Ready-to-blossom bases and the existence of geometrically continuous piecewise Chebyshevian B-splines

Abstract

Existence of blossoms is crucial for design. In a single space, we recently characterised it in terms of ready-to-blossom bases. Such bases are magic, for their use makes existence of blossoms visible at first sight. A similar characterisation is given here for geometrically continuous piecewise Chebyshevian splines (sections in different Extended Chebyshev spaces, connection matrices at the knots). This enables us to re-prove the equivalence between existence of blossoms and existence of B-spline bases under the least possible differentiability assumptions. The existing proof of the latter result was totally different and it strongly relied on the fact that all spline sections were supposed to be C ∞ . To cite this article: M.-L. Mazure, C. R. Acad. Sci. Paris, Ser. I 347 (2009).