Skip to main content
SpringerLink
Log in

Relative weighted almost convergence based on fractional-order difference operator in multivariate modular function spaces

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In the present paper, we introduce the concept of relative weighted almost convergence and its weighted statistical extensions in multivariate modular function spaces based on a new type fractional-order double difference operator \(\nabla ^{a,b,c}_h\). We first define the concepts of weighted almost \(\nabla \)-statistical convergence and statistical weighted almost \(\nabla \)-convergence of double sequences. By using the notion of relative uniform convergence involving a scale function \(\sigma \), we introduce modular relative weighted almost statistical convergence and modular relative statistical weighted almost convergence of double sequences. We then obtain some inclusion relations between these proposed methods and provide some counterexamples that show that these are non-trivial and proper extensions of the existing literature on this topic. Moreover, we apply the relative statistical weighted almost convergence of a double sequence of positive linear operators to prove some Korovkin-type theorems in multivariate modular spaces by considering several kinds of test functions. Finally, we present a non-trivial application to generalized Boolean sum (GBS) operators of bivariate generalized Bernstein–Durrmeyer operators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Banach, S.: Theorie des operations lineaires. Monografie Mat. PWN, Warsaw (1932)

    MATH  Google Scholar 

  2. Lorentz, G.G.: A contribution to the theory of divergent series. Acta Math. 80(1), 167–190 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  3. King, J.P.: Almost summable sequences. Proc. Am. Math. Soc. 17(6), 1219–1225 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  4. Schaefer, P.: Almost convergent and almost summable sequences. Proc. Am. Math. Soc. 20(1), 51–54 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  5. Maddox, I.J.: Spaces of strongly summable sequences. Q. J. Math. 18(1), 345–355 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  6. Candan, M.: Almost convergence and double sequential band matrix. Acta Math. Sci. 34(2), 354–366 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Başar, F., Kirişçi, M.: Almost convergence and generalized difference matrix. Comput. Math. Appl. 61, 602–611 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kadak, U., Mohiuddine, S.A.: Generalized statistically almost convergence based on the difference operator which includes the (p, q)-Gamma function and related approximation theorems. Results Math. 73, 9 (2018). https://doi.org/10.1007/s00025-018-0789-6

    Article  MathSciNet  MATH  Google Scholar 

  9. Mohiuddine, S.A., Alotaibi, A.: Weighted almost convergence and related infinite matrices. J. Inequal. Appl. 2018, 15 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kirişci, M., Kadak, U.: The method of almost convergence with operator of the form fractional order and applications. J. Nonlinear Sci. Appl. 10, 828–842 (2017)

    Article  MathSciNet  Google Scholar 

  11. Moricz, F., Rhoades, B.E.: Almost convergence of double sequences and strong regularity of summability matrices. Math. Proc. Camb. Philos. Soc. 104(2), 283–294 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. Pringsheim, A.: Zur theorie der zweifach unendlichen Zahlenfolgen. Math. Ann. 53, 289–321 (1900)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Çakan, C., Altay, B.: Statistical boundedness and statistical core of double sequences. J. Math. Anal. Appl. 317, 690–697 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fridy, J.A., Orhan, C.: Statistical limit superior and inferior. Proc. Am. Math. Soc. 125, 3625–3631 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Moore, E.H.: An Introduction to a Form of General Analysis, The New Haven Mathematical Colloquium. Yale University Press, New Haven (1910)

    MATH  Google Scholar 

  19. Chittenden, E.W.: On the limit functions of sequences of continuous functions converging relatively uniformly. Trans. AMS 20, 179–184 (1919)

    Article  MathSciNet  MATH  Google Scholar 

  20. Demirci, K., Orhan, S.: Statistically relatively uniform convergence of positive linear operators. Results Math. 69, 359–367 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Musielak, J.: Orlicz spaces and modular spaces, Lecture Notes in mathematics, p. 1034. Springer, Berlin (1983)

    Book  Google Scholar 

  22. Bardaro, C., Musielak, J., Vinti, G.: Nonlinear Integral Operators and Applications, de Gruyter Series in Nonlinear Analysis and Appl., vol. 9, p. 201. Walter de Gruyter Publ, Berlin (2003)

