Research Article
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Power series methods and statistical limit superior

Year 2022, Volume: 71 Issue: 3, 752 - 758, 30.09.2022
https://doi.org/10.31801/cfsuasmas.1036338

Abstract

Given a real bounded sequence $x=(x_{j})$ and an infinite matrix $A=(a_{nj})$ Knopp core theorem is equivalent to study the inequality $limsup{Ax} ≤ limsup{x}.$ Recently Fridy and Orhan [6] have considered some variants of this inequality by replacing $limsup{x}$ with statistical limit superior $st - limsup{x}$. In the present paper we examine similar type of inequalities by employing a power series method $P$; a non-matrix sequence-to-function transformation, in place of $A =(a_{nj})$ .

References

  • Belen, C., Yildirim, M., S¨umb¨ul, C., On statistical and strong convergence with respect to a modulus function and a power series method, Filomat, 34(12) (2020), 3981-3993. https://doi.org/10.2298/FIL2012981B
  • Boos, J., Classical and Modern Methods in Summability, Oxford University Press, 2000.
  • Connor, J., The statistical and strong p−Cesaro convergence of sequences, Analysis, 8 (1988), 47-63. https://doi.org/10.1524/anly.1988.8.12.47
  • Demirci, K., Khan, M. K., Orhan, C., Strong and A-statistical comparisons for sequences, J. Math. Anal. Appl., 278 (2003) , 27-33. https://doi.org/10.1016/S0022-247X(02)00456-0
  • Fridy, J. A., On statistical convergence, Analysis, 5 (1985), 301-313. https://doi.org/10.1524/anly.1985.5.4.301
  • Fridy, J. A., Statistical limit points, Proc. Amer. Math. Soc., 118 (1993) , 1187-1192.
  • Fridy, J. A., Orhan, C., Statistical limit superior and limit inferior, Proc. Amer. Math. Soc., 125 (1997) , 3625-3631. Doi: S 0002-9939(97)04000-8.
  • Hardy, G. H., Divergent Series, Oxford Univ. Press, London, 1949.
  • Khan, M. K., Orhan, C., Matrix characterization of A-statistical convergence, J. Math. Anal. Appl., 335 (2007) , 406-417. https://doi.org/10.1016/j.jmaa.2007.01.084
  • Knopp, K., Zur Theorie der Limitierungsverfahren (Erste Mittilung), Math. Zeit., 31 (1930), 97-127.
  • Kolk, E., Matrix summability of statistically convergent sequences, Analysis, 1993.
  • Maddox, I. J., Some analogues of Knopp’s core theorem, Inter. J. Math. and Math. Sci., 2 (1979) , 605-614. https://doi.org/10.1155/S0161171279000454
  • Maddox, I. J., Steinhaus type theorems for summability matrices, Proc. Amer. Math. Soc., 45 (1974), 209-213.
  • Miller, H. I., A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc., 347 (1995) , 1881-1819.
  • Orhan, C., Sublinear functionals and Knopp’s core theorem, Internat. J. Math. and Math. Sci., 2 (1979) , 605-614. https://doi.org/10.1155/S0161171290000680
  • Salat, T., On statistically convergent sequences of real numbers, Math Slovaca, 30(2) (1980), 139-140.
  • Ünver, M., Abel summability in topological spaces, Monatshefte fur Mathematik, 178(4) (2015), 633-643. https://doi.org/10.1007/s00605-014-0717-0
  • Ünver, M., Orhan, C., Statistical convergence with respect to power series methods and applications to approximation theory, Numerical and Functional Analysis and Optimization, 40(5) (2019), 533-547. https://doi.org/10.1080/01630563.2018.1561467
Year 2022, Volume: 71 Issue: 3, 752 - 758, 30.09.2022
https://doi.org/10.31801/cfsuasmas.1036338

