Shape Preserving Interpolatory Subdivision Schemes for Nonuniform Data

This article is concerned with a class of shape preserving four-point subdivision schemes which are stationary and which interpolate nonuniform univariate data {(x_i, f_i)}. These data are functional data, i.e., x_i≠x_j if i≠j. Subdivision for the strictly monotone x-values is performed by a subdivision scheme that makes the grid locally uniform. This article is concerned with constructing suitable subdivision methods for the f-data which preserve convexity; i.e., the data at the kth level, {x^(k)_i, f_i(k)} is a convex data set for all k provided the initial data are convex. First, a sufficient condition for preservation of convexity is presented. Additional conditions on the subdivision methods for convergence to a C¹ limit function are given. This leads to explicit rational convexity preserving subdivision schemes which generate continuously differentiable limit functions from initial convex data. The class of schemes is further restricted to schemes that reproduce quadratic polynomials. It is proved that these schemes are third order accurate. In addition, nonuniform linear schemes are examined which extend the well-known linear four-point scheme to the case of nonuniform data. Smoothness of the limit function generated by these linear schemes is proved by using the well-known smoothness criteria of the uniform linear four-point scheme.