Research Article
BibTex RIS Cite

A survey on recent results in Korovkin’s approximation theory in modular spaces

Year 2021, Volume: 4 Issue: 1, 48 - 60, 01.03.2021
https://doi.org/10.33205/cma.804697

Abstract

In this paper we give a survey about recent versions of Korovkin-type theorems for modular function spaces, a class which includes $L^p$, Orlicz, Musielak-Orlicz spaces and many others. We consider various kinds of modular convergence, using certain summability processes, like triangular matrix statistical convergence, and filter convergence (which are generalizations of the statistical convergence). Finally, wwe consider an abstract axiomatic convergence which includes the previous ones and even almost convergence, which is not generated by any filter, as we show by an example.

Supporting Institution

Gruppo Nazionale per l'Analisi Matematica e Applicazioni (GNAMPA)'' of the ``Instituto nazionale di alta matematica (INDAM)'' `

Project Number

not applicable

Thanks

Dedicated to Professor Francesco Altomare on occasion of his 70th Birthday

References

  • O. Agratini: On statistical approximation in spaces of continuous functions, Positivity, 13 (4) (2009), 735–743.
  • F. Altomare: Korovkin-type theorems and approximation by positive linear operators, Surv. Approx. Theory, 5 (2010), 92–164.
  • F. Altomare, M. Campiti: Korovkin-type approximation theory and its applications, Walter de Gruyter, Berlin, New York, (1994).
  • G. A. Anastassiou, O. Duman: Towards Intelligent Modeling: Statistical Approximation Theory, Intelligent System Reference Library 14, Springer-Verlag, Berlin, Heidelberg, New York, (2011).
  • C. Bardaro, A. Boccuto, K. Demirci, I. Mantellini and S. Orhan: Triangular A-statistical approximation by double sequences of positive linear operators, Results. Math., 68 (2015), 271–291.
  • C. Bardaro, A. Boccuto, K. Demirci, I. Mantellini and S. Orhan: Korovkin-type theorems for modular Ψ-A-statistical convergence, J. Function Spaces, 2015, Article ID 160401.
  • C. Bardaro, A. Boccuto, X. Dimitriou and I. Mantellini: Abstract Korovkin-type theorems in modular spaces and applications, Cent. Eur. J. Math., 11 (10) (2013), 1774–1784.
  • C. Bardaro, I. Mantellini: Korovkin theorem in modular spaces, Comment. Math., 47 (2) (2007), 239–253.
  • C. Bardaro, I. Mantellini: A Korovkin theorem in multivariate modular spaces, J. Function Spaces Appl., 7 (2) (2009), 105–120.
  • C. Bardaro, J. Musielak and G. Vinti: Nonlinear integral operators and applications, De Gruyter Series in Nonlinear Analysis and Appl., Berlin, Vol. 9, (2003).
  • C. Belen, M. Yildirim: Statistical approximation in multivariate modular function spaces, Comment. Math., 51 (1) (2011), 39-53.
  • H. Berens, G. G. Lorentz: Theorems of Korovkin type for positive linear operators on Banach lattices, in: Approximation Theory (Proc. Internat. Sympos. Univ. Texas,Austin,Texas (1973), 1–30; Academic Press, New York, (1973).
  • S. N. Bernstein, Demonstration du théorème de Weierstrass fondéee sur le calcul de probabilities, Com. of the Kharkov Math. Soc., 13 (1912), 1–2.
  • A. Boccuto, D. Candeloro: Integral and ideals in Riesz spaces, Inf. Sci., 179 (2009), 647–660.
  • A. Boccuto, K. Demirci and S. Yildiz: Abstract Korovkin-type theorems in the filter setting with respect to relative uniform convergence, Turkish J. Math, 44 (2020), 1238–1249.
  • A. Boccuto, X. Dimitriou: Convergence theorems for lattice group-valued measures, Bentham Science Publ., Sharjah, United Arab Emirates, (2015).
