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Results in Mathematics

, 74:198 | Cite as

Convergence of Bi-shift Localized Szász–Mirakjan Operators

  • Linsen XieEmail author
  • Tingfan Xie
Article
  • 36 Downloads

Abstract

Let \(\{\delta _{n} \}^{\infty }_{n=1}\)   and   \(\{\delta ^{\prime }_{n}\}^{\infty }_{n=1}\) be two sequences of positive numbers, and
$$\begin{aligned} C_{n,x}= \{ k: k\in N\cup \{0\} \,\,\text{ and }\,\, n(x-\delta ^{\prime }_{n})\le k\le n(x+\delta _{n}) \}. \end{aligned}$$
For any continuous function \(f: [0,\infty ) \rightarrow \mathbb {{\mathbb {R}}}\), we define a new localized Szász–Mirakjan operator as follows:
$$\begin{aligned} S_{n,\delta _{n},\delta ^{\prime }_{n}}(f,x)=e^{-nx}\sum _{k\in C_{n,x}}\frac{(nx)^{k}}{k!}f\left( \frac{k}{n}\right) , \,\, x\ge 0. \end{aligned}$$
We call this bi-shift localized Szász–Mirakjan operators. Certain new convergence theorems are obtained for such operators when the limits both \(\lim _{n\rightarrow \infty }\delta _{n}\sqrt{n}\) and \(\lim _{n\rightarrow \infty }\delta ^{\prime }_{n}\sqrt{n}\) exist.

Keywords

Bi-shift localization Szász–Mirakjan convergence 

Mathematics Subject Classification

41A36 

Notes

Acknowledgements

The authors are grateful to the anonymous referee for her/his very careful reading of the paper and her/his advice. This work was partially done when the first author was visiting St. Francis Xavier University. He would express his gratitude to the hospitality there. Supported by National Natural Science Foundation of China (11771194).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsLishui UniversityLishuiChina
  2. 2.Department of MathematicsChina Jiliang UniversityHangzhouChina

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