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ρ-STATISTICAL CONVERGENCE DEFINED BY A MODULUS FUNCTION OF ORDER (α,β)

Year 2022, Volume: 4 Issue: 1, 15 - 23, 30.04.2022
https://doi.org/10.47087/mjm.1092599

Abstract

The concept of strong w [ρ, f, q] −summability of order (α, β) for sequences of complex (or real) numbers is introduced in this work. We also give some inclusion relations between the sets of ρ-statistical convergence of order (α, β), strong wαβ [ρ, f, q] −summability and strong wαβ (ρ, q) −summability.

References

  • Y. Altın, Properties of some sets of sequences defined by a modulus function, Acta Math. Sci. Ser. B Engl. Ed. 29(2) (2009), 427–434.
  • N. D. Aral and H. Sengul Kandemir, I-Lacunary statistical Convergence of order β of difference sequences of fractional order, Facta Universitatis (NIS) Ser. Math. Inform. 36(1) (2021), 43–55.
  • A. Caserta, Di M. Giuseppe and L. D. R. Koˇcinac, Statistical convergence in function spaces, Abstr. Appl. Anal. 2011, Art. ID 420419, 11 pp.
  • J. S. Connor, The Statistical and Strong p-Cesaro Convergence of Sequences, Analysis, 8, pp. (1988), 47-63.
  • H. Cakallı, H. S ̧engu ̈l Kandemir and M. Et, ρ-statistical convergence of order beta, American Institute of Physics., https://doi.org/10.1063/1.5136141.
  • H. Cakallı, A study on statistical convergence, Funct. Anal. Approx. Comput. 1(2) (2009), 19–24.
  • R. Colak, Statistical convergence of order α, Modern Methods in Analysis and Its Applications, New Delhi, India: Anamaya Pub, 2010: 121–129.
  • M. Et, Generalized Ces`aro difference sequence spaces of non-absolute type involving lacunary sequences, Appl. Math. Comput. 219(17) (2013), 9372–9376.
  • M. Et, M. Cınar and H. Sengul, On ∆m−asymptotically deferred statistical equivalent sequences of order α, Filomat, 33(7) (2019), 1999–2007.
  • M. Et and H. Sengul, Some Cesaro-type summability spaces of order α and lacunary statistical convergence of order α, Filomat, 28(8) (2014), 1593–1602.
  • M. Et, Strongly almost summable difference sequences of order m defined by a modulus, Studia Sci. Math. Hungar. 40(4) (2003), 463–476.
  • M. Et, Spaces of Ces`aro difference sequences of order r defined by a modulus function in a locally convex space, Taiwanese J. Math. 10(4) (2006), 865-879.
  • H. Fast, Sur La Convergence Statistique, Colloq. Math., 2, pp. (1951), 241–244.
  • J. Fridy, On Statistical Convergence, Analysis, 5, pp. (1985), 301-313.
  • A. D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32(1) (2002), 129-138.
  • M. Isık, Strongly almost (w, λ, q)−summable sequences, Math. Slovaca 61(5) (2011), 779– 788.
  • E. Kolk, The statistical convergence in Banach spaces, Acta Comment. Univ. Tartu, 928 (1991), 41-52.
  • I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, 1970.
  • I. J. Maddox, Sequence spaces defined by a modulus, Math. Proc. Camb. Philos. Soc, 1986, 100:161-166.
  • A. K. Gaur and M. Mursaleen, Difference sequence spaces defined by a sequence of moduli, Demonstratio Math. 31(2) (1998), 275–278.
  • H. Nakano, Modulared sequence spaces, Proc. Japan Acad. 27 (1951), 508–512.
  • F. Nuray and E. Savas, Some new sequence spaces defined by a modulus function, Indian J. Pure Appl. Math. 24(11) (1993), 657–663.
  • S. Pehlivan and B. Fisher, Lacunary strong convergence with respect to a sequence of mod- ulus functions, Comment. Math. Univ. Carolin. 36(1) (1995), 69-76.
  • S. Pehlivan and B. Fisher, Some sequence spaces defined by a modulus, Mathematica Slo- vaca, 45(3) (1995), 275–280.
  • W. H. Ruckle, FK spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math. 25 (1973), 973–978.
  • T. Salat, On Statistically Convergent Sequences of Real Numbers, Math. Slovaca. 30 (1980), 139-150.
  • I. J. Schoenberg, The Integrability of Certain Functions and Related Summability Methods, Amer. Math. Monthly 66 (1959), 361–375.
  • H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951) 73-74.
  • H. Sengul, Some Ces`aro summability spaces defined by a modulus function of order (α,β), Commun. Fac .Sci. Univ. Ank. Series A1 66(2) (2017), 80–90.
  • H. Sengul, On Wijsman I−lacunary statistical equivalence of order (η,μ), J. Inequal. Spec. Funct. 9(2) (2018), 92–101.
  • H. Sengul and M. Et, f−lacunary statistical convergence and strong f−lacunary summability of order α, Filomat 32(13) (2018), 4513–4521.
  • H. Sengul and M. Et, On (λ,I)−statistical convergence of order α of sequences of function, Proc. Nat. Acad. Sci. India Sect. A 88(2) (2018), 181–186.
  • H.Sengul and O. Koyun, On (λ,A)−statistical convergence of order α, Commun.Fac.Sci. Univ. Ank. Ser. A1. Math. Stat. 68(2) (2019), 2094–2103.
  • H. Sengul and M. Et, On lacunary statistical convergence of order α, Acta Math. Sci. Ser. B Engl. Ed. 34(2) (2014), 473–482.
Year 2022, Volume: 4 Issue: 1, 15 - 23, 30.04.2022
https://doi.org/10.47087/mjm.1092599

