Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces

We study the existence and shape preserving properties of a generalized Bernstein operator Bn fixing a strictly positive function f0, and a second function f1 such that f1/f0 is strictly increasing, within the framework of extended Chebyshev spaces Un. The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator Bn : C[a, b] → Un with strictly increasing nodes, fixing f0, f1 ∈ Un. If Un ⊂ Un+1 and Un+1 has a non-negative Bernstein basis, then there exists a Bernstein operator Bn+1 : C[a, b] → Un+1 with strictly increasing nodes, fixing f0 and f1. In particular, if f0, f1, ..., fn is a basis of Un such that the linear span of f0, .., fk is an extended Chebyshev space over [a, b] for each k = 0, ..., n, then there exists a Bernstein operator Bn with increasing nodes fixing f0 and f1. The second main result says that under the above assumptions the following inequalities hold Bnf ≥ Bn+1f ≥ f for all (f0, f1)-convex functions f ∈ C [a, b] . Furthermore, Bnf is (f0, f1)-convex for all (f0, f1)-convex functions f ∈ C [a, b] . In the specific case of exponential polynomials we give alternative proofs of shape preserving properties by computing derivatives of the generalized Bernstein polynomials.