Advertisement

Some Theorems of Approximation Theory in Weighted Smirnov Classes with Variable Exponent

  • Ahmet TesticiEmail author
Article
  • 22 Downloads

Abstract

Let \({ G\subset {\mathbb {C}} }\) be a Jordan domain with rectifiable Dini smooth boundary \(\varGamma \). In this work, we investigate approximation properties of matrix transforms constructed via Faber series in weighted Smirnov classes with variable exponent. Moreover, direct and inverse theorems of approximation theory in weighted Smirnov classes with variable exponent are proved and some results related to constructive characterization in generalized Lipschitz classes are obtained.

Keywords

Muckenhoupt weights Matrix transforms Weighted variable exponent Smirnov classes Direct and inverse theorems Faber series Faber operators 

Mathematics Subject Classification

30E10 41A10 41A30 

Notes

Acknowledgements

The author would like to thank the referees for their precious and constructive suggestions and comments to improve the paper. This work was supported by TUBITAK Grant 114F422: Approximation Problems In The Variable Exponent Lebesgue Spaces.

References

  1. 1.
    Akgun, R.: Polynomial approximation of functions in weighted Lebesgue and Smirnov spaces with nonstandard growth. Georgian Math. J. 18, 203–235 (2011)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alper, S. Y.: Approximation in the mean of analytic functions of class \(E^{p}\). (Russian), In: Investigations on the modern problems of the function theory of a complex variable, Gos. Izdat. Fiz. Mat. Lit., 273-286, (1960)Google Scholar
  3. 3.
    Andersson, J.E.: On the degree of polynomial approximation in E\(^{p}\left( D\right) \). J. Approx. Theory 19, 61–68 (1977)CrossRefGoogle Scholar
  4. 4.
    Cruz-Uribe, D.V., Diening, L., Hästö, P.: The maximal operator on weighted variable Lebesgue spaces. Fract. Calc. Appl. Anal. 14(3), 361–374 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cruz-Uribe, D.V., Fiorenza, A.: Variable Lebesgue Spaces Foundation and Harmonic Analysis. Birkhäsuser, Basel (2013)CrossRefGoogle Scholar
  6. 6.
    Cruz Uribe, D., Wang, D.L.: Extrapolation and weighted norm inequalities in the variable Lebesgue spaces. Trans. Am. Math. Soc. 369(2), 1205–1235 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    David, G.: Operateurs integraux singulers sur certaines courbes du plan complexe. Ann. Sci. Ecole Norm. Sup. 4(17), 157–189 (1984)CrossRefGoogle Scholar
  8. 8.
    Diening, L., Harjulehto, P, Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Springer, New York (2011)CrossRefGoogle Scholar
  9. 9.
    Diening, L.: Maximal function on Musileak-Orlicz space and generalized Lebesgue spaces. Bull. Sci. Math. 129(8), 657–700 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Duren, P.L.: Theory of \(H^{p}\) Spaces. Academic Press, New York (1970)zbMATHGoogle Scholar
  11. 11.
    Goluzin G. M.: Geometric Theory of Functions of a Complex Variable. Translation of Mathematical Monographs 26, AMS, Rhode Island (1969)Google Scholar
  12. 12.
    Guven, A.: Trigonometric approximation by matrix transforms in \(L^{p\left( x\right) }\) space. Anal. Appl. 10(1), 47–65 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Guven, A., Israfilov, D.M.: Trigonometric approximation in generalized Lebesgue spaces \( L^{p\left( x\right) }\). J. Math. Inequal. 4(2), 285–299 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Havin, V.P.: Boundary properties of integrals of Cauchy type and conjugate harmonic functions of in regions with rectifiable boundary (Russian). Math. Sb. 68(110), 499–517 (1965)Google Scholar
  15. 15.
    Israfilov, D.M.: Approximation by \(p-\) faber polynomials in the weighted Smirnov class \(E^{p}\left( G,\omega \right) \) and the Bieberbach polynomials. Constr. Approx. 17, 335–351 (2001)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Israfilov, D.M., Guven, A.: Approximation in weighted Smirnov classes. East J. Approx. 11(1), 91–102 (2005)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Israfilov, D.M., Akgun, R.: Approximation in weighted Smirnov–Orlicz classes. J. Math. Kyoto Univ. 46(4), 755–770 (2006)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Israfilov, D.M., Kokilashvili, V.M., Samko, S.: Approximation in weighted Lebesgue and Smirnov spaces with variable exponents. Proc. A. Razmadze Math. Institute. 143, 25–35 (2007)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Israfilov, D.M., Testici, A.: Approximation in weighted Smirnov classes. Complex Var. Elliptic Equ. 60(1), 45–58 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Israfilov, D.M., Testici, A.: Approximation in Smirnov classes with variable exponent. Complex Var. Elliptic Equ. 60(9), 1243–1253 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Israfilov, D.M., Testici, A.: Approximation by matrix transform in weighted Lebesgue spaces with variable exponent. Results Math. 73(1), 8 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Israfilov, D.M., Testici, A.: Multiplier and approximation theorems in Smirnov classes with variable exponent. Turk. J. Math. 42, 1442–1456 (2018)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Israfilov, D.M., Testici, A.: Some inverse theorem of approximation theory in weighted Lebesgue space with variable exponent. Anal. Math. 44(4), 475–492 (2018)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Muckenhoupt, B.: Weighted norm inequalities of the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker, New York (1991)zbMATHGoogle Scholar
  26. 26.
    Rudin, W.: Real and Complex Analysis. Mc-Graw-Hill, Singapore (1987)zbMATHGoogle Scholar
  27. 27.
    Sharapudinov, I. I.: Approximation of functions in \( L_{2\pi }^{p\left( x\right) }\) by trigonometric polynomials. Izvestiya RAN : Ser. Math. 77(2), 197-224 (2013); English transl., Izvestiya : Mathematics 77(2), 407-434 (2013)Google Scholar
  28. 28.
    Sharapudinov, I.I.: On direct and inverse theorems of approximation theory in variable Lebesgue and Sobolev spaces. Azerb. J. Math. 4(1), 55–72 (2014)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Sharapudinov, I. I.: Some questions of approximation theory in the Lebesgue spaces with variable exponent. Vladikavkaz (2012)Google Scholar
  30. 30.
    Suetin, P. K.: Series of Faber Polynomials. Moscow : Nauka, Newyork: Gordon and Breach Science Publishers, Singapore (1998)Google Scholar
  31. 31.
    Warschawski, S.: Über das randverhalten der ableitung der abbildungsfunktionen bei konformer abbildung. Math. Z. 35, 321–456 (1932)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsBalikesir UniversityBalikesirTurkey

Personalised recommendations