On dimension elevation in Quasi Extended Chebyshev spaces

Abstract

Via blossoms we analyse the dimension elevation process from ${{\mathcal E}_n^{p,q}}$ to ${{\mathcal E}_{n+1}^{p,q}}$ , where ${{\mathcal E}_n^{p,q}}$ is spanned over [0, 1] by 1, x,..., x ^n-2, x ^p, (1 − x)^q, p, q being any convenient real numbers. Such spaces are not Extended Chebyshev spaces but Quasi Extended Chebyshev spaces. They were recently introduced in CAGD for shape preservation purposes (Costantini in Math Comp 46:203–214; 1986, Costantini in Advanced Course on FAIRSHAPE, pp. 87–114 in 1996; Costantini in Curves and Surfaces with Applications in CAGD, pp. 85–94, 1997). Our results give a new insight into the special case p = q for which dimension elevation had already been considered, first when p = q was supposed to be an integer (Goodman and Mazure in J Approx Theory 109:48–81, 2001), then without the latter requirement (Costantini et al. in Numer Math 101:333–354, 2005). The question of dimension elevation in more general Quasi Extended Chebyshev spaces is also addressed.