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A general sampling theorem for multiwavelet subspaces

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Abstract

An orthogonal scaling function ϕ(t) can realize perfect A/D (Analogue/Digital) and D/A if and only if ϕ(t) is cardinal in the case of scalar wavelet. But it is not true when it comes to multiwavelets. Even if a multiscaling function φ(t) is not cardinal, it also holds for perfect A/D and D/A. This property shows the limitation of Selesnick’s sampling theorem. In this paper, we present a general sampling theorem for multiwavelet subspaces by Zak transform and make a large family of multiwavelets with some good properties (orthogonality, compact support, symmetry, high approximation order, etc.), but not necessarily with cardinal property, realize perfect A/D and D/A. Moreover, Selesnick’s result is just the special case of our theorem. And our theorem is suitable for some symmetrical or nonorthogonal multiwavelets.

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References

  1. Unser, M., Sampling-50 years after Shannon, Proceedings of IEEE, April 2000, 88(4): 569–587.

    Article  Google Scholar 

  2. Walter, G. G., A sampling theorem for wavelet subspaces, IEEE Trans. Inform. Theory, March 1992, 38(2): 881–884.

    Article  MathSciNet  Google Scholar 

  3. Janssen, A. J. E. M., The Zak transform and sampling theorem for wavelet subspaces, IEEE Trans. Signal Processing, December 1993, 41: 3360–3365.

    Article  MATH  Google Scholar 

  4. Chen, W., Itoh, S., A sampling theorem for shift-invariant subspace, IEEE Trans. Signal Processing, 1998, 46(10): 2822–2924.

    Article  MATH  MathSciNet  Google Scholar 

  5. Xia. X. G., On sampling theorem, wavelet, and wavelet transforms, IEEE Trans. Signal Processing, December 1993, 41(12): 3524–3535.

    Article  Google Scholar 

  6. Geronimo, J. S., Hardin, D. P., Massopust, P. R., Fractal function and wavelet expansions based on several scaling functions, Jour. Appr. Theory, September 1994, 78(3): 373–401.

    Article  MATH  MathSciNet  Google Scholar 

  7. Chui, C. K., Lian, J., A study of orthonormal multi-wavelets, Applied Numer. Math, March 1996, 20(3): 273–298.

    Article  MathSciNet  Google Scholar 

  8. Plonka, G., Strela, V., Construction of multiscaling function with approximation and symmetry, SIAM J. Math. Anal, March 1998, 29(2): 481–510.

    Article  MATH  MathSciNet  Google Scholar 

  9. Selesnick, I. W., Interpolating multiwavelet bases and the sampling theorem, IEEE Trans. Signal Processing, June 1999, 47(6): 1615–1621.

    Article  MATH  MathSciNet  Google Scholar 

  10. Blu, T., Unser, M., Approximation error for quasi-interpolators and (multi) wavelet expansions, Applied and Computation Harmonic Analysis, 1999, 6: 219–251.

    Article  MATH  MathSciNet  Google Scholar 

  11. Shu, S., Jin, J. C., Yu, H. Y., et al., A sampling theorem for shift-invariant subspace generated by several scaling function inL 2(R), Proceedings of ICSP’96, Beijing: IEEE Press, 24–27.

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Authors and Affiliations

  1. The Key Laboratory of Intelligent Information Processing, Institute of Computing Technology, Chinese Academy of Sciences, 100080, Beijing, China

    Jia Caiyan

  2. Information Engineering College, Xiangtan University, 411105, Xiangtan, China

    Gao Xieping

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  1. Jia Caiyan
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  2. Gao Xieping
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Correspondence to Jia Caiyan.

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Jia, C., Gao, X. A general sampling theorem for multiwavelet subspaces. Sci China Ser F 45, 365–372 (2002). https://doi.org/10.1007/BF02714093

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  • DOI: https://doi.org/10.1007/BF02714093

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