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An intermediate Voronovskaja type theorem

  • Dumitru PopaEmail author
Original Paper
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Abstract

We prove the following intermediate Voronovskaja type theorem: Let \(V_{n}:C \left[ a,b\right] \rightarrow C\left[ a,b\right] \) be a sequence of positive linear operators such that \(\lim \nolimits _{n\rightarrow \infty }V_{n}\left( f\right) =f\) uniformly on \(\left[ a,b\right] \) for every \(f\in C\left[ a,b \right] \). Let \(x\in \left[ a,b\right] \) and suppose that there exists a sequence \(\left( \lambda _{n}\right) _{n\in {\mathbb {N}}}\subset \left( 0,\infty \right) \) such that \(\lim \nolimits _{n\rightarrow \infty }\lambda _{n}=\infty \) and there exists \(1<p<\infty \) such that the sequence \(\left( \lambda _{n}^{p}V_{n}\left( \left| \cdot -x\right| ^{p}\right) \left( x\right) \right) _{n\in {\mathbb {N}}}\) is bounded. Then, for every continuous function \(f:\left[ a,b\right] \rightarrow {\mathbb {R}}\) differentiable at \(x\in \left[ a,b\right] \) we have
$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\lambda _{n}\left[ V_{n}\left( f\right) \left( x\right) -f\left( x\right) V_{n}\left( {\mathbf {1}}\right) \left( x\right) -f^{\prime }\left( x\right) V_{n}\left( \left( \cdot -x\right) \right) \left( x\right) \right] =0. \end{aligned}$$
As an application we prove a Voronovskaja type theorem for the sequences of the operators \(V_{n}:C\left[ 0,1\right] \rightarrow C\left[ 0,1\right] \) defined by
$$\begin{aligned} V_{n}\left( f\right) \left( x\right) =a_{n}b_{n}\int _{0}^{1}\frac{ t^{a_{n}}f\left( \left( 1-t\right) \xi +tx\right) }{t+b_{n}}dt \end{aligned}$$
where \(\xi \in \left[ 0,1\right] \), \(\left( a_{n}\right) _{n\in {\mathbb {N}}}\) is a sequence of natural numbers with \(\lim \nolimits _{n\rightarrow \infty }a_{n}=\infty \), \(\left( b_{n}\right) _{n\in {\mathbb {N}}}\subset \left( 0,\infty \right) \) with \(\lim \nolimits _{n\rightarrow \infty }b_{n}=\infty \) and \(\lim \nolimits _{n\rightarrow \infty }\frac{a_{n}}{b_{n}}\in \left[ 0,\infty \right) \).

Keywords

Korovkin approximation theorem Positive linear operators Asymptotic formula Voronovskaia type theorem 

Mathematics Subject Classification

41A35 41A36 41A25 

Notes

Acknowledgements

We would like to thank the two referees of our paper for carefully reading the manuscript and for such constructive comments, remarks and suggestions which helped improving the first version of the paper.

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Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.Department of MathematicsOvidius University of ConstantaConstantaRomania

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