Uniform Lacunary Statistical Convergence on Time Scales

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E. Yilmaz, S.A. Mohiuddine, Y. Altin, H. Koyunbakan


We introduce (θ,m)-uniform lacunary density of any set and (θ,m)-uniform lacunary statistical convergence on an arbitrary time scale. Moreover, (θ,m)-uniform strongly p-lacunary summability and some inclusion relations about these new concepts are also presented.

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  1. Y. Altin, H. Koyunbakan and E. Yilmaz, Uniform statistical convergence on time scales, J. Appl. Math. 2014 (2014), Art. ID 471437.
  2. V. Baláz, and T. Salát, Uniform density u and corresponding I u -convergence, Math. Commun. 11(1) (2006), 1-7.
  3. C. Belen and S. A. Mohiuddine, Generalized weighted statistical convergence and application, Appl. Math. Comput. 219 (2013), 9821-9826.
  4. M. Bohner and A. Peterson, Dynamic equations on time scales, an introduction with applications, Birkhauser, Boston, 2001.
  5. N. L. Braha, H. M. Srivastava and S. A. Mohiuddine, A Korovkin's type approximation theorem for periodic functions via the statistical summability of the generalized de la Vall ´ ee Poussin mean, Appl. Math. Comput. 228 (2014) 162-169.
  6. 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, Math. Comp. Model. 43 (2006), 194-207.
  7. H. Cakalli, Lacunary statistical convergence in topological groups, Indian J. Pure Appl. Math. 26(2) (1995), 113-119.
  8. J. S. Connor, The statistical and strong p-Cesà ro convergence of sequences, Analysis 8 (1988), 47-63.
  9. J. S. Connor and E. Sava ¸ s, Lacunary statistical and sliding window convergence for measurable functions, Acta Math. Hung. 145(2) (2015), 416-432.
  10. O. H. H. Edely, S. A. Mohiuddine and A. K. Noman, Korovkin type approximation theorems obtained through generalized statistical convergence, Appl. Math. Letters 23 (2010) 1382-1387.
  11. M. Et, Generalized Cesà ro difference sequence spaces of non-absolute type involving lacunary sequences, Appl. Math. Comput. 219(17) (2013), 9372-9376.
  12. M. Et, S. A. Mohiuddine and A. Alotaibi, On λ-statistical convergence and strongly λ-summable functions of order α, J. Inequal. Appl. 2013 (2013), Art. ID 469.
  13. H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241-244.
  14. A. R. Freedman, J. J. Sember and M. Raphael, Some Cesà ro-type summability spaces, Proc. London Math. Soc. 37(3) (1978), 508-520.
  15. J. A. Fridy, On statistical convergence, Analysis 5 (1985), 301-313.
  16. J. A. Fridy and C. Orhan, Lacunary statistical convergence, Pac. J. Math. 160(1) (1993), 43-51.
  17. J. A. Fridy and C. Orhan, Lacunary statistical summability, J. Math. Anal. Appl. 173(2) (1993), 497-504.
  18. T. Gulsen and E. Yilmaz, Spectral theory of Dirac system on time scales, Appl. Anal. 1-11 (2016), Doi:10.1080/00036811.2016.1236923.
  19. G. Sh. Guseinov, Integration on time scales, J. Math. Anal. Appl. 285(1) (2003), 107-227.
  20. B. Hazarika, S. A. Mohiuddine and M. Mursaleen, Some inclusion results for lacunary statistical convergence in locally solid Riesz spaces, Iranian J. Sci. Tech. 38 (A1) (2014), 61-68.
  21. S. Hilger, Analysis on measure chains-A unified approach to continuous and discrete calculus, Results Math. 18 (1990), 18-56.
  22. S. Hilger, Ein Makettenkalkl mit Anwendung auf Zentrumsmannigfaltigkeiten Ph.D. Thesis, Universtat Wurzburg, 1988.
  23. I. J. Maddox, Spaces of strongly summable sequences, Quart. J. Math. 18(1) (1967), 345-355.
  24. S. A. Mohiuddine and M. A. Alghamdi, Statistical summability through lacunary sequence in locally solid Riesz spaces, J. Inequal. Appl. 2012 (2012), Art. ID 225.
  25. S. A. Mohiuddine and M. Aiyub, Lacunary statistical convergence in random 2-normed spaces, Appl. Math. Inform. Sci. 6(3) (2012), 581-585.
  26. S. A. Mohiuddine, A. Alotaibi and M. Mursaleen, Statistical summability (C,1) and a Korovkin type approximation theorem, J. Inequal. Appl. 2012 (2012), Art. ID 172.
  27. S. A. Mohiuddine and Q. M. D. Lohani, On generalized statistical convergence in intuitionistic fuzzy normed space, Chaos Solitons Fract. 42 (2009), 1731-1737.
  28. F. Moricz, Statistical limits of measurable functions, Analysis, 24 (2004), 1-18.
  29. M. Mursaleen and S. A. Mohiuddine, On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, Journal of Computational and Applied Mathematics, 233(2) (2009), 142-149.
  30. F. Nuray, Uniform statistical convergence, Sci. Engineer. J. Firat Univ. 11(3) (1999), 219-222.
  31. F. Nuray and B. Aydin, Strongly summable and statistically convergent functions, Inform. Tech. Valdymas 30(1) (2004), 74-76.
  32. R. A. Raimi, Convergence, density, and Ï„-density of bounded sequences, Proc. Amer. Math. Soc. 14 (1963), 708-712.
  33. D. Rath and B. C. Tripathy, On statistically convergent and statistically Cauchy sequences, Indian J. Pure Appl. Math. 25(4) (1994), 381-386.
  34. E. Sava ¸ s and F. Nuray, On σ-statistically convergence and lacunary σ-statistically convergence, Math. Slovaca 43(3) (1993), 309-315.
  35. M. S. Seyyidoglu and N. O. Tan, A note on statistical convergence on time scale, J. Inequal. Appl. 2012 (2012), Art. ID 219.
  36. M. S. Seyyidoglu and N. O. Tan, On a generalization of statistical cluster and limit points, Adv. Difference Equ. 2015 (2015), Art. ID 55.
  37. B. C. Tripathy, On statistical convergence, Proc. Est. Aca. Sci. Phy. 47(4) (1998), 299-303.
  38. 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 (2013), 57-71.
  39. C. Turan and O. Duman, Convergence Methods on Time Scales, 11th international conference of numerical analysis and Applied Mathematics, AIP Conference Proc. 1558 (2013), 1120-1123.
  40. H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), 73-74.
  41. E. Yilmaz, Y. Altin and H. Koyunbakan, λ-Statistical convergence on time scales, Dyn. Cont. Disc. Impul. Syst. Ser. A: Math. Anal. 23(2016), 69-78.
  42. A. Zygmund, Trigonometrical Series, Monogr. Mat., vol. 5. Warszawa-Lwow 1935.