Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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FRAMES AND SAMPLING THEOREMS IN MULTIWAVELET SUBSPACES

  • Liu, Zhanwei (School of Information Engineering, Zhengzhou University) ;
  • Wu, Guochang (College of Information, Henan University of Finance and Economics) ;
  • Yang, Xiaohui (Institute of Applied Mathematics, School of Mathematics and Information Sciences, Henan University)
  • Received : 2009.09.10
  • Accepted : 2009.10.26
  • Published : 2010.05.30
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Abstract

In this paper, we investigate the sampling theorem for frame in multiwavelet subspaces. By the frame satisfying some special conditions, we obtain its dual frame with explicit expression. Then, we give an equivalent condition for the sampling theorem to hold in multiwavelet subspaces. Finally, a sufficient condition under which the sampling theorem holds is established. Some typical examples illustrate our results.

Keywords

Acknowledgement

Supported by : NSF

References

  1. G. G. Walter, A sampling theorem for wavelet subspaces, IEEE Trans. Inform. Theory, 38(2)(1992), pp. 881-884. https://doi.org/10.1109/18.119745
  2. X. G. Xia and Z. Zhang, On sampling theorem, wavelet and wavelet transforms, IEEE Trans. Signal Processing, 41(12)1993, pp. 3524-3535. https://doi.org/10.1109/78.258090
  3. A. J. E. M. Janssen, The Zak transform and sampling theorems for wavelet subspaces , IEEE Trans. Signal Processing, 41(12)1993, pp. 3360-3364. https://doi.org/10.1109/78.258079
  4. Y. Liu, Irregular sampling for spline wavelet subspaces, IEEE Trans. Inform. Theory, 42(2)(1996), pp. 623-627. https://doi.org/10.1109/18.485731
  5. W. Chen, S. Itoh, J. Shiki. A sampling theorem for shift-invariant spaces, IEEE Trans Signal Processing, 46(3)(1998), pp. 2802-2810.
  6. J. S. Geronimo, D. P. Hardin, and P. R. Massopust, Fractal functions and wavelet expansions based on several scaling functions, J. Approx. Theory, 78(3)(1994), pp. 373-401. https://doi.org/10.1006/jath.1994.1085
  7. I. W. Selesnick, Interpolating multiwavelet bases and the sampling theorem, IEEE Trans. Signal Processing, 47(6)(1999), pp. 1615-1621. https://doi.org/10.1109/78.765131
  8. W. C. Sun and X. W. Zhou, Sampling theorem for multiwavelet subspaces, Chin. Sci. Bull., 44(14)(1999), pp. 1283-1286. https://doi.org/10.1007/BF02885844
  9. D. Zhou, Interpolatory orthogonal multiwavelets and refinable functions, IEEE Trans. Signal Processing, 50(3)(2002), 520-527, 2002. https://doi.org/10.1109/78.984728
  10. C. Zhao, P. Zhao Sampling Theorem and Irregular Sampling Theorem for Multiwavelet subspaces, IEEE Trans Signal Processing, 53(3)(2005), pp. 705-713.
  11. X. W. Zhou, W. C. Sun, On the Sampling Theorem for Wavelet Subspaces, J. Fourier. Anal. Appl, 5(4)(1999), pp. 347-354. https://doi.org/10.1007/BF01259375
  12. X. W. Zhou, W. C. Sun, Sampling Theorem for Wavelet Subspaces: Error Estimate and Irregular Sampling, IEEE Trans Signal Processing, 48(1)(2000), pp. 223-226. https://doi.org/10.1109/78.815492
  13. O. Christensen, An Introduction to Frames and Riesz bases, Birkhauser, Boston, MA, 2003.
  14. C. Blanco, C. Cabrelli and S. Heineken, Functions in Sampling Spaces, Sampling Theory In Signal and Image Processing, 5(3)2006, pp. 275-295.
  15. C. K. Chui and J. Lian, A study of orthonormal multi-wavelets, Appl, Numer. Math., 20(3)1996, pp. 273-298. https://doi.org/10.1016/0168-9274(95)00111-5