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Asimptotik Lacunary İstatistiksel ɸ-Denk Üç İndisli Diziler

Year 2021, Volume: 16 Issue: 1, 137 - 146, 27.05.2021
https://doi.org/10.29233/sdufeffd.866055

Abstract

İstatistiksel yakınsaklık kavramı ilk kez, Fast [2] tarafından verilmiştir. Bu kavram, hem uygulamalı matematikte hem de matematiği içeren diğer bilim dallarında önemli rol oynar. Marouf [13] ise 1993’te asimptotik denk dizilerde yeni kavramlar vermiştir. 1980’de yaptığı çalışmada ise Patterson [16] asimptotik denk diziler için istatistiksel benzerlerini sunmuştur. 2006’da Patterson ve Savaş [17] lacunary dizileri kullanarak bu kavramlara yeni bir boyut kazandırmıştır. Diğer taraftan, üç indisli diziler için istatistiksel yakınsaklık kavramı Şahiner vd. [20] tarafından sunulmuştur. Aynı zamanda literatürdeki bazı çalışmalarda, herhangi bir reel dizinin istatistiksel yakınsaklığı mutlak değere göre belirlenir. Reel sayıların mutlak değeri özel bir Orlicz fonksiyonu olarak bilinir [19]. Bu makalenin temel amacı, üç indisli diziler için asimptotik olarak istatistiksel ϕ-denk ve asimptotik olarak lacunary istatistiksel ϕ-denk kavramlarını tanımlamaktır. Belli özel koşul altında Orlicz fonksiyonundan yararlanarak, yeni ispatlar vermek ve yeni kavramları literatüre kazandırmaya çalışmaktır. Ayrıca bu yeni notasyonlar arasındaki ilişkiler de çalışmamızda verilmiştir.

Thanks

Bu çalışmanın yazarı olarak, çalışmanın okunabilirliğinin gelişmesine katkı sağlayan hakemlere teşekkür ederim.

References

  • [1] A. Esi, E. Savaş, “On lacunary statistically convergent triple sequences in probabilistic normed space,” Appl. Math. Inf. Sci., 9 (5), 2529-2534, 2015.
  • [2] H. Fast, “Sur la convergence statistique,” Colloq. Math., 2, 241-244, 1951.
  • [3] J. A. Fridy, “On statistical convergence,” Analysis (Munich), 5, 301-313, 1985.
  • [4] M. Gürdal, “Some types of convergence”, PhD. Thesis, Süleyman Demirel Univ., Isparta, Turkey, 2004.
  • [5] M. Gürdal, M. B. Huban, “On I-convergence of double sequences in the topology induced by random 2-norms,” Mat. Vesnik, 66, 73-83, 2014.
  • [6] M. Gürdal, M. B. Huban, “Statistical convergence and C^*-operator algebras,” Theory Appl. Math. Comput. Sci., 7 (2), 41-50, 2017.
  • [7] M. B. Huban, M. Gürdal, “Wijsman lacunary invariant statistical convergence for triple sequences via Orlicz function,” J. Classical Anal., 2021, in press.
  • [8] M. B. Huban, M. Gürdal, H. Baytürk, “On asymptotically lacunary statistical equivalent triple sequences via ideals and Orlicz function,” Honam Math. J., 2021, in press.
  • [9] M. Gürdal, A. Şahiner, “Extremal I-limit points of double sequences,” Appl. Math. E-Notes, 8, 131-137, 2008.
