LAUSR

We prove a general Bernstein type asymptotic evaluation for linear positive operators on the space $$C\left[ \alpha ,\beta \right] $$. Our proof is entirely different than those known in the literature. As application, we prove a Bernstein type asymptotic evaluation for some general Bernstein kind of linear positive operators, which extend the classical Bernstein extension of Voronovskaja’s theorem. We use then these general results to get, for many concrete linear positive operators, the Voronovskaja–Bernstein type results.