Generalized Bernstein Operators on the Classical Polynomial Spaces

摘要

We study generalizations of the classical Bernstein operators on the polynomial spaces P-n[a, b], where instead of fixing 1 and x, we reproduce exactly 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 1 and f(1). These operators are defined by non-decreasing sequences of nodes precisely when f(1) 0 on (a, b), but even if f(1) vanishes somewhere inside (a, b), they converge to the identity.