  • [10] B. Hazarika, V. Kumar, “On asymptotically double lacunary statistical equivalent sequences in ideal context,” J. Inequal. Appl.,543, 1-15, 2013.
  • [11] M. B. Huban, M. Gürdal, E. Savaş, “I-statistical limit superior and I-statistical limit inferior of triple sequences”, 7th International Conference on Recent Advances in Pure and Applied Mathematics (ICRAPAM 2020), Proceeding Book of ICRAPAM (2020), Bodrum/Muğla, 2020, 42-49.
  • [12] J. Li, “Asymptotic equivalence of sequences and summability,” Internat J. Math. Math. Sci., 20 (4), 749-758, 1997.
  • [13] M. Marouf, “Asymptotic equivalence and summability,” Internat J. Math. Math. Sci., 16 (4), 755-762, 1993.
  • [14] M. Mursaleen, O. H. Edely, “Statistical convergence of double sequences,” J. Math. Anal. Appl., 288, 223-231, 2003.
  • [15] A. A. Nabiev, E. Savaş, M. Gürdal, “Statistically localized sequences in metric spaces,” J. Appl. Anal. Comput., 9 (2), 739-746, 2019.
  • [16] R. F. Patterson, “On asymptotically statistically equivalent sequences,” Demonstr. Math., 36 (1), 149-153, 2003.
  • [17] R. F. Patterson, E. Savaş, “On asymptotically lacunary statistical equivalent sequences,” Thai J. Math., 4 (2), 267-272, 2006.
  • [18] I. P. Pobyvanets, “Asymptotic equivalence of some linear transformations defined by a nonnegative matrix and reduced to generalized equivalence on the sense of Cesaro and Abel,” Mat. Fiz., 28 (123), 83-87, 1980.
  • [19] M. M. Rao, Z. D. Ren, Applications of Orlicz spaces (Book style). Marcel Dekker Inc., 2002.
  • [20] A. Şahiner, M. Gürdal, F. K. Düden, “Triple sequences and their statistical convergence,” Selçuk J. Appl. Math., 8 (2), 49-55, 2007.
  • [21] T. Šalát, “On statistically convergent sequences of real numbers,” Math. Slovaca, 30 (2), 139-150, 1980.
  • [22] R. Savaş, “Multiple -statistically convergence via -functions,” Math. Meth. Appl. Sci., doi: 10.1002/mma. 7027, 1-8, 2020.
  • [23] R. Savaş, “Matrix characterization of asymptotically deferred equivalent sequences,”,Ouaest. Math., doi: 10.2989/16073606.2020.1829151, 2020.
  • [24] I. J. Schoenberg, “The integrability of certain functions and related summability methods,” Amer. Math. Monthly., 66, 361-375, 1959.
  • [25] E. Savaş, R. F. Patterson, “An extension asymptotically lacunary statistical equivalent sequences,”Aligarh Bull. Math., 27 (2), 109-113, 2008.
  • [26] U. Yamancı, M. Gürdal, “On lacunary ideal convergence in random n-normed space,” J. Math., Article ID 868457, v. 2013, 8 pages, 2013.
  • [27] U. Yamancı, M. Gürdal, “I-statistical convergence in 2-normed space,” Arab J. Math. Sci., 20 (1), 41-47, 2014.
  • [28] U. Yamancı, M. Gürdal, “On asymptotically generalized statistical equivalent double sequences via ideals,” Electron. J. Math. Anal. Appl., 3 (1), 89-96, 2015.

