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Korovkin-Type Theorems for Statistically Convergent Sequences of Monotone and Sublinear Operators

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Abstract

In this paper, we study the statistical convergence almost everywhere, in measure and in \(L^{p}\) in Korovkin-type theorem for monotone and sublinear operators. Our results are illustrated by a series of concrete examples.

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References

  1. Agratini, O.: Statistical convergence applied to Korovkin-type approximation theory. WSEAS Trans. Math. 16, 183–186 (2017)

    Google Scholar 

  2. Altomare, F.: Korovkin-type theorems and positive operators. Surv. Approx. Theory. 6, 92–164 (2010)

    MATH  Google Scholar 

  3. Altomare, F., Campiti, M.: Korovkin-Type Approximation Theory and Its Applications. de Gruyter Studies in Mathematics, 17, Berlin (1994, reprinted 2011)

  4. Anastassiou, G.A.: Approximations by sublinear operators. Acta Math. Univ. Comenian. (N.S.) 87(2), 237–250 (2018)

    MathSciNet  MATH  Google Scholar 

  5. Anastassiou, G.A.: Quantitative Approximations. Chapman and Hall/CRC, Boca Raton (2001). https://doi.org/10.1201/9781482285796

    Book  MATH  Google Scholar 

  6. Bede, B., Coroianu, L., Gal, S.G.: Approximation by Max-Product Type Operators. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-319-34189-7

    Book  MATH  Google Scholar 

  7. Cárdenas-Morales, D., Garancho, P.: Optimal linear approximation under general statistical convergence. In: Advances in Summability and Approximation Theory. Springer, 191–202 (2018). https://doi.org/10.1007/978-981-13-3077-3_12

  8. Connor, J.: The statistical and strong p-Cesaro convergence of sequences. Analysis 8, 47–63 (1988). https://doi.org/10.1524/anly.1988.8.12.47

    Article  MathSciNet  MATH  Google Scholar 

  9. Dellacherie, C.: Quelques Commentaires sur les Prolongements de Capacités. Séminaire Probabilités V, Strasbourg. Lecture Notes in Math. 191. Springer, Berlin (1970)

  10. Dirik, F.: Weighted statistical convergence of Bögel continuous functions by positive linear operator. In: Advances in Summability and Approximation Theory. Springer, pp. 181–189 (2012). https://doi.org/10.1007/978-981-13-3077-3_11

  11. Duman, O.: A-statistical convergence of sequences of convolution operators. Taiwan. J. Math. 12(2), 523–536 (2008). https://doi.org/10.11650/twjm/1500574172

    Article  MathSciNet  MATH  Google Scholar 

  12. Duman, O.: Statistical approximation theorems by k-positive linear operators. Arch. Math. (Basel) 86(6), 569–576 (2006). https://doi.org/10.1007/s00013-006-1476-5

    Article  MathSciNet  MATH  Google Scholar 

  13. Duman, O., Khan, M.K., Orhan, C.: A-statistical convergence of approximating operators. Math. Ineq. Appl. 6(4), 689–699 (2003). https://doi.org/10.7153/mia-06-62

    Article  MathSciNet  MATH  Google Scholar 

  14. Erdös, P., Jenenbaüm, G.: Sur les densities de certaines suites d’entiers. Proc. Lond. Math. Soc. 59(3), 417–438 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Fridy, J.A.: On statistical convergence. Analysis 5, 301–313 (1985). https://doi.org/10.1524/anly.1985.5.4.301

    Article  MathSciNet  MATH  Google Scholar 

  17. Fridy, J.A., Orhan, C.: Lacunary statistical summability. J. Math. Anal. Appl. 173, 497–504 (1993). https://doi.org/10.1006/jmaa.1993.1082

    Article  MathSciNet  MATH  Google Scholar 

  18. Fridy, J.A., Khan, M.K.: Tauberian theorems via statistical convergence. J. Math. Anal. Appl. 228, 73–95 (1998). https://doi.org/10.1006/jmaa.1998.6118

    Article  MathSciNet  MATH  Google Scholar 

  19. Gadjev, A.D.: The convergence problem for a sequence of positive linear operators on unbounded sets, and theorems analogous to that of P.P. Korovkin. Soviet Math. Dokl. 15, 1433–1436 (1974). http://mi.mathnet.ru/eng/dan/v218/i5/p1001

  20. Gadjev, A.D., Orhan, C.: Some approximation theorems via statistical convergence. Rocky Mt. J. Math. 32(1), 129–138 (2002). https://doi.org/10.1216/rmjm/1030539612

    Article  MathSciNet  MATH  Google Scholar 

  21. Gal, S.G., Niculescu, C.P.: A nonlinear extension of Korovkin’s theorem. Mediterr. J. Math. (2020). https://doi.org/10.1007/s00009-020-01583-7

