Convergence of dimension elevation in Chebyshev spaces versus approximation by Chebyshevian Bernstein operators

Rachid Ait Haddou (1) , Marie-Laurence Mazure (2)

1 IMAG - Institut d'Informatique et de Mathématiques Appliquées de Grenoble
2 MGMI - Modélisation Géométrique & Multirésolution pour l'Image

A nested sequence of extended Chebyshev spaces possessing Bernstein bases generates an infinite dimension elevation algorithm transforming control polygons of any given level into control polygons of the next level. In this talk, we present our recent results on the convergence of dimension elevation to the underlying Chebyshev-Bézier curve for the case of Müntz spaces and rational function spaces. Moreover, we reveal an equivalence between the convergence of dimension elevation to the underlying curve and the convergence of the corresponding Chebyshevian Bernstein operators to the identity. Applications to Polya type theorems on positive polynomials will be presented.

Mots clés: Extended Chebyshev spaces; Müntz spaces; rational function spaces; dimension elevation; Bernstein operators; Polya theorem on positive polynomials.

Domaines: Mathématiques [math]

Rachid Ait Haddou, Marie-Laurence Mazure. Convergence of dimension elevation in Chebyshev spaces versus approximation by Chebyshevian Bernstein operators. MAMERN VI–2015: 6th International Conference on Approximation Methods and Numerical Modelling in Environment and Natural Resources, Univ. Pau, Jun 2015, Pau, France. ⟨hal-01164458⟩