# CHARACTERIZATION OF ORTHONORMAL HIGH-ORDER BALANCED MULTIWAVELETS IN TERMS OF MOMENTS

• Kwon, Soon-Geol
• Published : 2009.01.31
• 31 4

#### Abstract

In this paper, we derive a characterization of orthonormal balanced multiwavelets of order p in terms of the continuous moments of the multiscaling function $\phi$. As a result, the continuous moments satisfy the discrete polynomial preserving properties of order p (or degree p - 1) for orthonormal balanced multiwavelets. We derive polynomial reproduction formula of degree p - 1 in terms of continuous moments for orthonormal balanced multiwavelets of order p. Balancing of order p implies that the series of scaling functions with the discrete-time monomials as expansion coefficients is a polynomial of degree p - 1. We derive an algorithm for computing the polynomial of degree p - 1.

#### Keywords

multiwavelets;balanced multiwavelets;characterization of balancing condition;polynomial preservation/annihilation;moments;orthonormal bases

#### References

1. S. Bacchelli, M. Cotronei, and D. Lazzaro, An algebraic construction of k-balanced multiwavelets via the lifting scheme, Numer. Algorithms 23 (2000), no. 4, 329–356. https://doi.org/10.1023/A:1019120621646
2. C. Chui and Q. Jiang, Balanced multi-wavelets in Rs, Math. Comp. 74 (2005), no. 251, 1323–1344 . https://doi.org/10.1090/S0025-5718-04-01681-3
3. F. Keinert, Wavelets and multiwavelets, Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2004.
4. S.-G. Kwon, Propreties of moments of orthonormal balanced multiwavelets, J. Appl. Math. & Computing 12 (2003), no. 1-2, 367–373.
5. J. Lebrun and M. Vetterli, Balanced multiwavelets theory and design, IEEE Trans. Signal Process. 46 (1998), no. 4, 1119–1125. https://doi.org/10.1109/78.668561
6. J. Lebrun and M. Vetterli, High order balanced multiwavelets, In Proc. IEEE ICASSP, Seatle, WA, 1529–1532, May 1998.
7. J. Lebrun and M. Vetterli, High-order balanced multiwavelets: theory, factorization, and design, IEEE Trans. Signal Process. 49 (2001), no. 9, 1918–1930. https://doi.org/10.1109/78.942621
8. G. Plonka and V. Strela, From wavelets to multiwavelets. Mathematical methods for curves and surfaces, II, (Lillehammer, 1997), 375–399, Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, TN, 1998.
9. I. Selesnick, Multiwavelet bases with extra approximation properties, IEEE Trans. Signal Process. 46 (1998), no. 11, 2898–2908.
10. I. Selesnick, Interpolating multiwavelet bases and the sampling theorem, IEEE Trans. Signal Process. 47 (1999), no. 6, 1615–1621. https://doi.org/10.1109/78.765131
11. I. Selesnick, Balanced multiwavelet bases based on symmetric fir filters, IEEE Trans. Signal Process. 48 (2000), no. 1, 184–191. https://doi.org/10.1109/78.815488

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