    Article  MathSciNet  MATH  Google Scholar 

  22. Gal, S.G., Niculescu, C.P.: A note on the Choquet type operators. Aequ. Math. 95, 433–447 (2021). https://doi.org/10.1007/s00010-021-00803-z

    Article  MathSciNet  MATH  Google Scholar 

  23. Gal, S.G., Niculescu, C.P.: Choquet operators associated to vector capacities. J. Math. Anal. Appl. 479, 903–925 (2019). https://doi.org/10.1016/j.jmaa.2021.125153

    Article  MathSciNet  Google Scholar 

  24. Gal, S.G., Niculescu, C.P.: Nonlinear versions of Korovkin’s abstract theorems. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas. (2022). https://doi.org/10.1007/s13398-021-01187-0

    Article  MATH  Google Scholar 

  25. Gal, S.G., Niculescu, C.P.: Korovkin type theorems for weakly nonlinar and monotone operators. Submitted

  26. Grabisch, M.: Set Functions, Games and Capacities in Decision Making. Springer, New York (2016). https://doi.org/10.1007/978-3-319-30690-2

  27. Iancu, T.I.: Statistical Korovkin-type theorem for monotone and sublinear operators. submitted

  28. Lorentz, G.G.: Bernstein Polynomials. Chelsea Publishing Company, New York (1986)

    MATH  Google Scholar 

  29. Karakus, S., Demirci, K., Duman, O.: Statistical approximation by positive linear operators on modular spaces. Positivity 14(2), 321–334 (2010). https://doi.org/10.1007/s11117-009-0020-9

    Article  MathSciNet  MATH  Google Scholar 

  30. Korovkin, P.P.: On convergence of linear positive operators in the space of continuous functions.(Russian). Doklady Akad. Nauk. SSSR. (NS) 90, 961–964 (1953)

    Google Scholar 

  31. Korovkin, P.P.: Linear Operators and Approximation Theory. Fitzmatgiz, Moscow (1959) (in Russian) [English translation, Hindustan Publ. Corp., Delhi (1960)]

  32. Miller, H.I.: A measure theoretical subsequence characterization of statistical convergence. Trans. Am. Math. Soc. 347, 1811–1819 (1995). https://doi.org/10.1090/S0002-9947-1995-1260176-6

    Article  MathSciNet  MATH  Google Scholar 

  33. Niculescu, C.P., Popovici, F.: A note on the behavior of integrable functions at infinity. J. Math. Anal. Appl. 381(2), 742–747 (2011). https://doi.org/10.1016/j.jmaa.2011.03.045

    Article  MathSciNet  MATH  Google Scholar 

  34. Niculescu, C.P., Popovici, F.: The behavior at infinity of an integrable function. Expo. Math. 30(3), 277–282 (2012). https://doi.org/10.1016/j.exmath.2012.03.002

    Article  MathSciNet  MATH  Google Scholar 

  35. Niculescu, C.P., Popovici, F.: The asymptotic behavior of integrable functions. Real Anal. Exchange 38(1), 157–168 (2013). https://doi.org/10.14321/realanalexch.38.1.0157

    Article  MathSciNet  MATH  Google Scholar 

  36. Pehlivan, S., Mamedov, M.A.: Statistical cluster points and turnpike. Optimization 48, 93–106 (2000). https://doi.org/10.1080/02331930008844495

    Article  MathSciNet  MATH  Google Scholar 

  37. Sakaoglu, I., Unver, M.: Statistical approximation for multivariable integrable functions. Miskolc Math. Notes 13(2), 485–491 (2012). https://doi.org/10.18514/MMN.2012.336

  38. Salat, T.: On statistically convergent sequences of real numbers. Math. Slovaca 30, 139–150 (1980)

    MathSciNet  MATH  Google Scholar 

  39. Schoenberg, I.J.: The integrability of certain functions and related summability methods. Am. Math. Mon. 66, 361–375 (1959). https://doi.org/10.1080/00029890.1959.11989303

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang Z., Klir, G.J.: Generalized Measure Theory. Springer, New York (2009). https://doi.org/10.1007/978-0-387-76852-6

  41. Zygmund, A.: Trigonometric Series, 2nd edn. Cambridge University Press, Cambridge (1979)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Oradea, University Street No. 1, 410087, Oradea, Romania

    Sorin G. Gal

  2. Academy of Romanian Scientists, Splaiul Independentei nr. 54, 050094, Bucharest, Romania

    Sorin G. Gal & Ionut T. Iancu

Authors
  1. Sorin G. Gal
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  2. Ionut T. Iancu
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Correspondence to Sorin G. Gal.

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Gal, S.G., Iancu, I.T. Korovkin-Type Theorems for Statistically Convergent Sequences of Monotone and Sublinear Operators. Bull. Malays. Math. Sci. Soc. 46, 79 (2023). https://doi.org/10.1007/s40840-023-01471-9

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