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

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 [1] and rational function spaces [2]. 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 [4, 3]. Applications to Pólya type theorems on positive polynomials will be presented. 2 RACHID AIT-HADDOU AND MARIE-LAURENCE MAZURE AcknowledgementsThe research of the first author was supported by the KAUST Visual Com-puting Center. REFERENCES[1] Ait-Haddou R. (2014). Dimension elevation in Müntz spaces: a new emer-gence of the Müntz condition. J. Approx. Theory 181, 6–17.[2] Ait-Haddou R., Mazure M. L. Approximation by Chebyshevian BernsteinOperators versus Convergence of Dimension Elevation. (Submitted).[3] Ait-Haddou R., Mazure M. L. Dimension elevation for rational functionspaces and Pólya theorem on positive polynomials. (In preparation). [4] Mazure M. L. (2009). Bernstein-type operators in Chebyshev spaces. Nu-merical Algorithms, 52 (1), 93–128.