolatory subdivision schemes. Examples and construction procedures of convergent \(n_s\)-step interpolatory \(\textsf{M}\)-subdivision schemes are provided to illustrate our results with dilation factors \(\textsf{M}=2,3,4\). In addition, for the dyadic dilation \(\textsf{M}=2\) and \(r=2,3\), using r masks with only two-ring stencils, we provide examples of \(\mathscr {C}^r\)-convergent r-step interpolatory r-mask quasi-stationary dyadic subdivision schemes.

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Acknowledgements

Research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) under grant RGPIN 2024-04991.

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Correspondence to Bin Han.

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Communicated by: Tomas Sauer

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Han, B. Interpolating refinable functions and \(n_s\)-step interpolatory subdivision schemes. Adv Comput Math 50, 98 (2024). https://doi.org/10.1007/s10444-024-10192-x

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