  • A. Boccuto, X. Dimitriou: Korovkin-type theorems for abstract modular convergence, Results Math., 69 (2016), 477–495.
  • H. Bohman, On approximation of continuous and of analytic functions, Arkiv Math., 2 (3) (1952), 43–56.
  • J. Borsík, T. Šalát: On F-continuity of real functions, Tatra Mt. Math. Publ., 2 (1993), 37–42.
  • P. L. Butzer, R. J. Nessel: Fourier Analysis and Approximation I, Academic Press, New York, London, (1971).
  • E. Briem: Convergence of sequences of positive operators on Lp-spaces, in: Proc. of the Nineteenth Nordic Congress of Mathematicians,Reykjavik, (1984), 126–131, Visindafel. Isl., XLIV, Icel. Math. Soc. Reykjavik, 1985.
  • K. Demirci: I-limit superior and limit inferior, Math. Commun., 6 (2) (2001), 165–172.
  • K. Demirci, A. Boccuto, S. Yildiz and F. Dirik: Relative uniform convergence of a sequence of functions at a point and Korovkin-type approximation theorems, Positivity, 24 (2020), 1-11.
  • K. Demirci, K. F. Dirik: A Korovkin-type approximation theorem for double sequences of positive linear operators of two variables in A-statistical sense, Bull. Korean Math. Soc., 47 (4) (2010), 825–837.
  • R. A. DeVore, G. G. Lorentz: Constructive Approximation, Grund. Math. Wiss. 303, Springer Verlag, Berlin, Heidelberg, New York, (1993).
  • K. Donner: Korovkin theorems in Lp-spaces, J. Funct. Anal., 42 (1) (1981), 12–28.
  • O. Duman, M. Khan and C. Orhan: A-statistical convergence of approximation operators, Math. Ineq. Appl., 6 (4) (2003) 689–699.
  • H. Fast: Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244.
  • A. D. Gadjiev: The convergence problem for a sequence of positive linear operators on unbounded sets, and theorems analogous to that of P.P. Korovkin, (Russian), Dokl. Akad. Nauk SSSR, 218 (5) (1974), 1001–1004.
  • A. D. Gadjiev: Theorems of the type of P.P. Korovkin’s Theorems, Mat. Zametki, 20 (5) (1976), 781–786.
  • S. Karaku¸s, K. Demirci and O. Duman: Equi-statistical convergence of positive linear operators, J. Math. Anal. Appl., 339 (2008), 1065–1072.
  • W. Kitto, D. E. Wulbert: Korovkin approximations in Lp-spaces, Pacific J. Math., 63 (1) (1976), 153–167.
  • P. P. Korovkin: On convergence of linear positive operators in the spaces of continuous functions (Russian), Doklady Akad. Nauk. S.S.S.R., 90 (1953), 961–964.
  • P. P. Korovkin: Linear operators and approximation theory, Hindustan, New Delhi, (1960).
  • P. Kostyrko, T. Salat and W. Wilkzynski: I-convergence, Real Anal. Exchange, 26 (2) (2000/2001), 669–685.
  • W. M. Kozlowski: Modular Function Spaces, Pure Appl. Math., Marcel Dekker, New York, Basel, 1988.
  • L. Maligranda: Korovkin theorem in symmetric spaces, Comment. Math. Prace Mat., 27 (1) (1987), 135–140.
  • I. Mantellini: Generalized sampling operators in modular spaces, Comment. Math., 38 (1998), 77–92.
  • H. I. Miller: A-statistical convergence of subsequence of double sequences, Boll. U.M.I. 8 (2007), 727-739.
  • J. Musielak: Orlicz Spaces and Modular Spaces, Lecture Notes in Math., 1034, Springer-Verlag, Berlin, Heidelberg, New York, (1983).
  • H. Nakano: Modulared Semi-Ordered Linear Spaces, Maruzen Co., Ltd, Tokyo, (1950).
  • P. F. Renaud: A Korovkin theorem for abstract Lebesgue spaces, J. Approx. Theory, 102 (1) (2000), 13–20.
  • P. M. Soardi: On quantitative Korovkin’s theorem in multivariate Orlicz spaces, Math. Japonica, 48 (1998), 205–212.
  • H. Steinhaus: Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951), 73–74.
Year 2021, Volume: 4 Issue: 1, 48 - 60, 01.03.2021
https://doi.org/10.33205/cma.804697

