A class of nonlinear four-point subdivision schemes

Abstract

We consider four-point subdivision schemes of the form

$$ (Sf)_{2i} = f_i,\qquad (Sf)_{2i+1} = \frac{f_i+f_{i+1}}{2} - \frac{1}{{8}}M\!\left(\strut \Delta^2f_{i-1}, \Delta^2f_i\right) $$

with any M that is originally defined as a positive-valued function for positive arguments and is extended to the whole of ℝ² by setting $M(x,y):=- M(\left|x\right|,\left|y\right|)$ if x < 0, y < 0 and M(x, y) : = 0 if xy ≤ 0. For these schemes, we study analytic properties, such as convexity preservation, convergence, smoothness of the limit function, stability and approximation order, in terms of simple and easily verifiable conditions on M. Fourth-order approximation on intervals of strict convexity is also investigated. All the results known for the most frequently used schemes, the PPH scheme and the power-p schemes, are included as special cases or improved, and extended to more general situations. The various statements are illustrated by two examples and tested by numerial experiments.