Bernstein-type operators which preserve polynomials

Abstract

In this paper we present the sequence of linear Bernstein-type operators defined for f ∈ C [ 0 , 1 ] by B n ( f ∘ τ − 1 ) ∘ τ , B n being the classical Bernstein operators and τ being any function that is continuously differentiable ∞ times on [ 0 , 1 ] , such that τ ( 0 ) = 0 , τ ( 1 ) = 1 and τ ′ ( x ) > 0 for x ∈ [ 0 , 1 ] . We investigate its shape preserving and convergence properties, as well as its asymptotic behavior and saturation. Moreover, these operators and others of King type are compared with each other and with B n . We present as an interesting byproduct sequences of positive linear operators of polynomial type with nice geometric shape preserving properties, which converge to the identity, which in a certain sense improve B n in approximating a number of increasing functions, and which, apart from the constant functions, fix suitable polynomials of a prescribed degree. The notion of convexity with respect to τ plays an important role.