Approximation Order Equivalence Properties of Manifold-Valued Data Subdivision Schemes

There has been an emerging interest in developing an approximation theory for manifold-valued functions. In this paper, we address the following fundamental problem: Let M be a manifold with a metric d. For each smoothness factor r > 0 and approximation order R > 0, is there an approximation operator Ah = Ah;r,R that maps samples of any f : R→ M on a grid of size h to an approximant fh = Ah(f |hZ) : R→ M with the properties that a) supx d(fh(x), f(x)) = O(h) whenever f is a bounded C function, and b) fh is C smooth ? The case of M = R is of course well-studied. In the recent paper [14], the authors show that subdivision methods can be used to create arbitrarily smooth interpolants for M -valued data, addressing b) above. In this paper we further show that interpolatory subdivision schemes can be used to solve a) above. So, altogether, we establish the fact that if a linear interpolatory subdivision scheme possesses a smoothness order r and an approximation order R, then a nonlinear interpolatroy subdivision scheme for M -valued data constructed based on this linear scheme has the same smoothness and approximation orders. In other words, subdivision schemes furnish a constructive approximation method for solving the open problem posted above. We discuss the construction of quasi-interpolants of manifold-valued data based on general (not necessarily interpolatory) subdivision schemes. Acknowledgments. The work of this research was partially supported by the National Science Foundation grant DMS 0542237. The empirical observation on approximation equivalence (Figure 1) was first presented in the Sixth International Conference on Curves and Surfaces (Avignon, France, June 29-July 5, 2006.) and further discussed in a mini-symposium in the 12th International Conference in Approximation Theory (San Antonio, Texas, March 4-8, 2007) and the MAIA 2007 conference (Ålesund, Norway, August 22-26, 2007.) A preliminary effort on this problem was made in [5]. The second named author thanks Nira Dyn for inviting him to a workshop on “Subdivision and Refinability” (Pontignano, Siena, Italy, May 1-4, 2008), at which the main result of this paper was first presented.