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. We characterize convergence and smoothness of $n_s$-step interpolatory subdivision schemes and their interpolating M-refinable functions. Inspired by $n_s$-step interpolatory stationary subdivision schemes, we further introduce the notion of $n_s$-step interpolatory quasi-stationary subdivision schemes, and then we characterize their convergence and smoothness properties. Examples 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 $M=2$, using masks with two-ring stencils, we also provide examples of $C^2$-convergent $2$-step or $C^3$-convergent $3$-step interpolatory quasi-stationary subdivision schemes.

翻译：标准插值细分格式及其底层插值可细化函数在计算机辅助几何设计(CAGD)、数值偏微分方程和逼近理论中具有重要意义。推广这些概念，我们引入并研究了$n_s$步插值M-细分格式及其插值M-可细化函数，其中$n_s\in \mathbb{N} \cup\{\infty\}$，M为膨胀因子。我们刻画了$n_s$步插值细分格式及其插值M-可细化函数的收敛性与光滑性。受$n_s$步插值平稳细分格式的启发，进一步引入$n_s$步插值拟平稳细分格式的概念，并刻画其收敛性与光滑性。给出了收敛的$n_s$步插值M-细分格式的实例，以膨胀因子$M=2,3,4$阐释相关结果。此外，对于二进膨胀$M=2$，使用双环模板的掩模，还给出了$C^2$收敛的2步或$C^3$收敛的3步插值拟平稳细分格式的实例。