Abstract

References

  • Y. Altın, Properties of some sets of sequences defined by a modulus function, Acta Math. Sci. Ser. B Engl. Ed. 29(2) (2009), 427–434.
  • N. D. Aral and H. Sengul Kandemir, I-Lacunary statistical Convergence of order β of difference sequences of fractional order, Facta Universitatis (NIS) Ser. Math. Inform. 36(1) (2021), 43–55.
  • A. Caserta, Di M. Giuseppe and L. D. R. Koˇcinac, Statistical convergence in function spaces, Abstr. Appl. Anal. 2011, Art. ID 420419, 11 pp.
  • J. S. Connor, The Statistical and Strong p-Cesaro Convergence of Sequences, Analysis, 8, pp. (1988), 47-63.
  • H. Cakallı, H. S ̧engu ̈l Kandemir and M. Et, ρ-statistical convergence of order beta, American Institute of Physics., https://doi.org/10.1063/1.5136141.
  • H. Cakallı, A study on statistical convergence, Funct. Anal. Approx. Comput. 1(2) (2009), 19–24.
  • R. Colak, Statistical convergence of order α, Modern Methods in Analysis and Its Applications, New Delhi, India: Anamaya Pub, 2010: 121–129.
  • M. Et, Generalized Ces`aro difference sequence spaces of non-absolute type involving lacunary sequences, Appl. Math. Comput. 219(17) (2013), 9372–9376.
  • M. Et, M. Cınar and H. Sengul, On ∆m−asymptotically deferred statistical equivalent sequences of order α, Filomat, 33(7) (2019), 1999–2007.
  • M. Et and H. Sengul, Some Cesaro-type summability spaces of order α and lacunary statistical convergence of order α, Filomat, 28(8) (2014), 1593–1602.
  • M. Et, Strongly almost summable difference sequences of order m defined by a modulus, Studia Sci. Math. Hungar. 40(4) (2003), 463–476.
  • M. Et, Spaces of Ces`aro difference sequences of order r defined by a modulus function in a locally convex space, Taiwanese J. Math. 10(4) (2006), 865-879.
  • H. Fast, Sur La Convergence Statistique, Colloq. Math., 2, pp. (1951), 241–244.
  • J. Fridy, On Statistical Convergence, Analysis, 5, pp. (1985), 301-313.
  • A. D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32(1) (2002), 129-138.
  • M. Isık, Strongly almost (w, λ, q)−summable sequences, Math. Slovaca 61(5) (2011), 779– 788.
  • E. Kolk, The statistical convergence in Banach spaces, Acta Comment. Univ. Tartu, 928 (1991), 41-52.
  • I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, 1970.
  • I. J. Maddox, Sequence spaces defined by a modulus, Math. Proc. Camb. Philos. Soc, 1986, 100:161-166.
  • A. K. Gaur and M. Mursaleen, Difference sequence spaces defined by a sequence of moduli, Demonstratio Math. 31(2) (1998), 275–278.
  • H. Nakano, Modulared sequence spaces, Proc. Japan Acad. 27 (1951), 508–512.
  • F. Nuray and E. Savas, Some new sequence spaces defined by a modulus function, Indian J. Pure Appl. Math. 24(11) (1993), 657–663.
  • S. Pehlivan and B. Fisher, Lacunary strong convergence with respect to a sequence of mod- ulus functions, Comment. Math. Univ. Carolin. 36(1) (1995), 69-76.
  • S. Pehlivan and B. Fisher, Some sequence spaces defined by a modulus, Mathematica Slo- vaca, 45(3) (1995), 275–280.
  • W. H. Ruckle, FK spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math. 25 (1973), 973–978.
  • T. Salat, On Statistically Convergent Sequences of Real Numbers, Math. Slovaca. 30 (1980), 139-150.
  • I. J. Schoenberg, The Integrability of Certain Functions and Related Summability Methods, Amer. Math. Monthly 66 (1959), 361–375.
  • H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951) 73-74.
  • H. Sengul, Some Ces`aro summability spaces defined by a modulus function of order (α,β), Commun. Fac .Sci. Univ. Ank. Series A1 66(2) (2017), 80–90.
  • H. Sengul, On Wijsman I−lacunary statistical equivalence of order (η,μ), J. Inequal. Spec. Funct. 9(2) (2018), 92–101.
  • H. Sengul and M. Et, f−lacunary statistical convergence and strong f−lacunary summability of order α, Filomat 32(13) (2018), 4513–4521.
  • H. Sengul and M. Et, On (λ,I)−statistical convergence of order α of sequences of function, Proc. Nat. Acad. Sci. India Sect. A 88(2) (2018), 181–186.
  • H.Sengul and O. Koyun, On (λ,A)−statistical convergence of order α, Commun.Fac.Sci. Univ. Ank. Ser. A1. Math. Stat. 68(2) (2019), 2094–2103.
  • H. Sengul and M. Et, On lacunary statistical convergence of order α, Acta Math. Sci. Ser. B Engl. Ed. 34(2) (2014), 473–482.
There are 34 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Nazlım Deniz Aral

