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

Metin Basarır

Abstract

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).

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).
There are 27 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Metin Basarır

Publication Date April 15, 2020
Submission Date February 25, 2020
Acceptance Date April 11, 2020
Published in Issue Year 2020 Volume: 8 Issue: 1

Cite

APA Basarır, M. (2020). A Note on the $(\theta ,\varphi )$-Statistical Convergence of the Product Time Scale. Konuralp Journal of Mathematics, 8(1), 192-196.
AMA Basarır M. A Note on the $(\theta ,\varphi )$-Statistical Convergence of the Product Time Scale. Konuralp J. Math. April 2020;8(1):192-196.
Chicago Basarır, Metin. “A Note on the $(\theta ,\varphi )$-Statistical Convergence of the Product Time Scale”. Konuralp Journal of Mathematics 8, no. 1 (April 2020): 192-96.
EndNote Basarır M (April 1, 2020) A Note on the $(\theta ,\varphi )$-Statistical Convergence of the Product Time Scale. Konuralp Journal of Mathematics 8 1 192–196.
IEEE M. Basarır, “A Note on the $(\theta ,\varphi )$-Statistical Convergence of the Product Time Scale”, Konuralp J. Math., vol. 8, no. 1, pp. 192–196, 2020.
ISNAD Basarır, Metin. “A Note on the $(\theta ,\varphi )$-Statistical Convergence of the Product Time Scale”. Konuralp Journal of Mathematics 8/1 (April 2020), 192-196.
JAMA Basarır M. A Note on the $(\theta ,\varphi )$-Statistical Convergence of the Product Time Scale. Konuralp J. Math. 2020;8:192–196.
MLA Basarır, Metin. “A Note on the $(\theta ,\varphi )$-Statistical Convergence of the Product Time Scale”. Konuralp Journal of Mathematics, vol. 8, no. 1, 2020, pp. 192-6.
Vancouver Basarır M. A Note on the $(\theta ,\varphi )$-Statistical Convergence of the Product Time Scale. Konuralp J. Math. 2020;8(1):192-6.
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