Skip to main content
SpringerLink
Log in

The spectrum sequences of periodic frame multiresolution analysis

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

The periodic multiresolution analysis (PMRA) and the periodic frame multiresolution analysis (PFMRA) provide a general recipe for the construction of periodic wavelets and periodic wavelet frames, respectively. This paper addresses PFMRAs by the introduction of the notion of spectrum sequence. In terms of spectrum sequences, the scaling function sequences generating a normalized PFMRA are characterized; a characterization of the spectrum sequences of PFMRAs is obtained, which provides a method to construct PFMRAs since its proof is constructive; a necessary and sufficient condition for a PFMRA to admit a single wavelet frame sequence is obtained; a necessary and sufficient condition for a PFMRA to be contained in a given PMRA is also obtained. What is more, it is proved that an arbitrary PFMRA must be contained in some PMRA. In the meanwhile, some examples are provided to illustrate the general theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Christensen, O.: An Introduction to Frames and Riesz Bases, Birkhäuser Boston, 2003

    MATH  Google Scholar 

  2. Daubechies, I.: Ten Lecture on Wavelets, Philadelphia, 1992

  3. Young, R. M.: An Introduction to Nonharmonic Fourier Series, Academic Press, New York, 1980

    MATH  Google Scholar 

  4. Goh, S. S., Teo, K. M.: Wavelet frames and shift-invariant subspaces of periodic functions. Appl. Comput. Harmon. Anal., 20, 326–344 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Peng, S. L., Li, D. F., Chen, Q. H.: Theory and Applications of Periodic Wavelets (Chinese edition), Academic Press, P. R. China, 2003

    Google Scholar 

  6. Benedetto, J. J., Li, S. D.: The theory of multiresolution analysis frames and applications to filter banks. Appl. Comput. Harmonic Anal., 5, 389–427 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. Debnath, L.: Wavelet Transforms and Time-Frequency Signal Analysis, Birkhäuser Boston, 2001

  8. Plonka, G., Tasche, M.: A unified approach to periodic wavelets, In C. K. Chui, L. Montefusco, L. Puccio (eds), Wavelets: Theory, Algorithms and Applications, 1994, 137–151

  9. Atteia, M., Jaouni, H.: The construction of a basis of periodic wavelets, Mathematical methods for curves and surfaces (Ulvik, 1994), 21–30, Vanderbilt Univ. Press, Nashville, TN, 1995

    Google Scholar 

  10. Benedetto, J. J., Pfander, G.: Periodic wavelet transforms and periodicity detection. J. Appl. Math., 62, 1329–1368 (2002)

    MATH  MathSciNet  Google Scholar 

  11. Chen, H. L.: Complex Harmonic Splines, Periodic Quasi-Wavelets: Theory and Applications, Kluwer Academic Publishers, 2000

  12. Chen, H. L., Liang, X. Z., Peng, S. L., Xiao, S. L.: Real-valued periodic wavelets: construction and relation with Fourier series. J. Comput. Math., 17, 509–522 (1999)

    MATH  MathSciNet  Google Scholar 

  13. Glenn, J.: A continued fraction analysis of periodic wavelet coefficients. Proc. Amer. Math. Soc., 132, 1367–1375 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Goh, S. S., Yeo, C. H.: Uncertainty products of local periodic wavelets. Adv. Comput. Math., 13, 319–333 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lin, W., Wang, X.: Numerical solutions to acoustic scattering in shallow oceans by periodic wavelets, Direct and inverse problems of mathematical physics (Newark, DE, 1997), 267–279, Int. Soc. Anal. Appl. Comput., 5, Kluwer Acad. Publ., Dordrecht, 2000

    Google Scholar 

  16. Walter, G. G., Cai, L.: Periodic wavelets from scratch. J. Comput. Anal. Appl., 1, 25–41 (1999)

    MathSciNet  Google Scholar 

  17. Kim, H. O., Kim, R. Y., Lim, J. K.: On the spectrums of frame multiresolution analyses. J. Math. Anal. Appl., 305, 528–545 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Jia, R. Q., Zhou, D. X.: Convergence of subdivision schemes associated with nonnegative masks. SIAM J. Matrix Anal. Appl., 21, 418–430 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  19. Zhou, D. X.: Self-similar lattice tilings and subdivision schemes. SIAM J. Math. Anal., 33, 1–15 (2001)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Applied Sciences, Beijing University of Technology, Beijing, 100124, P. R. China

    Yun Zhang Li

  2. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, P. R. China

    Qiao Fang Lian

Authors
  1. Yun Zhang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qiao Fang Lian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qiao Fang Lian.

Additional information

Supported by the National Natural Science Foundation of China (Grant No. 10671008) and the first author is also partially supported by the Excellent Talents Foundation of Beijing, China (20051D0501022), PHR(IHLB), the Project-sponsored by SRF for ROCS, SEM of China and the Scientific Research Foundation for the Excellent Returned Overseas Chinese Scholars, Beijing

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, Y.Z., Lian, Q.F. The spectrum sequences of periodic frame multiresolution analysis. Acta. Math. Sin.-English Ser. 25, 403–418 (2009). https://doi.org/10.1007/s10114-008-7077-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-008-7077-4

Keywords

MR(2000) Subject Classification

Search

Navigation