A Note on the $(\theta ,\varphi )$-Statistical Convergence of the Product Time Scale

Year 2020, Volume: 8 Issue: 1, 192 - 196, 15.04.2020

Metin Basarır

Abstract

In this paper, we introduce the concepts $(\theta ,\varphi )$-density of a subset of the product time scale $\mathbb{T}^{2}$ and $(\theta ,\varphi )$ -statistical convergence of $\Delta $- measurable function $f$ \ defined on the product time scale $\mathbb{T}^{2}$ with the help of lacunary sequences. Later, we have discussed the connection between classical convergence and $ (\theta ,\varphi )$-statistical convergence. In addition, we have seen that $ f$ is strongly $(\theta ,\varphi )$-Cesaro summable on $\mathbb{T}^{2}$ then $f$ is $(\theta ,\varphi )$-statistical convergent$.$

Keywords

delta-convergence , statistical convergence , density , product time scale , lacunary sequences , p-Cesaro summable

References

• [1] H. Aktuğlu and Ş. Bekar, q-Cesaro matrix and q-statistical convergence, J. Comput. Appl. Math. 235 (2011) 4717–4723.
• [2] Y. Altın, H. Koyunbakan and E. Yılmaz, Uniform Statistical Convergence on Time Scales, Journal of Applied Mathematics, Volume 2014, Article ID 471437, 6 pages.
• [3] G. Aslim, G. Sh. Guseinov, Weak semirings, w-semirings, and measures, Bull. Allahabad Math. Soc. 14 (1999) 1–20.
• [4] B. Aulbach, S. Hilger, A unified approach to continuous and discrete dynamics. J. Qual. Theory Diff. Equ. (Szeged, 1988), Colloq. Math. Soc. J`anos Bolyai, North-Holland Amsterdam 53, 37–56 (1990).
• [5] M. Bohner and G. Sh. Guseinov, Partial differentation on time scales, Dynamic systems and Applications 13, No.3 (2004) 351-379.
• [6] M. Bohner and G. Sh. Guseinov, Double integral calculus of variations on time scales, Computers and Mathematics with Applications 54, No.1 (2007) 45-57.
• [7] A. Cabada and D. R. Vivero, Expression of the Lebesque D-integral on time scales as a usual Lebesque integral; application to the calculus of D-antiderivates, Math. Comput. Modelling, 43 (2006) 194–207.
• [8] A. Cabada and D. R. Vivero, Expression of the Lebesque integral on time scales as a usual Lebesque integral; application to the calculus of antiderivates, Mathematical and Computer Modelling, 43 (2006) 194-207.
• [9] H. Çakallı, A new approach to statistically quasi Cauchy sequences, Maltepe Journal of Mathematics, 1 (1) (2019) 1-8.
• [10] Muhammed Çınar, Emrah Yılmaz, Yavuz Altın, Mikail Et, (l,n)-Statistical Convergence on a Product Time Scale, Punjab University Journal of Mathematics, 51 (11) (2019) 41-52.
• [11] H. Fast, Sur la convergence statitique, Colloq. Math. 2 (1951) 241–244.
• [12] A. R. Freedman, J. J. Sember and M. Raphael, Some Cesaro type summability spaces, Proc. London Math. Soc. 37 (1978) 508–520.
• [13] J. A. Fridy, On statistical convergence, Analysis, 5 (1985) 301–313.
• [14] A. D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32 (1) (2002), 129-138.
• [15] G. Sh. Guseinov, Integration on time scales, J. Math. Anal. Appl. 285 (1) (2003) 107–127.
• [16] S. Hilger, Ein maßkettenkalkul mit anwendung auf zentrumsmanningfaltigkeilen Ph.D thesis, Universitat, W¨urzburg (1989).
• [17] S. Hilger, Analysis on measure chains-A unified a approach to continuous and discrete calculus, Results Math. 18 (1990) 19–56.
• [18] F. Moricz, Statistical limit of measurable functions, Analysis, 24 (2004), 1-18.
• [19] T. Rzezuchowski, A note on measures on time scales, Demonstr. Math. 33 (2009) 27–40.
• [20] M. S. Seyyidoglu and N. O. Tan, A note on statistical convergence on time scales, J. Inequal. Appl. (2012) 219–227.
• [21] H. Steinhaus, Sur la convergence ordinarie et la convergence asimptotique, Colloq. Math. 2 (1951) 73–74.
• [22] N. Tok and M. Başarır, On the $\lambda _{h}^{\alpha }$-statistical convergence of the functions defined on the time scale, Proceedings of International Mathematical Sciences,1 (1) (2019) 1-10.
• [23] C. Turan and O. Duman, Fundamental Properties of Statistical Convergence and Lacunary Statistical Convergence on Time Scales, Filomat, 31(14) (2017) 4455–4467.
• [24] C. Turan and O. Duman, Statistical convergence on time scales and its characterizations, Advances in Applied Mathematics and Approximation Theory, Springer, Proceedings in Mathematics & Statistics, 41 (2013) 57-71.
• [25] N. Turan and M. Başarır, A note on quasi-statistical convergence of order a in rectangular cone metric space, Konuralp J. Math., 7 (1) (2019) 91-96.
• [26] N. Turan and M. Başarır, On the Dg-statistical convergence of the function defined time scale, AIP Conference Proceedings, 2183, 040017 (2019); https://doi.org/10.1063/1.5136137.
• [27] A. Zygmund, Trigonometric Series, United Kingtom: Cambridge Univ. Press (1979).