Abstract Korovkin-type theorems in modular spaces and applications

Carlo Bardaro; Antonio Boccuto; Xenofon Dimitriou; Ilaria Mantellini

Open Mathematics (2013)

  • Volume: 11, Issue: 10, page 1774-1784
  • ISSN: 2391-5455

Abstract

top
We prove some versions of abstract Korovkin-type theorems in modular function spaces, with respect to filter convergence for linear positive operators, by considering several kinds of test functions. We give some results with respect to an axiomatic convergence, including almost convergence. An extension to non positive operators is also studied. Finally, we give some examples and applications to moment and bivariate Kantorovich-type operators, showing that our results are proper extensions of the corresponding classical ones.

How to cite

top

Carlo Bardaro, et al. "Abstract Korovkin-type theorems in modular spaces and applications." Open Mathematics 11.10 (2013): 1774-1784. <http://eudml.org/doc/268937>.

@article{CarloBardaro2013,
abstract = {We prove some versions of abstract Korovkin-type theorems in modular function spaces, with respect to filter convergence for linear positive operators, by considering several kinds of test functions. We give some results with respect to an axiomatic convergence, including almost convergence. An extension to non positive operators is also studied. Finally, we give some examples and applications to moment and bivariate Kantorovich-type operators, showing that our results are proper extensions of the corresponding classical ones.},
author = {Carlo Bardaro, Antonio Boccuto, Xenofon Dimitriou, Ilaria Mantellini},
journal = {Open Mathematics},
keywords = {Modular space; Linear operator; Korovkin theorem; Filter convergence; Almost convergence; modular space; linear operator; filter convergence; almost convergence},
language = {eng},
number = {10},
pages = {1774-1784},
title = {Abstract Korovkin-type theorems in modular spaces and applications},
url = {http://eudml.org/doc/268937},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Carlo Bardaro
AU - Antonio Boccuto
AU - Xenofon Dimitriou
AU - Ilaria Mantellini
TI - Abstract Korovkin-type theorems in modular spaces and applications
JO - Open Mathematics
PY - 2013
VL - 11
IS - 10
SP - 1774
EP - 1784
AB - We prove some versions of abstract Korovkin-type theorems in modular function spaces, with respect to filter convergence for linear positive operators, by considering several kinds of test functions. We give some results with respect to an axiomatic convergence, including almost convergence. An extension to non positive operators is also studied. Finally, we give some examples and applications to moment and bivariate Kantorovich-type operators, showing that our results are proper extensions of the corresponding classical ones.
LA - eng
KW - Modular space; Linear operator; Korovkin theorem; Filter convergence; Almost convergence; modular space; linear operator; filter convergence; almost convergence
UR - http://eudml.org/doc/268937
ER -