Abstract

References

  • Belen, C., Yildirim, M., S¨umb¨ul, C., On statistical and strong convergence with respect to a modulus function and a power series method, Filomat, 34(12) (2020), 3981-3993. https://doi.org/10.2298/FIL2012981B
  • Boos, J., Classical and Modern Methods in Summability, Oxford University Press, 2000.
  • Connor, J., The statistical and strong p−Cesaro convergence of sequences, Analysis, 8 (1988), 47-63. https://doi.org/10.1524/anly.1988.8.12.47
  • Demirci, K., Khan, M. K., Orhan, C., Strong and A-statistical comparisons for sequences, J. Math. Anal. Appl., 278 (2003) , 27-33. https://doi.org/10.1016/S0022-247X(02)00456-0
  • Fridy, J. A., On statistical convergence, Analysis, 5 (1985), 301-313. https://doi.org/10.1524/anly.1985.5.4.301
  • Fridy, J. A., Statistical limit points, Proc. Amer. Math. Soc., 118 (1993) , 1187-1192.
  • Fridy, J. A., Orhan, C., Statistical limit superior and limit inferior, Proc. Amer. Math. Soc., 125 (1997) , 3625-3631. Doi: S 0002-9939(97)04000-8.
  • Hardy, G. H., Divergent Series, Oxford Univ. Press, London, 1949.
  • Khan, M. K., Orhan, C., Matrix characterization of A-statistical convergence, J. Math. Anal. Appl., 335 (2007) , 406-417. https://doi.org/10.1016/j.jmaa.2007.01.084
  • Knopp, K., Zur Theorie der Limitierungsverfahren (Erste Mittilung), Math. Zeit., 31 (1930), 97-127.
  • Kolk, E., Matrix summability of statistically convergent sequences, Analysis, 1993.
  • Maddox, I. J., Some analogues of Knopp’s core theorem, Inter. J. Math. and Math. Sci., 2 (1979) , 605-614. https://doi.org/10.1155/S0161171279000454
  • Maddox, I. J., Steinhaus type theorems for summability matrices, Proc. Amer. Math. Soc., 45 (1974), 209-213.
  • Miller, H. I., A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc., 347 (1995) , 1881-1819.
  • Orhan, C., Sublinear functionals and Knopp’s core theorem, Internat. J. Math. and Math. Sci., 2 (1979) , 605-614. https://doi.org/10.1155/S0161171290000680
  • Salat, T., On statistically convergent sequences of real numbers, Math Slovaca, 30(2) (1980), 139-140.
  • Ünver, M., Abel summability in topological spaces, Monatshefte fur Mathematik, 178(4) (2015), 633-643. https://doi.org/10.1007/s00605-014-0717-0
  • Ünver, M., Orhan, C., Statistical convergence with respect to power series methods and applications to approximation theory, Numerical and Functional Analysis and Optimization, 40(5) (2019), 533-547. https://doi.org/10.1080/01630563.2018.1561467
There are 18 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Nilay Şahin Bayram 0000-0003-3263-8589

Publication Date September 30, 2022
Submission Date December 14, 2021
Acceptance Date March 16, 2022
Published in Issue Year 2022 Volume: 71 Issue: 3

Cite

APA Şahin Bayram, N. (2022). Power series methods and statistical limit superior. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(3), 752-758. https://doi.org/10.31801/cfsuasmas.1036338
AMA Şahin Bayram N. Power series methods and statistical limit superior. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. September 2022;71(3):752-758. doi:10.31801/cfsuasmas.1036338
Chicago Şahin Bayram, Nilay. “Power Series Methods and Statistical Limit Superior”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71, no. 3 (September 2022): 752-58. https://doi.org/10.31801/cfsuasmas.1036338.
EndNote Şahin Bayram N (September 1, 2022) Power series methods and statistical limit superior. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71 3 752–758.
IEEE N. Şahin Bayram, “Power series methods and statistical limit superior”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 71, no. 3, pp. 752–758, 2022, doi: 10.31801/cfsuasmas.1036338.
ISNAD Şahin Bayram, Nilay. “Power Series Methods and Statistical Limit Superior”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71/3 (September 2022), 752-758. https://doi.org/10.31801/cfsuasmas.1036338.
JAMA Şahin Bayram N. Power series methods and statistical limit superior. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71:752–758.
MLA Şahin Bayram, Nilay. “Power Series Methods and Statistical Limit Superior”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 71, no. 3, 2022, pp. 752-8, doi:10.31801/cfsuasmas.1036338.
Vancouver Şahin Bayram N. Power series methods and statistical limit superior. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71(3):752-8.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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