On the Generalized Szász–Mirakyan Operators

Abstract

In this paper, we construct sequences of Szasz–Mirakyan operators which are based on a function ρ. This function not only characterizes the operators but also characterizes the Korovkin set $${\left \{ 1,\rho ,\rho ^{2} \right \}}$$ in a weighted function space. We give theorems about convergence of these operators to the identity operator on weighted spaces which are constructed using the function ρ and which are subspaces of the space of continuous functions on $${\mathbb{R} ^{+}}$$ . We give quantitative type theorems in order to obtain the degree of weighted convergence with the help of a weighted modulus of continuity constructed using the function ρ. Further, we prove some shape-preserving properties of the operators such as the ρ-convexity and the monotonicity. Our results generalize the corresponding ones for the classical Szasz operators.