In this paper we introduce a general family of Sz\'asz--Mirakjan--Durrmeyer type operators depending on an integer parameter $j \in \mathbb{Z}$. They can be viewed as a generalization of the Sz\'asz--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.

翻译：本文引入了一类依赖于整数参数$j \in \mathbb{Z}$的广义Szász–Mirakjan–Durrmeyer型算子族。该算子族可视为Szász–Mirakjan–Durrmeyer算子[9]、Phillips算子[11]以及相应高阶Kantorovich修正形式的推广。对于$j\in {\mathbb{N}}$，这类算子具有保持常数项及单项式$x^{j}$的独特性质。研究表明，该算子族的扩展涵盖了先前研究的若干经典算子，从而使得相关结果得以统一。我们给出了该算子序列的完整渐近展开式，其所有系数均以简洁形式表示。为了证明该展开式对于正半轴上指数增长的局部可积函数类成立，我们推导出了一个本身具有独立意义的局部化结果。