Convexity preservation of the four-point interpolatory subdivision scheme

In this note we examine the convexity preserving properties of the (linear) fourpoint interpolatory subdivision scheme of Dyn, Gregory and Levin when applied to functional univariate strictly convex data. Conditions on the tension parameter guaranteeing preservation of convexity are derived. These conditions depend on the initial data. The resulting scheme is the four-point scheme with tension parameter bounded from above by a bound smaller than 1/16. Thus the scheme generates C1 limit functions and has approximation order two.