2004/10/07 by Marie-Madeleine Derriennic, Derriennic, Marie-Madeleine
Mathematics · #41A10 #41A25 #41A36 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:41A10 #msc:41A25 #msc:41A36
paper · pdf · doi:10.48550/arxiv.math/0410206
arxiv created 2004/10/22 · arxiv updated 2009/12/01
We introduce here a generalization of the modified Bernstein polynomials for Jacobi weights using the q-Bernstein basis proposed by G.M. Phillips to generalize classical Bernstein Polynomials. The function is evaluated at points which are in geometric progression in ]0,1[. Numerous properties of the modified Bernstein Polynomials are extended to their q-analogues: simultaneous approximation, pointwise convergence even for unbounded functions, shape-preserving property, Voronovskaya theorem, self-adjointness. Some properties of the eigenvectors, which are q-extensions of Jacobi polynomials, are given.