Shape-preserving \(C^1\) Hermite interpolants generated by a Gori-Pitolli subdivision scheme (Q939573)

The paper starts by recalling some basic properties of Gori-Pitolli (GP) B-splines (Section 2) and cubic GP B-splines (Section 3) [\textit{L. Gori} and \textit{F. Pitolli}, Rend. Mat. Appl., VII. Ser. 20, No.1--4, 305--322 (2000; Zbl 0989.65153); together with \textit{E. Santi}, Numer. Algorithms 28, No.1--4, 199--213 (2001; Zbl 0993.65157)]. Next (Section 4), the local Bernstein-Bezier representation of cubic GP B-splines is constructed. Section 5 is devoted to the description of the main properties of quadratic GP splines and polynomials, while in Section 6, the family of Hermite subdivision schemes depending on a parameter is described. This family contains as a particular case the cubic Hermite interpolants. Next, a monotone Hermite problem and a convex Hermite interpolation problem are studied. The authors purpose is to give algorithms for the construction of monotone interpolants to arbitrary nondecreasing data and also for the construction of convex interpolants to arbitrary convex data. These algorithms are illustrated by some examples.