Abstract
Bernstein–Chlodovsky polynomials approximate continuous functions up to a certain growth on the positive half-axis. This paper is devoted to a study of their asymptotic simultaneous approximation properties. We present a Voronovskaya-type theorem for the derivatives of the Bernstein–Chlodovsky polynomials. This result answers a conjecture of Professor Gonska raised in 2018 to the affirmative.
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Acknowledgements
The author is grateful to the both anonymous referees for an extremely thorough reading of the manuscript. Their valuable advice led to several improvements. Furthermore, a slight error in the proof of Lemma 3.3 was discovered accompanied by a hint how to repair it. One referee suggested some considerations which were incorporated into the introduction. The other referee suggested to study the point \(x=0\) in a separate case. This is done in Theorem 2.2. In particular, the author thanks for pointing out reference [12].
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Dedicated to the 70th birthday of Professor Heiner Gonska.
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Abel, U. A Voronovskaya-Type Result for Simultaneous Approximation by Bernstein–Chlodovsky Polynomials. Results Math 74, 117 (2019). https://doi.org/10.1007/s00025-019-1036-5
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DOI: https://doi.org/10.1007/s00025-019-1036-5