Sufficient Conditions for Smooth Non-Uniform Variational Refinement Curves

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Approximation Theory XI




Sufficient conditions are given for C1 and C2 (subdivision) curves generated by a particular non-uniform, interpolatory, variational refinement scheme. The 'energy' functional being minimized is a discretization of the standard linearized spline functional over piece wise linear curves { a generalization of the minimizing function a used for the uniform scheme in (10). The conditions used are uniform bounds on either the energy functional or certain divided differences, along with a condition on the knots, forcing them to be dense and uniform in the limit. To establish C2, a certain 'bootstrap' argument is applied. The argument is based on a generalization of a result in (6) used to show smoothness of curves generated by nonuniform corner cutting. x1. Introduction: In this paper we investigate the smoothness of the limiting curves generated by a particular non-uniform, interpolatory variational refinement scheme. The problem is a generalization of the uniform scheme given in (10) to non-uniform subdivision, while the analysis is more in-line with that given in (6) and (4). The major difference between the analysis in (6) and here is that our scheme is not convex, as they are in corner cutting. To get around this, we exploit the variational nature of the problem explicitly to establish contraction of certain differences. To do so, we require that either the functional or certain divided differences are uniformly bounded, and that the knots become nearly uniform during the refinement process. This paper does not investigate whether these bounds are achievable in practice, or the effect on the knots due to the bounds. We leave those questions for future work.