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f-Statistical Convergence of Double Sequences of Order \(\widetilde{\alpha }\)

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Abstract

In this paper, we present new concepts of f-statistical convergence for double sequences of order \( \widetilde{\alpha }\) and strong f-Cesàro summability for double sequences of order \( \widetilde{\alpha }\) for sequences of (complex or real) numbers. Besides, we give the relationship between the spaces \( w_{\tilde{\alpha },0} ^{2}\left( f\right) , w_{\tilde{\alpha }}^{2}\left( f\right) \) and \( w_{\tilde{\alpha },\infty }^{2}\left( f\right) \). Furthermore, we express the properties of the strong f-Cesàro summability of order \( \widetilde{\beta }\) which is related to strong f-Cesàro summability of order \( \tilde{\alpha }\). Also some relations between f-statistical convergence of order \( \widetilde{\alpha }\) and strong f-Cesàro summability of order \( \widetilde{\alpha }\) are given. The main purpose of this paper is to introduce and examine the concept of f-double statistical convergence of order \( \widetilde{\alpha },\) where f-is an unbounded function and give relations between f-double statistical convergence of order \( \widetilde{\alpha }\) and strong f-Cesàro summability for double sequence of order \( \widetilde{\alpha }\) so as to fill up the existing gaps in the literature.

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References

  1. Zygmund A (1997) Trigonometric series, vol I, II. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  2. Steinhaus H (1951) Sur la convergence ordinaire et la convergence asymptotique. Colloq Math 2:73–74

    Article  MathSciNet  Google Scholar 

  3. Fast H (1951) Sur la convergence statistique. Colloq Math 2:241–244

    Article  MathSciNet  Google Scholar 

  4. Schoenberg IJ (1959) The integrability of certain functions and related summability methods. Am Math Mon 66:361–375

    Article  MathSciNet  Google Scholar 

  5. Fridy JA (1985) On statistical convergence. Analysis 5:301–313

    Article  MathSciNet  Google Scholar 

  6. Connor JS (1988) The statistical and strong \( p\)-Cesaro convergence of sequences. Analysis 8(1–2):47–63

    MathSciNet  MATH  Google Scholar 

  7. Maddox IJ (1988) Statistical convergence in a locally convex space. Math Proc Camb Philos Soc 104(1):141–145

    Article  MathSciNet  Google Scholar 

  8. Rath D, Tripathy BC (1996) Matrix maps on sequence spaces associated with sets of integers. Indian J Pure Appl Math 27(2):197–206

    MathSciNet  MATH  Google Scholar 

  9. Tripathy BC (1998) On statistical convergence. Proc Est Acad Sci Phys Math 47(4):299–303

    MathSciNet  MATH  Google Scholar 

  10. Tripathy BC (1997) Matrix transformation between some classes of sequences. J Math Anal Appl 206(2):448–450

    Article  MathSciNet  Google Scholar 

  11. Šalát T (1980) On statistically convergent sequences of real numbers. Math Slovaca 30:139–150

    MathSciNet  MATH  Google Scholar 

  12. Móricz F (2003) Statistical convergence of multiple sequences. Arch Math 81:82–89

    Article  MathSciNet  Google Scholar 

  13. Niven I, Zuckerman HS (1960) An introduction to the theory of numbers. Wiley, New York

    MATH  Google Scholar 

  14. Gadjiev AD, Orhan C (2002) Some approximation theorems via statistical convergence. Rocky Mt. J Math 32(1):129–138

    Article  MathSciNet  Google Scholar 

  15. Çolak R (2010) Statistical convergence of order \( \alpha \). In: Mursaleen M (ed) Modern methods in analysis and its applications. Anamaya Pub., New Delhi, pp 121–129

    Google Scholar 

  16. Altin Y, Altinok H, Çolak R (2015) Statistical convergence of order \( \alpha \) for difference sequences. Quaest Math 38(4):505–514

    Article  MathSciNet  Google Scholar 

  17. Et M, Çolak R, Altin Y (2014) Strongly almost summable sequences of order \( \alpha \). Kuwait J Sci 41(2):35–47

    MathSciNet  MATH  Google Scholar 

  18. Mursaleen M, Osama HHE (2003) Statistical convergence of double sequences. J Math Anal Appl 288:223–231

    Article  MathSciNet  Google Scholar 

  19. Tripathy BC, Sarma B (2008) Statistically convergent difference double sequence spaces. Acta Math Sin (Engl Ser) 24(5):737–742

    Article  MathSciNet  Google Scholar 

  20. Tripathy BC (2003) Statistically convergent double sequences. Tamkang J Math 34(3):231–237

    Article  MathSciNet  Google Scholar 

  21. Mursaleen M, Belen C (2014) On statistical lacunary summability of double sequences. Filomat 28(2):231–239

    Article  MathSciNet  Google Scholar 

  22. Bhunia S, Das P, Pal SK (2012) Restricting statistical convergence. Acta Math Hungar 134(1–2):153–161

    Article  MathSciNet  Google Scholar 

  23. Çolak R, Altin Y (2013) Statistical convergence of double sequences of order \( \widetilde{\alpha }\). J Funct Spaces Appl 5:682823

    MathSciNet  MATH  Google Scholar 

  24. Mohiuddine SA, Alotaibi A, Mursaleen M (2012) Statistical convergence of double sequences in locally solid Riesz spaces. Abstr Appl Anal 2012:719729

    MathSciNet  MATH  Google Scholar 

  25. Mursaleen M, Mohiuddine SA (2014) Convergence methods for double sequences and applications. Springer, Berlin

    Book  Google Scholar 

  26. Pringsheim A (1900) Zur Ttheorie der zweifach unendlichen Zahlenfolgen. Math Ann 53:289–321

    Article  MathSciNet  Google Scholar 

  27. Nakano H (1953) Concave modulars. J Math Soc Jpn 5:29–49

    Article  MathSciNet  Google Scholar 

  28. Ruckle WH (1973) FK spaces in which the sequence of coordinate vectors is bounded. Can J Math 25:973–978

    Article  MathSciNet  Google Scholar 

  29. Maddox IJ (1986) Sequence spaces defined by a modulus. Math Proc Camb Philos Soc 100:161–166

    Article  MathSciNet  Google Scholar 

  30. Aizpuru A, Listán-García MC, Rambla-Barreno F (2014) Density by moduli and statistical convergence. Quaest Math 37(4):525–530

    Article  MathSciNet  Google Scholar 

  31. Bhardwaj VK, Dhawan S (2015) \( f\) -statistical convergence of order \( \alpha \) and strong Cesàro summability of order \( \alpha \) with respect to a modulus. J Inequal Appl 332:14

    MATH  Google Scholar 

  32. Aizpuru A, Listán-García MC, Rambla-Barreno F (2012) Double density by moduli and statistical convergence. Bull Belg Math Soc Simon Stevin 19(4):663–673

    Article  MathSciNet  Google Scholar 

  33. Maddox IJ (1987) Inclusions between \(FK-\)spaces and Kuttner’s theorem. Math Proc Camb Philos Soc 101(3):523–527

    Article  MathSciNet  Google Scholar 

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  1. Department of Mathematics, Firat University, 23119, Elazığ, Turkey

    Birgül Torgut & Yavuz Altin

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  1. Birgül Torgut
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  2. Yavuz Altin
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Torgut, B., Altin, Y. f-Statistical Convergence of Double Sequences of Order \(\widetilde{\alpha }\). Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 90, 803–808 (2020). https://doi.org/10.1007/s40010-019-00629-0

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