Generalized Bernstein operators on the classical polynomial spaces

We study generalizations of the classical Bernstein operators on polynomial spaces, where instead of fixing $\mathbf{1}$ and $x$, we require that $\mathbf{1}$ and a strictly increasing polynomial $f_1$ be fixed. Via several examples, we exhibit the diversity of behaviours in this more general setting. We also prove that for sufficiently large dimensions, there always exist generalized Bernstein operators fixing $\mathbf{1}$ and $f_1$, and converging to the identity.