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Journal of Mathematics
Volume 2013 (2013), Article ID 254694, 7 pages
http://dx.doi.org/10.1155/2013/254694
Research Article

Convergence and Characterization of Wavelets Associated with Dilation Matrix in Variable Spaces

Department of Mathematics, Research and Post Graduate Studies, M. M. H. College, Model Town, Ghaziabad, Uttar Pradesh 20/001, India

Received 11 November 2012; Revised 17 January 2013; Accepted 18 January 2013

Academic Editor: Ding-Xuan Zhou

Copyright © 2013 Devendra Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer, “The maximal function on variable Lp spaces,” Annales Academiæ Scientiarum Fennicæ, vol. 28, no. 1, pp. 223–238, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer, “Corrections to: “The maximal function on variable Lp spaces”,” Annales Academiæ Scientiarum Fennicæ, vol. 29, no. 1, pp. 247–249, 2004. View at Google Scholar · View at MathSciNet
  3. L. Diening, “Maximal function on generalized Lebesgue spaces Lp(·),” Mathematical Inequalities & Applications, vol. 7, no. 2, pp. 245–253, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. L. Diening, “Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces,” Bulletin des Sciences Mathématiques, vol. 129, no. 8, pp. 657–700, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  5. A. Nekvinda, “Hardy-Littlewood maximal operator on Lp(x)(),” Mathematical Inequalities & Applications, vol. 7, no. 2, pp. 255–265, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. M. Izuki, “Wavelets and modular inequalities in variable Lp spaces,” Georgian Mathematical Journal, vol. 15, no. 2, pp. 281–293, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. T. S. Kopaliani, “Greediness of the wavelet system in Lp(t)() spaces,” East Journal on Approximations, vol. 14, no. 1, pp. 59–67, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. D. V. Cruz-Uribe, J. M. Martell, and C. Pérez, Weights, Extrapolation and the Theory of Rubio de Francia: Operator Theory: Advances and Applications, vol. 215, Birkhäuser/Springer, Basel, Switzerland, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  9. Y. Meyer, Wavelets and Operators, vol. 37 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1992. View at MathSciNet
  10. G. G. Walter, “Pointwise convergence of wavelet expansions,” Journal of Approximation Theory, vol. 80, no. 1, pp. 108–118, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. S. E. Kelly, M. A. Kon, and L. A. Raphael, “Pointwise convergence of wavelet expansions,” American Mathematical Society, vol. 30, no. 1, pp. 87–94, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. S. E. Kelly, M. A. Kon, and L. A. Raphael, “Local convergence for wavelet expansions,” Journal of Functional Analysis, vol. 126, no. 1, pp. 102–138, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. L. Carleson, “On convergence and growth of partial sums of Fourier series,” Acta Mathematica, vol. 116, pp. 135–157, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. R. A. Hunt, “On the convergence of Fourier series,” in Proceedings of the Conference on Orthogonal Expansions and Their Continuous Analogues, D. T. Haimo, Ed., pp. 235–255, Southern Illinois University Press, 1968.
  15. N. N. Reyes, “Behavior of partial sums of wavelet series,” Journal of Approximation Theory, vol. 103, no. 1, pp. 55–60, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. T. Tao, “On the almost everywhere convergence of wavelet summation methods,” Applied and Computational Harmonic Analysis, vol. 3, no. 4, pp. 384–387, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. P. Manchanda, A. A. S. Mukheimer, and A. H. Siddiqi, “Pointwise convergence of wavelet expansions associated with dilation matrix,” Applicable Analysis, vol. 76, no. 3-4, pp. 301–308, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. I. I. Sharapudinov, “The basis property of the Haar system in the space Lp(t)([0,1]) and the principle of localization in the mean,” Matematicheskiĭ Sbornik, vol. 130 (172), no. 2, pp. 275–283, 1986. View at Google Scholar · View at MathSciNet
  19. I. I. Sharapudinov, “The basis property of the Haar system in the space Lp(t)([0,1]) and the principle of localization in the mean,” Sbornik: Mathematics, vol. 58, no. 2, pp. 279–287, 1987. View at Google Scholar · View at MathSciNet
  20. P. Wojtaszczyk, A Mathematical Introduction to Wavelets, vol. 37 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, UK, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  21. E. Hernández and G. Weiss, A first Course on Wavelets. With a Foreword by Yves Meyer, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  22. D. Kumar, “Convergence of a class of non-orthogonal wavelet expansions in Lp(), 1p,” Panamerican Mathematical Journal, vol. 19, no. 4, pp. 61–70, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. O. Kováčik and J. Rákosník, “On spaces Lp(x) and WK,p(x),” Czechoslovak Mathematical Journal, vol. 41 (116), no. 4, pp. 592–618, 1951. View at Google Scholar
  24. A. K. Lerner, “On modular inequalities in variable Lp spaces,” Archiv der Mathematik, vol. 85, no. 6, pp. 538–543, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. D. Cruz-Uribe, A. Fiorenza, J. M. Martell, and C. Pérez, “The boundedness of classical operators on variable Lp spaces,” Annales Academiæ Scientiarum Fennicæ, vol. 31, no. 1, pp. 239–264, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. D. E. Edmunds, J. Lang, and A. Nekvinda, “On Lp(x) norms,” The Royal Society of London A, vol. 455, no. 1981, pp. 219–225, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. C. Bennett and R. Sharpley, Interpolation of Operators, vol. 129 of Pure and Applied Mathematics, Academic Press, Boston, Mass, USA, 1988. View at Zentralblatt MATH · View at MathSciNet
  28. J. García-Cuerva and J. M. Martell, “Wavelet characterization of weighted spaces,” The Journal of Geometric Analysis, vol. 11, no. 2, pp. 241–264, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. P. G. Lemarié-Rieusset, “Ondelettes et poids de Muckenhoupt,” Studia Mathematica, vol. 108, no. 2, pp. 127–147, 1994. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet