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Mathematical Problems in Engineering
Volume 2010, Article ID 128294, 26 pages
http://dx.doi.org/10.1155/2010/128294
Research Article

Characterizations of Tight Frame Wavelets with Special Dilation Matrices

1Institute of Information and System Science, Beifang University of Nationalities, Yinchuan 750021, China
2Department of Foundation, Beifang University of Nationalities, Yinchuan 750021, China

Received 22 May 2010; Revised 29 October 2010; Accepted 25 November 2010

Academic Editor: Angelo Luongo

Copyright © 2010 Huang Yongdong and Zhu Fengjuan. 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. I. Daubechies, Ten Lectures on Wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1992.
  2. L. Debnath, Wavelet Transforms and Their Applications, Birkhäuser, Boston, Mass, USA, 2002.
  3. E. Hernández and G. Weiss, A First Course on Wavelets, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1996. View at Publisher · View at Google Scholar
  4. F. Keinert, Wavelets and Multiwavelets, Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004.
  5. O. Christensen, An Introduction to Frames and Riesz Bases, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, Mass, USA, 2003.
  6. J. J. Benedetto and S. Li, “The theory of multiresolution analysis frames and applications to filter banks,” Applied and Computational Harmonic Analysis, vol. 5, no. 4, pp. 389–427, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. J. J. Benedetto and O. M. Treiber, “Wavelet frames: multiresolution analysis and extension principles,” in Wavelet Transforms and Time-Frequency Signal Analysis, L. Debnath, Ed., Applied and Numerical Haimonic Analysis, pp. 3–36, Birkhäuser, Boston, Mass, USA, 2001. View at Google Scholar · View at Zentralblatt MATH
  8. M. Bownik, “A characterization of affine dual frames in L2(Rn),” Applied and Computational Harmonic Analysis, vol. 8, no. 2, pp. 203–221, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. D. Bakić, I. Krishtal, and E. N. Wilson, “Parseval frame wavelets with En(2)-dilations,” Applied and Computational Harmonic Analysis, vol. 19, no. 3, pp. 386–431, 2005. View at Publisher · View at Google Scholar
  10. D. Bakić, “Semi-orthogonal Parseval frame wavelets and generalized multiresolution analyses,” Applied and Computational Harmonic Analysis, vol. 21, no. 3, pp. 281–304, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. M. Bownik, Z. Rzeszotnik, and D.n Speegle, “A characterization of dimension functions of wavelets,” Applied and Computational Harmonic Analysis, vol. 10, no. 1, pp. 71–92, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. C. K. Chui, W. He, J. Stöckler, and Q. Sun, “Compactly supported tight affine frames with integer dilations and maximum vanishing moments,” Advances in Computational Mathematics, vol. 18, no. 2-4, pp. 159–187, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. I. Daubechies, B. Han, A. Ron, and Z. Shen, “Framelets: MRA-based constructions of wavelet frames,” Applied and Computational Harmonic Analysis, vol. 14, no. 1, pp. 1–46, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. M. Ehler, “On multivariate compactly supported bi-frames,” The Journal of Fourier Analysis and Applications, vol. 13, no. 5, pp. 511–532, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. B. Han and Q. Mo, “Multiwavelet frames from refinable function vectors,” Advances in Computational Mathematics, vol. 18, no. 2–4, pp. 211–245, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. Y. Huang and Z. Cheng, “Minimum-energy frames associated with refinable function of arbitrary integer dilation factor,” Chaos, Solitons and Fractals, vol. 32, no. 2, pp. 503–515, 2007. View at Publisher · View at Google Scholar
  17. H. O. Kim and J. K. Lim, “On frame wavelets associated with frame multiresolution analysis,” Applied and Computational Harmonic Analysis, vol. 10, no. 1, pp. 61–70, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. M.-J. Lai and J. Stöckler, “Construction of multivariate compactly supported tight wavelet frames,” Applied and Computational Harmonic Analysis, vol. 21, no. 3, pp. 324–348, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. S. Li, “A theory of generalized multiresolution structure and pseudoframes of translates,” The Journal of Fourier Analysis and Applications, vol. 7, no. 1, pp. 23–40, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. M. Paluszyński, H. Šikić, G. Weiss, and S. Xiao, “Generalized low pass filters and MRA frame wavelets,” The Journal of Geometric Analysis, vol. 11, no. 2, pp. 311–342, 2001. View at Google Scholar · View at Zentralblatt MATH
  21. M. Paluszyński, H. Šikić, G. Weiss, and S. Xiao, “Tight frame wavelets, their dimension functions, MRA tight frame wavelets and connectivity properties,” Advances in Computational Mathematics, vol. 18, no. 2–4, pp. 297–327, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. A. Ron and Z. Shen, “Affine systems in L2(Rd): the analysis of the analysis operator,” Journal of Functional Analysis, vol. 148, no. 2, pp. 408–447, 1997. View at Publisher · View at Google Scholar
  23. A. Ron and Z. Shen, “Affine systems in L2(Rd). II. Dual systems,” The Journal of Fourier Analysis and Applications, vol. 3, no. 5, pp. 617–637, 1997. View at Publisher · View at Google Scholar
  24. H. Šikić, D. Speegle, and G. Weiss, “Structure of the set of dyadic PFW's,” in Frames and operator theory in analysis and signal processing, vol. 451 of Contemporary Mathematics, pp. 263–291, American Mathematical Society, Providence, RI, USA, 2008. View at Google Scholar
  25. G. Wu, X. Yang, and Z. Liu, “MRA Parseval frame wavelets and their multipliers in L2(Rn),” Mathematical Problems in Engineering, Article ID 492585, 17 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH