Bernstein-type operators which preserve polynomials

In this paper we present the sequence of linear Bernstein-type operators defined for f∈C[0,1]f∈C[0,1] by Bn(f∘τ−1)∘τBn(f∘τ−1)∘τ, BnBn being the classical Bernstein operators and ττ being any function that is continuously differentiable ∞∞ times on [0,1][0,1], such that τ(0)=0τ(0)=0, τ(1)=1τ(1)=1 and τ′(x)>0τ′(x)>0 for x∈[0,1]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 BnBn. 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 BnBn 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 ττ plays an important role.