Weighted approximations by sampling type operators: recent and new results

Abstract

In this paper, we collect some recent results on the approximation properties of generalized sampling operators and Kantorovich operators, focusing on pointwise and uniform convergence, rate of convergence, and Voronovskaya-type theorems in weighted spaces of functions. In the second part of the paper, we introduce a new generalization of sampling Durrmeyer operators including a special function $\rho$ which satisfies certain assumptions. For the family of newly constructed operators, we obtain pointwise convergence, uniform convergence and rate of convergence for functions belonging to weighted spaces of functions.