Abstract
Consider the Banach space m of real bounded sequences, x, with \({\Vert x\Vert =\sup_{k}|x_{k}|}\). A positive linear functional L on m is called an S-limit if \({L(\chi _{K})=0}\) for every characteristic sequence \({\chi _{K} }\) of sets, K, of natural density zero. We provide regular sublinear functionals that both generate as well as dominate S-limits. The paper also shows that the set of S-limits and the collection of Banach limits are distinct but their intersection is not empty. Furthermore, we show that the generalized limits generated by translative regular methods is equal to the set of Banach limits. Some applications are also provided.
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A part of this research was conducted when the first author was visiting Kent State University. This research was supported by the Higher Education Council of Turkey (YÖK).
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Yurdakadim, T., Khan, M.K., Miller, H.I. et al. Generalized Limits and Statistical Convergence. Mediterr. J. Math. 13, 1135–1149 (2016). https://doi.org/10.1007/s00009-015-0554-y
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DOI: https://doi.org/10.1007/s00009-015-0554-y