Approximation order equivalence properties of manifold-valued data subdivision schemes (Q2882362)

Given a function \(f : \mathbb R \to M\) with some manifold \(M\), the question addressed in this paper is: Is it possible to construct an approximation operator that maps samples of this function on a grid of size \(h\) to an approximant \(f_h : \mathbb R \to M\) such that (a) the distance between \(f\) and \(f_h\), measured uniformly in terms of a given metric on \(M\) is bounded by \(O(h^R)\) with a given \(R>0\), and (b) the approximant \(f_h\) is in \(C^r\) with a given \(r\)? The key result is that such approximation operators do indeed exist, and that suitably chosen subdivision schemes are a possible constructive solution to this problem.