Positive linear operators preserving certain monomials on [0, ∞)

J.P. King constructed a sequence of positive linear operators on C [0, 1] preserving the constant function 1 and the monomial x2. After that, several papers were devoted to positive linear operators preserving prescribed functions. In 2009, J.M. Aldaz, O. Kounchev, H. Render modified the classical Bernstein operators to get new operators fixing 1 and x j for a given j ∈ N. Our aim is to construct operators acting on functions defined on [0, ∞) and preserving 1 and x j . To this end we consider suitable modifications of the Post-Widder and Gamma operators. Using pointwise asymptotic relations for the moments we obtain convergence results for the sequences of operators. Moreover, we establish the associated Voronovskaja formulas.