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.
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
© 2020 Walter de Gruyter GmbH, Berlin/Boston