Browse > Article
http://dx.doi.org/10.5666/KMJ.2015.55.4.1053

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

KEINERT, FRITZ (Department of Mathematics, Iowa State University)
KWON, SOON-GEOL (Department of Mathematics Education, Sunchon National University)
Publication Information
Kyungpook Mathematical Journal / v.55, no.4, 2015 , pp. 1053-1067 More about this Journal
Abstract
A two-direction multiscaling function ${\phi}$ 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 ${\phi}$. Examples for illustrating the general theory are given.
Keywords
two-direction multiwavelets; point values; normalization; multi-wavelet derivatives;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math., 41(7)(1988), 909-996.   DOI
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.   DOI
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.   DOI
8 S.-G. Kwon, Two-direction multiwavelet moments, Appl. Math. Comput., 219(8)(2012), 3530-3540.   DOI
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.   DOI
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.   DOI
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.   DOI
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.   DOI
17 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.   DOI
18 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.   DOI