Approximation properties of modified gamma operators preserving \(t^{\vartheta}\) (Q2120578)

The rate of convergence is obtained in terms of the first and second order modulus of continuity for a Gamma type operator, which preserves the functions \(1\) and \(t^{\theta}\) (\(\theta\in\mathbb{N}\) is fixed). Furthermore, a quantitative Voronovskaja type theorem is established for functions belonging to a weighted space, and an estimate of the rate of pointwise convergence on a Lebesgue point is given for functions of bounded variation. Finally, it is deduced that the modified Gamma operator, which preserves the functions \(1\) and \(t^3\), provide a better approximation than the classical Gamma operator.