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 $M$-subdivision schemes and their interpolating $M$-refinable functions with $n_s\in \mathbb{N} \cup\{\infty\}$ and a dilation factor $M\in \mathbb{N}\backslash\{1\}$. We completely characterize $\mathscr{C}^m$-convergence and smoothness of $n_s$-step interpolatory subdivision schemes and their interpolating $M$-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 $M$-subdivision schemes are provided to illustrate our results with dilation factors $M=2,3,4$. In addition, for the dyadic dilation $M=2$ 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.

翻译：标准插值细分格式及其基础插值细化函数在计算机辅助几何设计、数值偏微分方程和逼近论中具有重要意义。通过推广这些概念，我们引入并研究了$n_s$步插值$M$细分格式及其对应的插值$M$细化函数，其中$n_s\in \mathbb{N} \cup\{\infty\}$，膨胀因子$M\in \mathbb{N}\backslash\{1\}$。我们完全通过掩码刻画了$n_s$步插值细分格式及其插值$M$细化函数的$\mathscr{C}^m$收敛性与光滑性。受$n_s$步插值静态细分格式启发，我们进一步引入了$r$掩码拟静态细分格式的概念，并仅通过掩码刻画了其$\mathscr{C}^m$收敛性与光滑性。此外，将$n_s$步插值细分格式与$r$掩码拟静态细分格式相结合，我们可以得到$r n_s$步插值细分格式。本文提供了收敛的$n_s$步插值$M$细分格式的算例与构造方法，其中膨胀因子$M=2,3,4$，以验证我们的理论结果。另外，针对二进膨胀$M=2$和$r=2,3$的情况，我们利用仅含双环模板的$r$个掩码，给出了$\mathscr{C}^r$收敛的$r$步插值$r$掩码拟静态二进细分格式的算例。