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,d₁,...,d_N (d_iR^q), 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: ({d}_i,d_i+1)D_iR^2q, where the sets D_i 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.