Critical Length for Design Purposes and Extended Chebyshev Spaces

We analyze the connection between two ideas of apparently different nature. On one hand, the existence of an extended Chebyshev basis, which means that the Hermite interpolation problem has always a unique solution. On the otherhand, the existence of a normalized totally positive basis, which means thatthe space is suitable for design purposes. We prove that the intervals wherethe existence of a normalized totally positive basis is guaranteed are those intervals wherethe existence of an extended Chebyshev basis of the space of derivatives can be ensured.We apply our results to the spaces C _n generated by 1,t, , t ^n-2, cos t, sin t. In particular, C ₅ is a space suitable for design which permits the exact reproduction ofremarkable parametric curves, including lines and circles with a single control polygon. We prove that this space has the minimal dimension for this purpose.