Abstract
The Lupaş q-analogue, \(R_{n,q}\), is historically the first known q-version of the Bernstein operator. It has been studied extensively in different aspects by a number of authors during the last decades. In this work, the following issues related to the image of the Lupaş q-analogue are discussed: A new explicit formula for the moments has been derived, independence of the image \(R_{\text {n,q}}\) from the parameter q has been examined, the diagonalizability of operator \(R_{\text {n,q}}\) has been proved, and the fact that \(R_{\text {n,q}}\) does not preserve modulus of continuity has been established.
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Acknowledgements
The authors express their sincere gratitude to Professor Heping Wang (Capital Normal University, Beijing, China) for providing them with a copy of his unpublished work which drawn their attention to the problem considered in Sect. 5. The first-named author gratefully acknowledges the support of Recep Tayyip Erdogan University as this work was completed while she was on post-doctoral leave at Atilim University. Last but not least the authors extend their appreciations to the anonymous referees for their thorough reading of the manuscript and beneficial comments.
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Communicated by Fuad Kittaneh.
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Gürel Yılmaz, Ö., Ostrovska, S. & Turan, M. On the Image of the Lupaş q-Analogue of the Bernstein Operators. Bull. Malays. Math. Sci. Soc. 47, 11 (2024). https://doi.org/10.1007/s40840-023-01614-y
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DOI: https://doi.org/10.1007/s40840-023-01614-y