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 ns-step interpolatory M-subdivision schemes and their interpolating M-refinable functions with ns∈N∪{∞}
Nearby in the stack
and a dilation factor
M∈N\{1}
. We completely characterize
Cm
-convergence and smoothness of
ns
-step interpolatory subdivision schemes and their interpolating
M
-refinable functions in terms of their masks. Inspired by
ns
-step interpolatory stationary subdivision schemes, we further introduce the notion of
r
-mask quasi-stationary subdivision schemes, and then we characterize their
Cm
-convergence and smoothness properties using only their masks. Moreover, combining
ns
-step interpolatory subdivision schemes with
r
-mask quasi-stationary subdivision schemes, we can obtain
rns
-step interpolatory subdivision schemes. Examples and construction procedures of convergent
ns
-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