Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter July 14, 2018

Statistical convergence of multiple sequences on a product time scale

  • Emrah Yilmaz ORCID logo EMAIL logo , Yavuz Altin ORCID logo and Hikmet Koyunbakan ORCID logo

Abstract

In this study, we extend the concepts and fundamental results on statistical convergence from a single time scale to any product time scale. Various characterizations about these new notions are also obtained.

MSC 2010: 40A35; 40B05; 34N05

Acknowledgements

The authors wish to thank the anonymous referees for their careful reading of the manuscript and their fruitful comments and suggestions.

References

[1] R. Agarwal, M. Bohner, D. O’Regan and A. Peterson, Dynamic equations on time scales: A survey, J. Comput. Appl. Math. 141 (2002), no. 1–2, 1–26. 10.1016/S0377-0427(01)00432-0Search in Google Scholar

[2] Y. Altin, H. Koyunbakan and E. Yilmaz, Uniform statistical convergence on time scales, J. Appl. Math. 2014 (2014), Article ID 471437. 10.1155/2014/471437Search in Google Scholar

[3] B. Aulbach and S. Hilger, A unified approach to continuous and discrete dynamics, Qualitative Theory of Differential Equations (Szeged 1988), Colloq. Math. Soc. János Bolyai 53, North-Holland, Amsterdam (1990), 37–56. Search in Google Scholar

[4] C. Belen and S. A. Mohiuddine, Generalized weighted statistical convergence and application, Appl. Math. Comput. 219 (2013), no. 18, 9821–9826. 10.1016/j.amc.2013.03.115Search in Google Scholar

[5] M. Bohner and G. S. Guseinov, Multiple integration on time scales, Dynam. Systems Appl. 14 (2005), no. 3–4, 579–606. 10.1007/978-3-319-47620-9_7Search in Google Scholar

[6] M. Bohner and G. S. Guseinov, Multiple Lebesgue integration on time scales, Adv. Difference Equ. 2006 (2006), Article ID 26391. 10.1155/ADE/2006/26391Search in Google Scholar

[7] M. Bohner and A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications, Birkhäuser, Boston, 2001. 10.1007/978-1-4612-0201-1Search in Google Scholar

[8] D. Borwein, Linear functionals connected with strong Cesàro summability, J. Lond. Math. Soc. 40 (1965), 628–634. 10.1112/jlms/s1-40.1.628Search in Google Scholar

[9] R. C. Buck, Generalized asymptotic density, Amer. J. Math. 75 (1953), 335–346. 10.2307/2372456Search in Google Scholar

[10] A. Cabada and D. R. Vivero, Expression of the Lebesgue Δ-integral on time scales as a usual Lebesgue integral: Application to the calculus of Δ-antiderivatives, Math. Comput. Model. 43 (2006), no. 1–2, 194–207. 10.1016/j.mcm.2005.09.028Search in Google Scholar

[11] J. S. Connor, The statistical and strong p-Cesàro convergence of sequences, Analysis 8 (1988), no. 1–2, 47–63. 10.1524/anly.1988.8.12.47Search in Google Scholar

[12] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244. 10.4064/cm-2-3-4-241-244Search in Google Scholar

[13] J. A. Fridy, On statistical convergence, Analysis 5 (1985), no. 4, 301–313. 10.1524/anly.1985.5.4.301Search in Google Scholar

[14] G. S. Guseinov, Integration on time scales, J. Math. Anal. Appl. 285 (2003), no. 1, 107–127. 10.1016/S0022-247X(03)00361-5Search in Google Scholar

[15] S. Hilger, Analysis on measure chains—a unified approach to continuous and discrete calculus, Results Math. 18 (1990), no. 1–2, 18–56. 10.1007/BF03323153Search in Google Scholar

[16] B. Jackson, Partial dynamic equations on time scales, J. Comput. Appl. Math. 186 (2006), no. 2, 391–415. 10.1016/j.cam.2005.02.011Search in Google Scholar

[17] I. J. Maddox, Spaces of strongly summable sequences, Quart. J. Math. Oxford Ser. (2) 18 (1967), 345–355. 10.1093/qmath/18.1.345Search in Google Scholar

[18] I. J. Maddox, Statistical convergence in a locally convex space, Math. Proc. Cambridge Philos. Soc. 104 (1988), no. 1, 141–145. 10.1017/S0305004100065312Search in Google Scholar

[19] F. Móricz, Statistical convergence of multiple sequences, Arch. Math. (Basel) 81 (2003), no. 1, 82–89. 10.1007/s00013-003-0506-9Search in Google Scholar

[20] F. Móricz, Statistical limits of measurable functions, Analysis (Munich) 24 (2004), no. 1, 1–18. 10.1524/anly.2004.24.14.1Search in Google Scholar

[21] I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers, 5th ed., John Wiley & Sons, New York, 1991. Search in Google Scholar

[22] F. Nuray and B. Aydin, Strongly summable and statistically convergent functions, Inform. Technol. Ir Valdymas. 30 (2004), no. 1, 74–76. Search in Google Scholar

[23] D. Rath and B. C. Tripathy, On statistically convergent and statistically Cauchy sequences, Indian J. Pure Appl. Math. 25 (1994), no. 4, 381–386. Search in Google Scholar

[24] T. Šalát, On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), no. 2, 139–150. Search in Google Scholar

[25] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361–375. 10.1080/00029890.1959.11989303Search in Google Scholar

[26] M. S. Seyyidoglu and N. O. Tan, A note on statistical convergence on time scale, J. Inequal. Appl. 2012 (2012), Paper No. 219. 10.1186/1029-242X-2012-219Search in Google Scholar

[27] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), no. 1, 73–74. Search in Google Scholar

[28] B. C. Tripathy, On statistical convergence, Proc. Est. Acad. Sci. Phys. Math. 47 (1998), no. 4, 299–303. 10.3176/phys.math.1998.4.06Search in Google Scholar

[29] C. Turan and O. Duman, Convergence methods on time scales, AIP Conf. Proc. 1558 (2013), no. 1, 1120–1123. 10.1063/1.4825704Search in Google Scholar

[30] C. Turan and O. Duman, Statistical convergence on time scales and its characterizations, Advances in Applied Mathematics and Approximation Theory, Springer Proc. Math. Stat. 41, Springer, New York (2013), 57–71. 10.1007/978-1-4614-6393-1_3Search in Google Scholar

[31] C. Turan and O. Duman, Fundamental properties of statistical convergence and lacunary statistical convergence on time scales, Filomat 31 (2017), no. 14, 4455–4467. 10.2298/FIL1714455TSearch in Google Scholar

[32] E. Yilmaz, Y. Altin and H. Koyunbakan, λ-statistical convergence on time scales, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 23 (2016), no. 1, 69–78. 10.1080/03610926.2021.2006716Search in Google Scholar

[33] A. Zygmund, Trigonometric Series. Vol. I, II, Cambridge Math. Lib., Cambridge University Press, Cambridge, 1988. Search in Google Scholar

Received: 2016-05-16
Accepted: 2016-10-20
Published Online: 2018-07-14
Published in Print: 2020-09-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 25.4.2024 from https://www.degruyter.com/document/doi/10.1515/gmj-2018-0051/html
Scroll to top button