Positive Bernstein-Sheffer Operators Original Research Article

Let h(t) = Σn ≥ 1hntn, h1 > 0, and exp(xh(t)) = Σn ≥ 0Pn(x) tn/n!. For f ∈ C[0,1], the associated Bernstein-Sheffer operator of degree n is defined by Bhnf(x) = Pn− 1 Σnk = 0f(k/n)(nk) Pk(x) Pn − k(1 − x) where pn = pn(1). We characterize functions h for which Bhn is a positive operator for all n ≥ 0. Then we give a necessary and sufficient condition insuring the uniform convergence of Bhnf to f. When h is a polynomial, we give an upper bound for the error ∥ f − Bhnf ∥∞. We also discuss the behavior of Bhnf when h is a series with a finite or infinite radius of convergence.