Blossoming beyond extended Chebyshev spaces

In a previous series of papers a theory of blossoming was developed for spa ces of functions on an interval I spanned by the constant functions and fun ctions Phi (1), ... Phi (n), where Phi (1)', .... Phi (n)' span an extended Chebyshev space. This theory was then used to construct a generalisation o f the Bernstein basis and the de Casteljau algorithm. Also considered were functions defined to be piecewise in such spaces, leading to generalisation s of B-splines and the de Boor algorithm. Here we relax the condition that Phi (1)', .... Phi (n)' span an extended Chebyshev space, while retaining a ll the nice properties of the earlier theory. This allows us to include a l arge variety of new spaces, including spaces of polynomials which have been found to be successful for tension methods for shape-preserving interpolat ion. (C) 2001 Academic Press.