On the composition and decomposition of positive linear operators III: A non-trivial decomposition of the Bernstein operator

Abstract: The central problem in this technical report is the question if the classical Bernstein operator can be decomposed into nontrivial building blocks where one of the factors is the genuine Beta operator introduced by M\"uhlbach and Lupa\c{s}. We collect several properties of the Beta operator such as injectivity, the eigenstructure and the images of the monomials under its inverse. Moreover, we give a decomposition of the form $B_n = \bar{\mathbb{B}}_n \circ F_n $ where $F_n$ is a nonpositive linear operator having quite interesting properties. We study the images of the monomials under $F_n$, its moments and various representations. Also an asymptotic formula of Voronovskaya type for polynomials is given and a connection with a conjecture of Cooper and Waldron is established. In an appendix numerous examples illustrate the approximation behaviour of $F_n$ in comparison to $B_n$.