Mediterranean Journal of Mathematics

, Volume 10, Issue 2, pp 823–842 | Cite as

Modular Convergence Theorems for Integral Operators in the Context of Filter Exhaustiveness and Applications

  • Antonio BoccutoEmail author
  • Xenofon Dimitriou


We prove some modular convergence theorems for nonlinear Urysohn-type integral operators, applying filter convergence of sequences of functions. We give some applications to Mellin operators, including moment, Mellin-Poisson-Cauchy and Mellin-Gauss-Weierstrass operators. We show that our results are proper extensions of the classical ones, and we pose an open problem.

Mathematics Subject Classification (2010)

Primary 41A35 Secondary 46E30 47G10 


filter convergence filter exhaustiveness integral operator moment operator Mellin operator Mellin-Gauss-Weierstrass operator Mellin-Poisson-Cauchy operator modular modular convergence filter singularity 


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  1. 1.
    C. Bardaro and I. Mantellini, Linear integral operators with homogeneuos kernel: approximation properties in modular spaces. Application to Mellin-type convolution operators and to some classes of fractional operators, Applied Math. Rev., Vol. I, World Scientific Publ. (2000), 45–67.Google Scholar
  2. 2.
    Bardaro C., Mantellini I.: On approximation properties of Urysohn integral operators. Int. J. Pure Appl. Math. 3(2), 129–148 (2002)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bardaro C., Mantellini I.: Uniform modular integrability and convergence properties for a class of Urysohn integral operators in function spaces. Math. Slovaca 56(4), 465–482 (2006)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bardaro C., Mantellini I.: Pointwise convergence theorems for nonlinear Mellin convolution operators. Int. J. Pure Appl. Math. 27(4), 431–447 (2006)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bardaro C., Mantellini I.: A note on the Voronovskaja theorem for nonlinear Mellin-F´ejer convolution operators. Appl. Math. Letters 24, 2064–2067 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bardaro C., Musielak J., Vinti G.: Nonlinear Integral Operators and Applications. de Gruyter, Berlin (2003)zbMATHCrossRefGoogle Scholar
  7. 7.
    A. Boccuto and X. Dimitriou, Modular filter convergence theorems for Urysohn integral operators and applications, Acta Math. Sin. (Engl. Ser.) (2012), to appear.Google Scholar
  8. 8.
    Boccuto A., Dimitriou X., Papanastassiou N.: Brooks-Jewett-type theorems for the pointwise ideal convergence of measures with values in (l)-groups. Tatra Mt. Math. Publ. 49, 17–26 (2011)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Boccuto A., Dimitriou X., Papanastassiou N., Wilczyński W.: Ideal exhaustiveness, continuity and (α)-convergence for lattice group-valued functions. Int. J. Pure Appl. Math. 70(2), 211–227 (2011)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Butzer P.L., Jansche S.: A direct approach to the Mellin transform. J. Fourier Anal. Appl. 3, 325–375 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Kadets V., Leonov A.: Dominated Convergence and Egorov Theorems for Filter Convergence. J. Math. Phys., Anal., Geom. 3(2), 196–212 (2007)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Katěetov M.: Products of filters. Comment. Math. Univ. Carolin. 9, 173–189 (1968)MathSciNetGoogle Scholar
  13. 13.
    Kuratowski K.: Topology Vols I and II. Academic Press, New York-London (1966/1968)Google Scholar
  14. 14.
    R. G. Mamedov, The Mellin transform and approximation theory (Russian), ELM, Baku, 1991.Google Scholar
  15. 15.
    Mantellini I.: Generalized sampling operators in modular spaces. Comment. Math. Prace Mat. 38, 77–92 (1998)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Moll V.H.: The integrals in Gradshteyn and Ryzhik. VI. The beta function. Sci. Ser. A Math. Sci. (N. S.) 16, 9–24 (2008)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Musielak J.: Orlicz Spaces and Modular spaces, Lecture Notes in Math. 1034. Springer-Verlag, New York (1983)Google Scholar
  18. 18.
    Rainville E.D.: Special Functions. The Macmillan Co., New York (1960)zbMATHGoogle Scholar
  19. 19.
    Steinhaus H.: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 2, 73–74 (1951)MathSciNetGoogle Scholar

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© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaPerugiaItaly
  2. 2.Department of MathematicsUniversity of AthensAthensGreece

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