Interpolating Refinable Functions and $n_s$-step Interpolatory Subdivision Schemes

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.