A class of tight framelet packets

  • Da-Yong Lu
  • Qi-Bin Fan


This paper obtains a class of tight framelet packets on L 2(ℝ d ) from the extension principles and constructs the relationships between the basic framelet packets and the associated filters.


wavelet frames framelet packets framelets extension principles 

MSC 2010

42C15 42C40 


  1. [1]
    B. Behera: Multiwavelet packets and frame packets of L 2(ℝd). Proc. Ind. Acad. Sci., Math. Sci. 111 (2001), 439–463.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    J. J. Benedetto, O.M. Treiber: Wavelet frames: multiresolution analysis and extension principles. In: L. Debnath, ed., Wavelet Transforms and Time-Frequency Signal Analysis, Birkhäuser, 2001, pp. 3–36.Google Scholar
  3. [3]
    C. Boor, R.A. DeVore, A. Ron: On the construction of multivariate (pre) wavelets. Construct. Approx. 9 (1993), 123–166.MATHCrossRefGoogle Scholar
  4. [4]
    D. Chen: On the splitting trick and wavelet frame packets. SIAM J. Math. Anal. 4 (2000), 726–739.CrossRefGoogle Scholar
  5. [5]
    Q. J. Chen, Z.X. Cheng: A study on compactly supported orthogonal vector-valued wavelets and wavelet packets. Chaos, Solitons and Fractals 31 (2007), 1024–1034.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    O. Christensen: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston, 2003.MATHGoogle Scholar
  7. [7]
    C. R. Chui, C. Li: Non-orthogonal wavelet packets. SIAM J. Math. Anal. 24 (1993), 712–738.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    A. Cohen, I. Daubechies: On the instability of arbitrary biorthogonal wavelet packets. SIAM J. Math. Anal. 24 (1993), 1340–1354.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    R. R. Coifman, Y. Meyer, M. V. Wickerhauser: Size properties of wavelet packets. In: M. B. Ruskai et al., eds., Wavelets and Their Applications. Jones and Bartlett, Boston, 1992, pp. 453–470.Google Scholar
  10. [10]
    R. R. Coifman, Y. Meyer, M. V. Wickerhauser: Wavelet analysis and signal processing. In: M. B. Ruskai et al., eds., Wavelets and Their Applications. Jones and Bartlett, Boston, 1992, pp. 153–178.Google Scholar
  11. [11]
    I. Daubechies, B. Han: Pairs of dual wavelet frames from any two refinable functions. Constr. Approx. 20 (2004), 325–352.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    I. Daubechies, B. Han, A. Ron, Z. Shen: Framelets: MRA-based constructions of wavelet frames. Appl. Comput. Harmon. Anal. 1 (2003), 1–46.MathSciNetCrossRefGoogle Scholar
  13. [13]
    B. Han: Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix. J. Comput. Appl. Math. 155 (2003), 43–67.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    B. Han: Dual multiwavelet frames with high balancing order and compact fast frame transform. Appl. Comput. Harmon. Anal. 26 (2009), 14–42.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    B. Han: On dual wavelet tight frames. Appl. Comput. Harmon. Anal. 4 (1997), 380–413.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    B. Han, Q. Mo: Symmetric MRA tight wavelet frames with three generators and high vanishing moments. Appl. Comput. Harmon. Anal. 18 (2005), 67–93.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    R. Long, W. Chen: Wavelet basis packets and wavelet frame packets. J. Fourier Anal. Appl. 3 (1997), 239–256.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    A. Ron, Z. Shen: Affine systems in L 2(ℝd): the analysis of the analysis operator. J. Functional Anal. Appl. 148 (1997), 408–447.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    A. Ron, Z. Shen: Compactly supported tight affine spline frames in L 2(ℝd). Math. Comput. 67 (1998), 191–207.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    I. W. Selesnick, A. F. Abdelnour: Symmetric wavelet tight frames with two generators. Appl. Comput. Harmon. Anal. 17 (2004), 211–225.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    Z. Shen: Nontensor product wavelet packets in L 2(ℝs). SIAM J. Math. Anal. 26 (1995), 1061–1074.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2011

Authors and Affiliations

  1. 1.WuhanP.R.China
  2. 2.School of Mathematics and StatisticsWuhan UniversityWuhanP.R.China

Personalised recommendations