Bernstein-type operators which preserve polynomials

In this paper we present the sequence of linear Bernstein-type operators defined for f∈C0,1] 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.