A general scheme for shape preserving planar interpolating curves

This paper describes the application of the so-called Abstract Schemes (AS) for the construction of shape preserving interpolating planar curves. The basic idea behind AS is given by observing that when we interpolate some data points by a spline, we can dispose of several free parameters d0, d 1,..., dN (di ∈ Rq), which are associated with the knots. If we now express shape constraints as conditions relative to each interval between two knots, they can be rewritten as a sequences of inclusion conditions: (di, di+1) ∈ Di ⊂ R2q, where the sets Di are the corresponding feasible domains. In this setting the problems of existence, construction and selection of an optimal solution can be studied with the help of Set Theory in a general way. The method is then applied for the construction of shape preserving, planar interpolating curves.