Abstract

Project Number

not applicable

References

  • O. Agratini: On statistical approximation in spaces of continuous functions, Positivity, 13 (4) (2009), 735–743.
  • F. Altomare: Korovkin-type theorems and approximation by positive linear operators, Surv. Approx. Theory, 5 (2010), 92–164.
  • F. Altomare, M. Campiti: Korovkin-type approximation theory and its applications, Walter de Gruyter, Berlin, New York, (1994).
  • G. A. Anastassiou, O. Duman: Towards Intelligent Modeling: Statistical Approximation Theory, Intelligent System Reference Library 14, Springer-Verlag, Berlin, Heidelberg, New York, (2011).
  • C. Bardaro, A. Boccuto, K. Demirci, I. Mantellini and S. Orhan: Triangular A-statistical approximation by double sequences of positive linear operators, Results. Math., 68 (2015), 271–291.
  • C. Bardaro, A. Boccuto, K. Demirci, I. Mantellini and S. Orhan: Korovkin-type theorems for modular Ψ-A-statistical convergence, J. Function Spaces, 2015, Article ID 160401.
  • C. Bardaro, A. Boccuto, X. Dimitriou and I. Mantellini: Abstract Korovkin-type theorems in modular spaces and applications, Cent. Eur. J. Math., 11 (10) (2013), 1774–1784.
  • C. Bardaro, I. Mantellini: Korovkin theorem in modular spaces, Comment. Math., 47 (2) (2007), 239–253.
  • C. Bardaro, I. Mantellini: A Korovkin theorem in multivariate modular spaces, J. Function Spaces Appl., 7 (2) (2009), 105–120.
  • C. Bardaro, J. Musielak and G. Vinti: Nonlinear integral operators and applications, De Gruyter Series in Nonlinear Analysis and Appl., Berlin, Vol. 9, (2003).
  • C. Belen, M. Yildirim: Statistical approximation in multivariate modular function spaces, Comment. Math., 51 (1) (2011), 39-53.
  • H. Berens, G. G. Lorentz: Theorems of Korovkin type for positive linear operators on Banach lattices, in: Approximation Theory (Proc. Internat. Sympos. Univ. Texas,Austin,Texas (1973), 1–30; Academic Press, New York, (1973).
  • S. N. Bernstein, Demonstration du théorème de Weierstrass fondéee sur le calcul de probabilities, Com. of the Kharkov Math. Soc., 13 (1912), 1–2.
  • A. Boccuto, D. Candeloro: Integral and ideals in Riesz spaces, Inf. Sci., 179 (2009), 647–660.
  • A. Boccuto, K. Demirci and S. Yildiz: Abstract Korovkin-type theorems in the filter setting with respect to relative uniform convergence, Turkish J. Math, 44 (2020), 1238–1249.
  • A. Boccuto, X. Dimitriou: Convergence theorems for lattice group-valued measures, Bentham Science Publ., Sharjah, United Arab Emirates, (2015).
  • A. Boccuto, X. Dimitriou: Korovkin-type theorems for abstract modular convergence, Results Math., 69 (2016), 477–495.
  • H. Bohman, On approximation of continuous and of analytic functions, Arkiv Math., 2 (3) (1952), 43–56.
  • J. Borsík, T. Šalát: On F-continuity of real functions, Tatra Mt. Math. Publ., 2 (1993), 37–42.
  • P. L. Butzer, R. J. Nessel: Fourier Analysis and Approximation I, Academic Press, New York, London, (1971).
  • E. Briem: Convergence of sequences of positive operators on Lp-spaces, in: Proc. of the Nineteenth Nordic Congress of Mathematicians,Reykjavik, (1984), 126–131, Visindafel. Isl., XLIV, Icel. Math. Soc. Reykjavik, 1985.
  • K. Demirci: I-limit superior and limit inferior, Math. Commun., 6 (2) (2001), 165–172.
  • K. Demirci, A. Boccuto, S. Yildiz and F. Dirik: Relative uniform convergence of a sequence of functions at a point and Korovkin-type approximation theorems, Positivity, 24 (2020), 1-11.
  • K. Demirci, K. F. Dirik: A Korovkin-type approximation theorem for double sequences of positive linear operators of two variables in A-statistical sense, Bull. Korean Math. Soc., 47 (4) (2010), 825–837.
  • R. A. DeVore, G. G. Lorentz: Constructive Approximation, Grund. Math. Wiss. 303, Springer Verlag, Berlin, Heidelberg, New York, (1993).
  • K. Donner: Korovkin theorems in Lp-spaces, J. Funct. Anal., 42 (1) (1981), 12–28.
  • O. Duman, M. Khan and C. Orhan: A-statistical convergence of approximation operators, Math. Ineq. Appl., 6 (4) (2003) 689–699.
  • H. Fast: Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244.
  • A. D. Gadjiev: The convergence problem for a sequence of positive linear operators on unbounded sets, and theorems analogous to that of P.P. Korovkin, (Russian), Dokl. Akad. Nauk SSSR, 218 (5) (1974), 1001–1004.
  • A. D. Gadjiev: Theorems of the type of P.P. Korovkin’s Theorems, Mat. Zametki, 20 (5) (1976), 781–786.
  • S. Karaku¸s, K. Demirci and O. Duman: Equi-statistical convergence of positive linear operators, J. Math. Anal. Appl., 339 (2008), 1065–1072.
  • W. Kitto, D. E. Wulbert: Korovkin approximations in Lp-spaces, Pacific J. Math., 63 (1) (1976), 153–167.
  • P. P. Korovkin: On convergence of linear positive operators in the spaces of continuous functions (Russian), Doklady Akad. Nauk. S.S.S.R., 90 (1953), 961–964.
  • P. P. Korovkin: Linear operators and approximation theory, Hindustan, New Delhi, (1960).
  • P. Kostyrko, T. Salat and W. Wilkzynski: I-convergence, Real Anal. Exchange, 26 (2) (2000/2001), 669–685.
  • W. M. Kozlowski: Modular Function Spaces, Pure Appl. Math., Marcel Dekker, New York, Basel, 1988.
  • L. Maligranda: Korovkin theorem in symmetric spaces, Comment. Math. Prace Mat., 27 (1) (1987), 135–140.
  • I. Mantellini: Generalized sampling operators in modular spaces, Comment. Math., 38 (1998), 77–92.
  • H. I. Miller: A-statistical convergence of subsequence of double sequences, Boll. U.M.I. 8 (2007), 727-739.
  • J. Musielak: Orlicz Spaces and Modular Spaces, Lecture Notes in Math., 1034, Springer-Verlag, Berlin, Heidelberg, New York, (1983).
  • H. Nakano: Modulared Semi-Ordered Linear Spaces, Maruzen Co., Ltd, Tokyo, (1950).
  • P. F. Renaud: A Korovkin theorem for abstract Lebesgue spaces, J. Approx. Theory, 102 (1) (2000), 13–20.
  • P. M. Soardi: On quantitative Korovkin’s theorem in multivariate Orlicz spaces, Math. Japonica, 48 (1998), 205–212.
  • H. Steinhaus: Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951), 73–74.
There are 44 citations in total.

