Generalized Bernstein Operators on the Classical Polynomial Spaces

We study generalizations of the classical Bernstein operators on the polynomial spaces (mathbb {P}_{n}[a,b]), where instead of fixing (mathbf {1}) and x, we reproduce exactly (mathbf {1}) and a polynomial (f_1), strictly increasing on [a, b]. We prove that for sufficiently large n, there always exist generalized Bernstein operators fixing (mathbf {1}) and (f_1). These operators are defined by non-decreasing sequences of nodes precisely when (f_1^prime > 0) on (a, b), but even if (f_1^prime ) vanishes somewhere inside (a, b), they converge to the identity.