Extended Chebyshev spaces in rationality

Abstract

On a closed bounded interval, a given Extended Chebyshev space can be defined by means of generalised derivatives associated with systems of weight functions. Only recently we could identify all such systems, describing an iterative process to build them. In the present work, we interpret the first step of this process as the construction of rational spaces based on Extended Chebyshev spaces. This construction establishes an interesting symmetry between all Extended Chebyshev spaces "good for design" (i.e., all those which contain constants and which possess blossoms) and the rational spaces based on them (Extended Chebyshev spaces in rationality). In particular, this symmetry results in a very simple relation between the corresponding blossoms. A special case is obtained when considering polynomial spaces as examples of Extended Chebyshev spaces. The classical rational spaces then appear as examples of Extended Chebyshev spaces good for design, that is, possessing blossoms. This offers interesting new insights on the famous so-called rational Bezier curves.