References

top
  1. [1] Agratini O., On statistical approximation in spaces of continuous functions, Positivity, 2009, 13(4), 735–743 http://dx.doi.org/10.1007/s11117-008-3002-4[WoS][Crossref] Zbl1179.41023
  2. [2] Agratini O., Statistical convergence of a non-positive approximation process, Chaos Solitons Fractals, 2011, 44(11), 977–981 http://dx.doi.org/10.1016/j.chaos.2011.08.003[Crossref][WoS] Zbl1278.41010
  3. [3] Altomare F., Korovkin-type theorems and approximation by positive linear operators, Surv. Approx. Theory, 2010, 5, 92–164 Zbl1285.41012
  4. [4] Altomare F., Campiti M., Korovkin-type Approximation Theory and its Applications, de Gruyter Stud. Math., 17, Walter de Gruyter, Berlin, 1994 http://dx.doi.org/10.1515/9783110884586[Crossref] Zbl0924.41001
  5. [5] Anastassiou G.A., Duman O., Towards Intelligent Modeling: Statistical Approximation Theory, Intell. Syst. Ref. Libr., 14, Springer, Berlin, 2011 http://dx.doi.org/10.1007/978-3-642-19826-7[Crossref] Zbl1295.41001
  6. [6] Bardaro C., Boccuto A., Dimitriou X., Mantellini I., Modular filter convergence theorems for abstract sampling-type operators, Appl. Anal. (in press), DOI: 10.1080/00036811.2012.738480 [Crossref] Zbl1286.41003
  7. [7] Bardaro C., Mantellini I., Multivariate moment type operators: approximation properties in Orlicz spaces, J. Math. Inequal., 2008, 2(2), 247–259 http://dx.doi.org/10.7153/jmi-02-22[Crossref] Zbl1152.41308
  8. [8] Bardaro C., Mantellini I., A Korovkin theorem in multivariate modular function spaces, J. Funct. Spaces Appl., 2009, 7(2), 105–120 http://dx.doi.org/10.1155/2009/863153[Crossref] Zbl1195.41021
  9. [9] Bardaro C., Musielak J., Vinti G., Nonlinear Integral Operators and Applications, De Gruyter Ser. Nonlinear Anal. Appl., 9, Walter de Gruyter, Berlin, 2003 http://dx.doi.org/10.1515/9783110199277[Crossref] Zbl1030.47003
  10. [10] Belen C., Yildirim M., Statistical approximation in multivariate modular function spaces, Comment. Math., 2011, 51(1), 39–53 Zbl1291.41005
  11. [11] Boccuto A., Candeloro D., Integral and ideals in Riesz spaces, Inform. Sci., 2009, 179(17), 2891–2902 http://dx.doi.org/10.1016/j.ins.2008.11.001[Crossref] Zbl1185.28016
  12. [12] Boccuto A., Dimitriou X., Modular filter convergence theorems for Urysohn integral operators and applications, Acta Math. Sinica, 2013, 29(6), 1055–1066 http://dx.doi.org/10.1007/s10114-013-1443-6[Crossref][WoS] Zbl1268.41018
  13. [13] Boccuto A., Dimitriou X., Modular convergence theorems for integral operators in the context of filter exhaustiveness and applications, Mediterr. J. Math., 2013, 10(2), 823–842 http://dx.doi.org/10.1007/s00009-012-0199-z[WoS][Crossref] Zbl1266.41017
  14. [14] Borsík J., Šalát T., On F-continuity of real functions, Tatra Mt. Math. Publ., 1993, 2, 37–42 Zbl0788.26004
  15. [15] Demirci K., I-limit superior and limit inferior, Math. Commun., 2001, 6(2), 165–172 Zbl0992.40002
  16. [16] Duman O., Özarslan M.A., Erkuş-Duman E., Rates of ideal convergence for approximation operators, Mediterr. J. Math., 2010, 7(1), 111–121 http://dx.doi.org/10.1007/s00009-010-0031-6[WoS][Crossref] Zbl1200.41022
  17. [17] Gadjiev A.D., Orhan C., Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 2002, 32(1), 129–138 http://dx.doi.org/10.1216/rmjm/1030539612[Crossref] Zbl1039.41018
  18. [18] Karakuş S., Demirci K., Duman O., Statistical approximation by positive linear operators on modular spaces, Positivity, 2010, 14(2), 321–334 http://dx.doi.org/10.1007/s11117-009-0020-9[Crossref][WoS] Zbl1193.41014
  19. [19] Katětov M., Product of filters, Comment. Math. Univ. Carolinae, 1968, 9(1), 173–189 Zbl0155.50301
  20. [20] Komisarski A., Pointwise I-convergence and I-convergence in measure of sequences of functions, J. Math. Anal. Appl., 2008, 340(2), 770–779 http://dx.doi.org/10.1016/j.jmaa.2007.09.016[Crossref] Zbl1139.40002
  21. [21] Korovkin P.P., On convergence of linear positive operators in the spaces of continuous functions, Doklady Akad. Nauk SSSR (N.S.), 1953, 90, 961–964 (in Russian) 
  22. [22] Kostyrko P., Šalát T., Wilczynski W., I-convergence, Real Anal. Exchange, 2000/01, 26(2), 669–685 
  23. [23] Kuratowski K., Topology I–II, Academic Press/PWN, New York-London/Warsaw, 1966/1968 
  24. [24] Lahiri B.K., Das P., I and I*-convergence in topological spaces, Math. Bohem., 2005, 130(2), 153–160 Zbl1111.40001
  25. [25] Lorentz G.G., A contribution to the theory of divergent sequences, Acta Math., 1948, 80, 167–190 http://dx.doi.org/10.1007/BF02393648[Crossref] Zbl0031.29501
  26. [26] Maligranda L., Korovkin theorem in symmetric spaces, Comment. Math. Prace Mat., 1987, 27(1), 135–140 Zbl0635.41030
  27. [27] Mantellini I., Generalized sampling operators in modular spaces, Comment. Math. Prace Mat., 1998, 38, 77–92 Zbl0984.47025
  28. [28] Musielak J., Orlicz Spaces and Modular Spaces, Lecture Notes in Math., 1034, Springer, Berlin, 1983 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.