On Asymptotically Lacunary Statistically ɸ-Equivalent Triple Sequences

Year 2021, Volume: 16 Issue: 1, 137 - 146, 27.05.2021
https://doi.org/10.29233/sdufeffd.866055

Abstract

The concept of statistical convergence was first given by Fast [2]. This concept plays an important role in both applied mathematics and other diciplines that include mathematics. Marouf [13] gave new concepts in asymptotic equivalent series in 1993. In his study in 1980, Patterson [16] presented statistical similarities for asymptotic equivalent sequences. In 2006, Patterson and Savaş [17] add a new dimension to these concepts by using the lacunary series. On the other hand, the concept of statistical convergence for triple sequences is presented by Şahiner et al. in study [20]. Also, in some studies in the literature, the statistical convergence of any real series is determined by absolute value. The absolute value of real numbers is known as a special Orlicz function [19]. The primary goal of this article is to introduce the concepts of asymptotically statistically ϕ-equivalent and asymptotically lacunary statistically ϕ-equivalent triple sequences. Using the Orlicz function under special condition, new proofs are given and new concepts are introduced into the literature. Also, the relationship between these new notations will be given.

References

  • [1] A. Esi, E. Savaş, “On lacunary statistically convergent triple sequences in probabilistic normed space,” Appl. Math. Inf. Sci., 9 (5), 2529-2534, 2015.
  • [2] H. Fast, “Sur la convergence statistique,” Colloq. Math., 2, 241-244, 1951.
  • [3] J. A. Fridy, “On statistical convergence,” Analysis (Munich), 5, 301-313, 1985.
  • [4] M. Gürdal, “Some types of convergence”, PhD. Thesis, Süleyman Demirel Univ., Isparta, Turkey, 2004.
  • [5] M. Gürdal, M. B. Huban, “On I-convergence of double sequences in the topology induced by random 2-norms,” Mat. Vesnik, 66, 73-83, 2014.
  • [6] M. Gürdal, M. B. Huban, “Statistical convergence and C^*-operator algebras,” Theory Appl. Math. Comput. Sci., 7 (2), 41-50, 2017.
  • [7] M. B. Huban, M. Gürdal, “Wijsman lacunary invariant statistical convergence for triple sequences via Orlicz function,” J. Classical Anal., 2021, in press.
  • [8] M. B. Huban, M. Gürdal, H. Baytürk, “On asymptotically lacunary statistical equivalent triple sequences via ideals and Orlicz function,” Honam Math. J., 2021, in press.
  • [9] M. Gürdal, A. Şahiner, “Extremal I-limit points of double sequences,” Appl. Math. E-Notes, 8, 131-137, 2008.
  • [10] B. Hazarika, V. Kumar, “On asymptotically double lacunary statistical equivalent sequences in ideal context,” J. Inequal. Appl.,543, 1-15, 2013.
  • [11] M. B. Huban, M. Gürdal, E. Savaş, “I-statistical limit superior and I-statistical limit inferior of triple sequences”, 7th International Conference on Recent Advances in Pure and Applied Mathematics (ICRAPAM 2020), Proceeding Book of ICRAPAM (2020), Bodrum/Muğla, 2020, 42-49.
  • [12] J. Li, “Asymptotic equivalence of sequences and summability,” Internat J. Math. Math. Sci., 20 (4), 749-758, 1997.
  • [13] M. Marouf, “Asymptotic equivalence and summability,” Internat J. Math. Math. Sci., 16 (4), 755-762, 1993.
  • [14] M. Mursaleen, O. H. Edely, “Statistical convergence of double sequences,” J. Math. Anal. Appl., 288, 223-231, 2003.
  • [15] A. A. Nabiev, E. Savaş, M. Gürdal, “Statistically localized sequences in metric spaces,” J. Appl. Anal. Comput., 9 (2), 739-746, 2019.
  • [16] R. F. Patterson, “On asymptotically statistically equivalent sequences,” Demonstr. Math., 36 (1), 149-153, 2003.
  • [17] R. F. Patterson, E. Savaş, “On asymptotically lacunary statistical equivalent sequences,” Thai J. Math., 4 (2), 267-272, 2006.
  • [18] I. P. Pobyvanets, “Asymptotic equivalence of some linear transformations defined by a nonnegative matrix and reduced to generalized equivalence on the sense of Cesaro and Abel,” Mat. Fiz., 28 (123), 83-87, 1980.
  • [19] M. M. Rao, Z. D. Ren, Applications of Orlicz spaces (Book style). Marcel Dekker Inc., 2002.
  • [20] A. Şahiner, M. Gürdal, F. K. Düden, “Triple sequences and their statistical convergence,” Selçuk J. Appl. Math., 8 (2), 49-55, 2007.
  • [21] T. Šalát, “On statistically convergent sequences of real numbers,” Math. Slovaca, 30 (2), 139-150, 1980.
  • [22] R. Savaş, “Multiple -statistically convergence via -functions,” Math. Meth. Appl. Sci., doi: 10.1002/mma. 7027, 1-8, 2020.
  • [23] R. Savaş, “Matrix characterization of asymptotically deferred equivalent sequences,”,Ouaest. Math., doi: 10.2989/16073606.2020.1829151, 2020.
  • [24] I. J. Schoenberg, “The integrability of certain functions and related summability methods,” Amer. Math. Monthly., 66, 361-375, 1959.
  • [25] E. Savaş, R. F. Patterson, “An extension asymptotically lacunary statistical equivalent sequences,”Aligarh Bull. Math., 27 (2), 109-113, 2008.
  • [26] U. Yamancı, M. Gürdal, “On lacunary ideal convergence in random n-normed space,” J. Math., Article ID 868457, v. 2013, 8 pages, 2013.
  • [27] U. Yamancı, M. Gürdal, “I-statistical convergence in 2-normed space,” Arab J. Math. Sci., 20 (1), 41-47, 2014.
  • [28] U. Yamancı, M. Gürdal, “On asymptotically generalized statistical equivalent double sequences via ideals,” Electron. J. Math. Anal. Appl., 3 (1), 89-96, 2015.
There are 28 citations in total.

Details

Primary Language Turkish
Subjects Mathematical Sciences
Journal Section Makaleler
Authors

Mualla Birgül Huban 0000-0003-2710-8487

Publication Date May 27, 2021
Published in Issue Year 2021 Volume: 16 Issue: 1

Cite

IEEE M. B. Huban, “Asimptotik Lacunary İstatistiksel ɸ-Denk Üç İndisli Diziler”, Süleyman Demirel University Faculty of Arts and Science Journal of Science, vol. 16, no. 1, pp. 137–146, 2021, doi: 10.29233/sdufeffd.866055.