SIAM Journal on Numerical Analysis
Convergence and C1 analysis of subdivision schemes on manifolds by proximity
Computer Aided Geometric Design - Special issue: Geometric modelling and differential geometry
Convergence and smoothness analysis of subdivision rules in Riemannian and symmetric spaces
Advances in Computational Mathematics
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Proximity conditions are used extensively in the analysis of smoothness and approximation order properties of subdivision schemes for manifold-valued data. While these properties under question are independent of choice of coordinates on the manifold, it is not known whether the proximity condition itself is invariant under arbitrary change of coordinates. In this note, we answer this question to the affirmative, i.e. we prove that the proximity condition is satisfied in one coordinate system if and only if it is satisfied in any other coordinate system. In passing, we prove a connection between the general proximity condition and an alternate proximity condition used in the interpolatory case. This interpolatory proximity condition also enjoys the same invariance under change of coordinates.