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.

标准插值细分方案及其底层插值可细化函数在 CAGD、数值偏微分方程和逼近理论中很有趣。概括这些概念，我们介绍并研究\(n_s\)步插值\(\textsf{M}\)细分方案及其插值\(\textsf{M}\) - 可细化函数，其中\(n_s\in \) mathbb {N}\cup \{\infty \}\)和膨胀因子\(\textsf{M}\in \mathbb {N}\backslash \{1\}\) 。我们根据掩码完全表征了\(\mathscr {C}^m\) -步插值细分方案及其插值\ (\textsf{M}\) -可细化函数的收敛性和平滑度。受\(n_s\)步插值平稳细分方案的启发，我们进一步引入了r -mask 准平稳细分方案的概念，然后表征了它们的\(\mathscr {C}^m\)收敛性和平滑性特性只使用他们的面具。此外，将\(n_s\)步插值细分方案与r -mask 准平稳细分方案相结合，我们可以得到\(r n_s\)步插值细分方案。提供了收敛\(n_s\)步插值\(\textsf{M}\)细分方案的示例和构造过程，以说明我们使用膨胀因子\(\textsf{M}=2,3,4\)的结果。 此外，对于二元膨胀\(\textsf{M}=2\)和\(r=2,3\) ，仅使用带有两环模板的r掩模，我们提供了\(\mathscr {C} ^r\) -收敛r -步插值r -mask 准平稳二进细分方案。