The n_s-step Interpolatory (Quasi)-Stationary Subdivision Schemes and Their Interpolating Refinable Functions
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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∈ℕ∪{∞} 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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