Publication Date April 30, 2022
Acceptance Date May 5, 2022
Published in Issue Year 2022 Volume: 4 Issue: 1

Cite

APA Aral, N. D. (2022). ρ-STATISTICAL CONVERGENCE DEFINED BY A MODULUS FUNCTION OF ORDER (α,β). Maltepe Journal of Mathematics, 4(1), 15-23. https://doi.org/10.47087/mjm.1092599
AMA Aral ND. ρ-STATISTICAL CONVERGENCE DEFINED BY A MODULUS FUNCTION OF ORDER (α,β). Maltepe Journal of Mathematics. April 2022;4(1):15-23. doi:10.47087/mjm.1092599
Chicago Aral, Nazlım Deniz. “ρ-STATISTICAL CONVERGENCE DEFINED BY A MODULUS FUNCTION OF ORDER (α,β)”. Maltepe Journal of Mathematics 4, no. 1 (April 2022): 15-23. https://doi.org/10.47087/mjm.1092599.
EndNote Aral ND (April 1, 2022) ρ-STATISTICAL CONVERGENCE DEFINED BY A MODULUS FUNCTION OF ORDER (α,β). Maltepe Journal of Mathematics 4 1 15–23.
IEEE N. D. Aral, “ρ-STATISTICAL CONVERGENCE DEFINED BY A MODULUS FUNCTION OF ORDER (α,β)”, Maltepe Journal of Mathematics, vol. 4, no. 1, pp. 15–23, 2022, doi: 10.47087/mjm.1092599.
ISNAD Aral, Nazlım Deniz. “ρ-STATISTICAL CONVERGENCE DEFINED BY A MODULUS FUNCTION OF ORDER (α,β)”. Maltepe Journal of Mathematics 4/1 (April 2022), 15-23. https://doi.org/10.47087/mjm.1092599.
JAMA Aral ND. ρ-STATISTICAL CONVERGENCE DEFINED BY A MODULUS FUNCTION OF ORDER (α,β). Maltepe Journal of Mathematics. 2022;4:15–23.
MLA Aral, Nazlım Deniz. “ρ-STATISTICAL CONVERGENCE DEFINED BY A MODULUS FUNCTION OF ORDER (α,β)”. Maltepe Journal of Mathematics, vol. 4, no. 1, 2022, pp. 15-23, doi:10.47087/mjm.1092599.
Vancouver Aral ND. ρ-STATISTICAL CONVERGENCE DEFINED BY A MODULUS FUNCTION OF ORDER (α,β). Maltepe Journal of Mathematics. 2022;4(1):15-23.

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