Convergence of dimension elevation algorithms : only a typical CAGD issue?
Résumé
A given nested sequence of Extended Chebyshev spaces possessing Bernstein bases generates an in nite
dimension elevation algorithm transforming control polygons of any given level into control polygons of the next
level. This situation is the natural generalisation of the so-called degree elevation for parametric polynomial
curves. Proving the possible convergence of such algorithms (in the rough sense of uniform convergence to
the underlying curves) is a typical CAGD issue. In this talk, we review our recent results on this difficult
question, with special focus on Müntz spaces and rational spaces. In particular we present our finndings on the
equivalence between convergence of a given dimension elevation algorithm and the uniform approximation of
continuous functions by positive operators of the Bernstein-type naturally associated with the nested sequence.
Surprisingly enough, in the case of rational spaces, we show that convergence is also connected with Polya-type
theorems on positive polynomials.