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.
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