Local smoothness of functions and Bernstein-Durrmeyer operators

Continuous functions satisfying a local Lipschitz condition of order α(0 < α < 1) on any subset of [0, 1] are characterized by the local rate of convergence of Bernstein-Durrmeyer operators. As an application, we give an algorithm for singular detection. A new direct estimate for the approximation of continuous functions by Bernstein-Durrmeyer operators and Kantorovich operators is also presented.