Abstract
Hardy’s well-known Tauberian theorem for number sequences states that if a sequence \(x=\left( x_{k}\right) \) satisfies \(\lim Cx=L\) and \(\Delta x_{k}:=x_{k+1}-x_{k}=O\left( 1/k\right) \), then \(\lim x=L\), where Cx denotes the Cesàro mean (arithmetic mean) of x. In this study, we extend this result to the theory of time scales. We also discuss the theory on some special time scales.
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References
Aktuglu, H., Bekar, S.: \(q\)-Cesáro matrix and \(q\)-statistical convergence. J. Comput. Appl. Math. 235, 4717–4723 (2011)
Aslim, G., Guseinov, G.S.: Weak semirings \(\omega \)-semirings, and measures. Bull. Allahabad Math. Soc. 14, 1–20 (1999)
Boos, J.: Classical and Modern Methods in Summability. Oxford University Press, Oxford (2000)
Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser Boston, Inc., Boston (2001)
Fast, H.: Sur la convergence statistique. Colloq. Math. 2, 241–244 (1951)
Fekete, Á., Móricz, F.: A characterization of the existence of statistical limit of real-valued measurable functions. Acta Math. Hung. 114, 235–246 (2007)
Fridy, J.A.: On statistical convergence. Analysis 5, 301–313 (1985)
Fridy, J.A., Khan, M.K.: Statistical extensions of some classical Tauberian theorems. Proc. Am. Math. Soc. 128, 2347–2355 (2000)
Guseinov, G.S.: Integration on time scales. J. Math. Anal. Appl. 285, 107–127 (2003)
Hardy, G.H.: Divergent Series. Clarendon Press, Oxford (1949)
Móricz, F.: Statistical limits of measurable functions. Analysis (Munich) 24, 1–18 (2004)
Turan, C., Duman, O.: Statistical convergence on timescales and its characterizations. In: Advances in Applied Mathematics and Approximation Theory. Springer Proceedings in Mathematics and Statistics, vol. 41, pp. 57–71. Springer, New York (2013)
Turan, C., Duman, O.: Convergence methods on time scales. AIP Conf. Proc. 1558, 1120–1123 (2013)
Turan, C., Duman, O.: Fundamental properties of statistical convergence and lacunary statistical convergence on time scales. Filomat 31(14), 4455–4467 (2017)
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Turan Yalçın, C., Duman, O. Hardy-Type Tauberian Conditions on Time Scales. Mediterr. J. Math. 15, 198 (2018). https://doi.org/10.1007/s00009-018-1245-2
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DOI: https://doi.org/10.1007/s00009-018-1245-2