Point Values and Normalization of Two-Direction Multi-wavelets and their Derivatives

• Journal title : Kyungpook mathematical journal
• Volume 55, Issue 4,  2015, pp.1053-1067
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2015.55.4.1053
Title & Authors
Point Values and Normalization of Two-Direction Multi-wavelets and their Derivatives
KEINERT, FRITZ; KWON, SOON-GEOL;

Abstract
A two-direction multiscaling function $\small{{\phi}}$ satisfies a recursion relation that uses scaled and translated versions of both itself and its reverse. This offers a more general and flexible setting than standard one-direction wavelet theory. In this paper, we investigate how to find and normalize point values and derivative values of two-direction multiscaling and multiwavelet functions. Determination of point values is based on the eigenvalue approach. Normalization is based on normalizing conditions for the continuous moments of $\small{{\phi}}$. Examples for illustrating the general theory are given.
Keywords
two-direction multiwavelets;point values;normalization;multi-wavelet derivatives;
Language
English
Cited by
References
1.
I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math., 41(7)(1988), 909-996.

2.
I. Daubechies, Ten lectures on wavelets, volume 61 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.

3.
S. Du and D. Yuan, The description of two-directional biorthogonal finitely supported wavelet packets with poly-scale dilation, Key Engineering Materials, 439-440(2010), 1171-1176.

4.
F. Keinert, Wavelets and multiwavelets, Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton, FL, 2004.

5.
S.-G. Kwon, Approximation order of two-direction multiscaling functions, Preprint.

6.
S.-G. Kwon, High order orthogonal two-direction scaling functions from orthogonal balanced multiscaling functions, In preparation.

7.
S.-G. Kwon, Characterization of orthonormal high-order balanced multiwavelets in terms of moments, Bull. Korean Math. Soc., 46(1)(2009), 183-198.

8.
S.-G. Kwon, Two-direction multiwavelet moments, Appl. Math. Comput., 219(8)(2012), 3530-3540.

9.
J. Lebrun and M. Vetterli. High order balanced multiwavelets, Proc. IEEE ICASSP, Seatle, WA, (1998), 1529-1532.

10.
J. Lebrun and M. Vetterli, High-order balanced multiwavelets: theory, factorization, and design, IEEE Trans. Signal Process., 49(9)(2001), 1918-1930.

11.
B. Lv and X. Wang, Design and properties of two-direction compactly supported wavelet packets with an integer dilation factor, In 2009 Third Inter. Symp. on Intel. Info. Tech. Appl.

12.
J. Morawiec, On $L^1$-solution of a two-direction refinable equation, J. Math. Anal. Appl., 354(2)(2009), 648-656.

13.
G. Plonka and V. Strela, From wavelets to multiwavelets, In Mathematical methods for curves and surfaces, II (Lillehammer, 1997), Innov. Appl. Math., pages 375-399. Vanderbilt Univ. Press, Nashville, TN, 1998.

14.
G. Wang, X. Zhou, and B. Wang, The construction of orthogonal two-direction multiwavelet from orthogonal two-direction wavelet, Preprint.

15.
C. Xie and S. Yang, Orthogonal two-direction multiscaling functions, Front. Math. China, 1(4)(2006), 604-611.

16.
S. Yang and Y. Li, Two-direction refinable functions and two-direction wavelets with dilation factor m, Appl. Math. Comput., 188(2)(2007), 1908-1920.

17.
S. Yang and Y. Li, Two-direction refinable functions and two-direction wavelets with high approximation order and regularity, Sci. China Ser. A, 50(12)(2007), 1687-1704.

18.
S. Yang and C. Xie, A class of orthogonal two-direction refinable functions and two-direction wavelets, Int. J. Wavelets Multiresolut. Inf. Process., 6(6)(2008), 883-894.