    MATH  Google Scholar 

  23. Musielak, J.: Nonlinear approximation in some modular function spaces I. Math. Jpn. 38, 83–90 (1993)

    MathSciNet  MATH  Google Scholar 

  24. Kadak, U.: On relative weighted summability in modular function spaces and associated approximation theorems. Positivity 21(4), 1593–1614 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Orhan, S., Demirci, K.: Statistical approximation by double sequences of positive linear operators on modular spaces. Positivity 19, 23–36 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yılmaz, B., Demirci, K., Orhan, S.: Relative modular convergence of positive linear operators. Positivity. (2016). https://doi.org/10.1007/s11117-015-0372-2

  27. Bardaro, C., Mantellini, I.: Approximation properties in abstract modular spaces for a class of general sampling-type operators. Appl. Anal. 85(4), 383–413 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Bardaro, C., Mantellini, I.: A Korovkin theorem in multivariate modular function spaces. J. Funct. Spaces Appl. 7(2), 105–120 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Bardaro, C., Boccuto, A., Dimitriou, X., Mantellini, I.: Abstract Korovkin-type theorems in modular spaces and applications. Cent. Eur. J. Math. 11(10), 1774–1784 (2013). https://doi.org/10.2478/s11533-013-0288-7

    MathSciNet  MATH  Google Scholar 

  30. Chapman, S.: On non-integral orders of summability of series and integrals. Proc. Lond. Math. Soc. 2, 369–409 (1911)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kadak, U., Baliarsingh, P.: On certain Euler difference sequence spaces of fractional order and related dual properties. J. Nonlinear Sci. Appl. 8, 997–1004 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kadak, U.: Generalized weighted invariant mean based on fractional difference operator with applications to approximation theorems for functions of two variables. Results Math. 72(3), 1181–1202 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kadak, U.: Generalized statistical convergence based on fractional order difference operator and its applications to approximation theorems, Iran J. Sci. Technol. Trans. Sci. (2017). https://doi.org/10.1007/s40995-017-0400-0

  34. Kadak, U.: Weighted statistical convergence based on generalized difference operator involving (p, q)-gamma function and its applications to approximation theorems. J. Math. Anal. Appl. 448, 1633–1650 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  35. Baliarsingh, P.: On a fractional difference operator. Alex. Eng. J. 55(2), 1811–1816 (2016)

    Article  Google Scholar 

  36. Aktuğlu, H.: Korovkin type approximation theorems proved via \(\alpha \beta \)-statistical convergence. J. Comput. Appl. Math. 259, 174–181 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mursaleen, M.: Almost strongly regular matrices and a core theorem for double sequences. J. Math. Anal. Appl. 293(2), 523–531 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  38. Savaş, E.: Double almost statistical convergence of order \(\alpha \). Adv. Differ. Equ. 2013, 62 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  39. Korovkin, P.P.: On convergence of linear positive operators in the spaces of continuous functions (Russian). Doklady Akad. Nauk. S.S.S.R. 90, 961–964 (1953)

    Google Scholar 

  40. Boccuto, A., Dimitriou, X.: Korovkin-type theorems for abstract modular convergence. Results Math. 69, 477–495 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. Kajla, A., Miclauş, D.: Blending type approximation by GBS operators of generalized Bernstein–Durrmeyer type. Results Math 73, 1 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  42. Chen, X., Tan, J., Liu, Z., Xie, J.: Approximation of functions by a new family of generalized Bernstein operators. J. Math. Anal. Appl. 450, 244–261 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  43. Bogel, K.: Uber die mehrdimensionale Differentiation. Jber. Deutsch. Math.-Verein. 65, 45–71 (1962)

    MathSciNet  MATH  Google Scholar 

  44. Badea, C., Badea, I., Cottin, C., Gonska, H.H.: Notes on the degree of approximation of \(B\)-continuous and \(B\)-differentiable functions. Approx. Theory Appl. 4, 95–108 (1988)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Gazi University, 06100, Ankara, Turkey

    Uğur Kadak

Authors
  1. Uğur Kadak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Uğur Kadak.

Ethics declarations

Conflict of interest

The author declares that he has no conflict of interest.

Human and animal rights

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kadak, U. Relative weighted almost convergence based on fractional-order difference operator in multivariate modular function spaces. RACSAM 113, 2201–2220 (2019). https://doi.org/10.1007/s13398-018-0613-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13398-018-0613-x

Keywords

Mathematics Subject Classification

Search

Navigation