Details

Primary Language English
Subjects Applied Mathematics
Journal Section Articles
Authors

Ilaria Mantellını 0000-0002-3731-659X

Antonio Boccuto This is me 0000-0003-3795-8856

Carlo Bardaro 0000-0002-7567-4005

Project Number not applicable
Publication Date March 1, 2021
Published in Issue Year 2021 Volume: 4 Issue: 1

Cite

APA Mantellını, I., Boccuto, A., & Bardaro, C. (2021). A survey on recent results in Korovkin’s approximation theory in modular spaces. Constructive Mathematical Analysis, 4(1), 48-60. https://doi.org/10.33205/cma.804697
AMA Mantellını I, Boccuto A, Bardaro C. A survey on recent results in Korovkin’s approximation theory in modular spaces. CMA. March 2021;4(1):48-60. doi:10.33205/cma.804697
Chicago Mantellını, Ilaria, Antonio Boccuto, and Carlo Bardaro. “A Survey on Recent Results in Korovkin’s Approximation Theory in Modular Spaces”. Constructive Mathematical Analysis 4, no. 1 (March 2021): 48-60. https://doi.org/10.33205/cma.804697.
EndNote Mantellını I, Boccuto A, Bardaro C (March 1, 2021) A survey on recent results in Korovkin’s approximation theory in modular spaces. Constructive Mathematical Analysis 4 1 48–60.
IEEE I. Mantellını, A. Boccuto, and C. Bardaro, “A survey on recent results in Korovkin’s approximation theory in modular spaces”, CMA, vol. 4, no. 1, pp. 48–60, 2021, doi: 10.33205/cma.804697.
ISNAD Mantellını, Ilaria et al. “A Survey on Recent Results in Korovkin’s Approximation Theory in Modular Spaces”. Constructive Mathematical Analysis 4/1 (March 2021), 48-60. https://doi.org/10.33205/cma.804697.
JAMA Mantellını I, Boccuto A, Bardaro C. A survey on recent results in Korovkin’s approximation theory in modular spaces. CMA. 2021;4:48–60.
MLA Mantellını, Ilaria et al. “A Survey on Recent Results in Korovkin’s Approximation Theory in Modular Spaces”. Constructive Mathematical Analysis, vol. 4, no. 1, 2021, pp. 48-60, doi:10.33205/cma.804697.
Vancouver Mantellını I, Boccuto A, Bardaro C. A survey on recent results in Korovkin’s approximation theory in modular spaces. CMA. 2021;4(1):48-60.