Asymptotic properties for a general class of Szasz-Mirakjan-Durrmeyer operators

In this paper we introduce a general family of Szász–Mirakjan–Durrmeyer type operators depending on an integer parameter $j\in \mathbb{Z}$. They can be viewed as a generalization of the Szász–Mirakjan–Durrmeyer operators [9], Phillips operators [11] and corresponding Kantorovich modifications of higher order. For $j\in \mathbb{N}$, these operators possess the exceptional property to preserve constants and the monomial $x^j$. It turns out, that an extension of this family covers certain well-known operators studied before, so that the outcoming results could be unified. We present the complete asymptotic expansion for the sequence of these operators. All its coefficients are given in a concise form. In order to prove the expansions for the class of locally integrable functions of exponential growth on the positive half-axis, we derive a localization result which is interesting in itself.