Balanced multi-wavelets in

Authors:
Charles K. Chui and Qingtang Jiang

Journal:
Math. Comp. **74** (2005), 1323-1344

MSC (2000):
Primary 42C40, 65T60; Secondary 94A08

Published electronically:
June 7, 2004

MathSciNet review:
2137005

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The notion of -balancing was introduced a few years ago as a condition for the construction of orthonormal scaling function vectors and multi-wavelets to ensure the property of preservation/annihilation of scalar-valued discrete polynomial data of order (or degree ), when decomposed by the corresponding matrix-valued low-pass/high-pass filters. While this condition is indeed precise, to the best of our knowledge only the proof for is known. In addition, the formulation of the -balancing condition for is so prohibitively difficult to satisfy that only a very few examples for and vector dimension 2 have been constructed in the open literature. The objective of this paper is to derive various characterizations of the -balancing condition that include the polynomial preservation property as well as equivalent formulations that facilitate the development of methods for the construction purpose. These results are established in the general multivariate and bi-orthogonal settings for any .

**1.**K. Attakitmongcol, D. P. Hardin, D. M. Wilkes,*Multiwavelet prefilters II: Optimal orthogonal prefilters*, IEEE Tran. Image Proc.,**10**(2001), 1476-1487.**2.**Charles K. Chui,*Multivariate splines*, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 54, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. With an appendix by Harvey Diamond. MR**1033490****3.**C. K. Chui, Q. T. Jiang,*Multivariate balanced vector-valued refinable functions*, In*Mordern Development in Multivariate Approximation*, ISNM 145, 71-102, V.W. Haussmann, K. Jetter, M. Reimer, and J. Stöckler (eds.), Birhhäuser Verlag, Basel, 2003.**4.**Charles K. Chui and Jian-ao Lian,*A study of orthonormal multi-wavelets*, Appl. Numer. Math.**20**(1996), no. 3, 273–298. Selected keynote papers presented at 14th IMACS World Congress (Atlanta, GA, 1994). MR**1402703**, 10.1016/0168-9274(95)00111-5**5.**George C. Donovan, Jeffrey S. Geronimo, Douglas P. Hardin, and Peter R. Massopust,*Construction of orthogonal wavelets using fractal interpolation functions*, SIAM J. Math. Anal.**27**(1996), no. 4, 1158–1192. MR**1393432**, 10.1137/S0036141093256526**6.**Jeffrey S. Geronimo, Douglas P. Hardin, and Peter R. Massopust,*Fractal functions and wavelet expansions based on several scaling functions*, J. Approx. Theory**78**(1994), no. 3, 373–401. MR**1292968**, 10.1006/jath.1994.1085**7.**D. P. Hardin, D. W. Roach,*Multiwavelet prefilters I: Orthogonal prefilters preserving approximation order*, IEEE Tran. Circuits and System-II**45**(1998), 1119-1125.**8.**Rong-Qing Jia and Qing-Tang Jiang,*Approximation power of refinable vectors of functions*, Wavelet analysis and applications (Guangzhou, 1999) AMS/IP Stud. Adv. Math., vol. 25, Amer. Math. Soc., Providence, RI, 2002, pp. 155–178. MR**1887509****9.**R. Q. Jia, Q. T. Jiang,*Spectral analysis of transition operators and its applications to smoothness analysis of wavelets*, SIAM J. Matrix Anal. Appl.**24**(2003), 1071-1109.**10.**Rong-Qing Jia, Qingtang Jiang, and Zuowei Shen,*Distributional solutions of nonhomogeneous discrete and continuous refinement equations*, SIAM J. Math. Anal.**32**(2000), no. 2, 420–434 (electronic). MR**1781224**, 10.1137/S0036141099350882**11.**Qingtang Jiang,*Multivariate matrix refinable functions with arbitrary matrix dilation*, Trans. Amer. Math. Soc.**351**(1999), no. 6, 2407–2438. MR**1650101**, 10.1090/S0002-9947-99-02449-6**12.**J. Lebrun,*Balancing Multiwavelets*, PhD Thesis, Swiss Federal Institute of Technology, Lausanne, Switzerland, 2000.**13.**J. Lebrun, M. Vetterli,*Balanced multiwavelets: Theory and design*, IEEE Trans. Signal Processing**46**(1998), 1119-1125.**14.**J. Lebrun, M. Vetterli,*High order balanced multiwavelets*, In*Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP)*, Seattle, 1998.**15.**J. A. Lian, C. K. Chui,*Balanced multi-wavelets with short filters*, IEEE Signal Processing Letters,**11**(2004), 75-78.**16.**Ivan W. Selesnick,*Multiwavelet bases with extra approximation properties*, IEEE Trans. Signal Process.**46**(1998), no. 11, 2898–2908. MR**1719939**, 10.1109/78.726804**17.**I. W. Selesnick,*Balanced multiwavelet bases based on symmetric FIR filters*, IEEE Trans. Signal Proc.**48**(2000), 184-191.**18.**G. Strang,*Eigenvalues of and convergence of cascade algorithm*, IEEE Trans. Signal Proc.**44**(1996), 233-238.**19.**V. Strela, P. Heller, G. Strang, P. Topiwala, C. Heil,*The application of multiwavelet filter banks to image processing*, IEEE Trans. Image Proc.**8**(1999), 548-563.**20.**X. G. Xia, D. P. Hardin, J. S. Geronimo, B. Suter,*Design of prefilters for discrete multiwavelet transforms*, IEEE Trans. Signal Proc.**44**(1996), 25-35.

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
42C40,
65T60,
94A08

Retrieve articles in all journals with MSC (2000): 42C40, 65T60, 94A08

Additional Information

**Charles K. Chui**

Affiliation:
Department of Mathematics & Computer Science, University of Missouri–St. Louis, St. Louis, Missouri 63121 and Department of Statistics, Stanford University, Stanford, California 94305

Email:
ckchui@stanford.edu

**Qingtang Jiang**

Affiliation:
Department of Mathematics & Computer Science, University of Missouri–St. Louis, St. Louis, Missouri 63121

Email:
jiang@math.umsl.edu

DOI:
https://doi.org/10.1090/S0025-5718-04-01681-3

Keywords:
Multi-wavelets,
characterization of balancing condition,
polynomial preservation/annihilation

Received by editor(s):
May 18, 2003

Received by editor(s) in revised form:
January 9, 2004

Published electronically:
June 7, 2004

Additional Notes:
The first author was supported in part by NSF Grants #CCR-9988289 and #CCR-0098331, and ARO Grant #DAAD 19-00-1-0512

The second author was supported in part by a University of Missouri–St. Louis research award

Article copyright:
© Copyright 2004
American Mathematical Society