Asymptotic properties for a general class of Szasz-Mirakjan-Durrmeyer operators
<|reference_start|>Asymptotic properties for a general class of Szasz-Mirakjan-Durrmeyer operators: 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.<|reference_end|>
@article{abel2024asymptotic,
    title={Asymptotic properties for a general class of Szasz-Mirakjan-Durrmeyer
  operators},
    author={Ulrich Abel, Ana Maria Acu, Margareta Heilmann, Ioan Rasa},
    journal={arXiv preprint arXiv:2407.16474},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.16474},
    primaryClass={math.CA cs.NA math.NA}
}