Polynomial splines as examples of Chebyshevian splines
Résumé
The results we present here concern geometrically continuous polynomial splines, in the sense that the left/right derivatives at the knots are linked by connection matrices. A classical suffi cient condition for spaces of such splines to be suitable for either Approximation or Geometric Design is the total positivity of all their connection matrices (i.e., all their minors are non-negative). Their entries can then serve as shape parameters. As an example, they can be used in spline interpolation to make up for the Gibbs phenomenon. We show how to obtain all the connection matrices leading to suitable spline spaces. The results in themselves are interesting in so far as they permit more effi cient shape e ffects than total positivity. Even more interesting is the way we achieve them. They follow from considering polynomial splines as special instances of Chebyshevian splines and polynomial spaces on a closed bounded interval as special instance of Extented Chebyshev spaces.