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Proceedings of the American Mathematical Society

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$ \alpha\sim l $ relatively dense sets and weighted pseudo almost periodic phenomenon


Authors: Jin Liang, G. M. N’Guérékata and Ti-Jun Xiao
Journal: Proc. Amer. Math. Soc. 148 (2020), 687-696
MSC (2010): Primary 34K14, 42A75, 43A60
DOI: https://doi.org/10.1090/proc/14818
Published electronically: October 18, 2019
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Abstract: In this paper, we propose a new idea of studying the weighted pseudo almost periodic phenomenon. We introduce a new mathematical concept of $ \alpha \sim l $ relatively dense set and show that the $ \alpha \sim l$ relatively dense set is more useful than the classical relatively dense set due to Bohr or Bochner in treating the weighted pseudo almost periodic phenomenon. We prove the uniqueness of decomposition of the weighted pseudo almost periodic functions with values in Banach spaces and with weights in a new weight function set, and show that the new weight set is larger than all the existing weight function sets which ensures the uniqueness of such decomposition in literature. Moreover, we reveal several properties of the vector-valued weighted pseudo almost periodic functions, and provide a series of concrete examples and remarks for illustrating our theoretical results.


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Additional Information

Jin Liang
Affiliation: School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
Email: jinliang@sjtu.edu.cn

G. M. N’Guérékata
Affiliation: School of Computer, Mathematical and Natural Sciences, Morgan State University, Baltimore, Maryland 21251

Ti-Jun Xiao
Affiliation: School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China

DOI: https://doi.org/10.1090/proc/14818
Keywords: Weighted pseudo almost periodic function, vector-valued function, Banach space, relatively dense, uniqueness of decomposition
Received by editor(s): May 21, 2019
Published electronically: October 18, 2019
Additional Notes: The first author is the corresponding author
This work was partially supported by the NSF of China (11571229, 11771091, 11971306)
Communicated by: Mourad Ismail
Article copyright: © Copyright 2019 American Mathematical Society