ANZIAM J. 49 (2007), no. 1, pp. 85–97.

Biorthogonal interpolatory multiscaling functions and corresponding multiwavelets

Shou Zhi Yang
Department of Mathematics
Shantou University
Shantou 515063
P.R. China
Received 2 May, 2004; revised 30 June, 2006


A method for constructing a pair of biorthogonal interpolatory multiscaling functions is given and an explicit formula for constructing the corresponding biorthogonal multiwavelets is obtained. A multiwavelet sampling theorem is also established. In addition, we improve the stability of the biorthogonal interpolatory multiwavelet frame by the linear combination of a pair of biorthogonal interpolatory multiwavelets. Finally, we give an example illustrating how to use our method to construct biorthogonal interpolatory multiscaling functions and corresponding multiwavelets.

Download the article in PDF format (size 136 Kb)

2000 Mathematics Subject Classification: primary 42C15; secondary 94A12
(Metadata: XML, RSS, BibTeX) MathSciNet: MR2378151 Z'blatt-MATH: pre05243893


  1. C. K. Chui and J. Lian, “A study on orthonormal multiwavelets”, J. Appl. Numer. Math. 20 (1996) 273–298. MR1402703
  2. C. K. Chui and J. Z. Wang, “A cardinal spline approach to wavelets”, Proc. Amer. Math. Soc. 113 (1991) 785–793. MR1077784
  3. I. Daubechies and J. C. Lagarias, “Two-scale difference equations. II. Local regularity, infinite products of matrices and fractals”, SIAM J. Math. Anal. 23 (1991) 1031–1079. MR1166574
  4. G. C. Donovan, J. S. Geronimo and D. P. Hardin, “Construction of orthogonal wavelets using fractal interpolation functions”, SIAM J. Math. Anal. 27 (1996) 1158–1192. MR1393432
  5. J. S. Geronimo, D. P. Hardin and P. R. Massopust, “Fractal functions and wavelet expansions based on several scaling functions”, J. Approx. Theory 78 (1994) 371–401. MR1292968
  6. S. S. Goh, Jiang Qingtang and Xia Tao, “Construction of biorthogonal multiwavelets using the lifting scheme”, Appl. Comput. Harmon. Anal. 9 (2000) 336–352. MR1793422
  7. T. N. T. Goodman, S. L. Lee and W. S. Tang, “Wavelets in wandering subspaces”, Trans. Amer. Math. Soc. 338 (1993) 639–654. MR1117215
  8. J. Lian, “Orthogonal criteria for multiscaling functions”, Appl. Comp. Harm. Anal. 5 (1998) 277–311. MR1632545
  9. G. C. Rota and G. Strang, “A note on the joint spectral radius”, Indag. Math. 22 (1960) 379–381. MR147922
  10. Yang Shouzhi, “A fast algorithm for constructing orthogonal multiwavelets”, ANZIAM J. 46 (2004) 185–202. MR2099504
  11. Yang Shouzhi, “An algorithm for constructing biorthogonal multiwavelets with higher approximation orders”, ANZIAM J. 47 (2006) 513–526. MR2234018
  12. Yang Shouzhi, Cheng Zhengxing and Wang Hongyong, “Construction of biorthogonal multiwavelets”, J. Math. Anal. Appl. 276 (2002) 1–12. MR1944332
  13. G. Strang and D. X. Zhou, “Inhomogeneous refinement equations”, J. Fourier Anal. Appl. 4 (1998) 733–747. MR1666013
  14. X. G. Xia and Z. Zhang, “On sampling theorem, wavelets, and wavelet transforms”, IEEE Trans. Signal Processing 41 (1993) 3524–3535.
  15. D. X. Zhou, “Existence of multiple refinable distributions”, Michigan Math. J. 44 (1997) 317–329. MR1460417
  16. D. X. Zhou, “The p-norm joint spectral radius for even integers”, Methods Appl. Anal. 5 (1998) 39–54. MR1631335
  17. D. X. Zhou, “Multiple refinable Hermite interpolants”, J.Approx. Theory 102 (2000) 46–71. MR1736045
  18. D. X. Zhou, “Interpolatory orthogonal multiwavelets and refinable functions”, IEEE Trans. Signal Processing 50 (2002) 520–527. MR1895060
Australian Mathematical Publishing Association Inc.

Valid XHTML 1.0 Transitional