High balanced biorthogonal multiwavelets with symmetry. (English) Zbl 1472.42056

Li, Youfa; Yang, Shouzhi; Shen, Yanfeng; Zhang, Gengrong

Abstr. Appl. Anal. 2014, Article ID 154269, 8 p. (2014).

Summary: Balanced multiwavelet transform can process the vector-valued data sparsely while preserving a polynomial signal. In [Sci. China, Ser. F 49, No. 4, 504–515 (2006; Zbl 1129.42019); Sci. China, Ser. A 49, No. 1, 86–97 (2006; Zbl 1193.42100)], S. Yang and L. Peng constructed balanced multiwavelets from the existing nonbalanced ones. It will be proved, however, in this paper that if the nonbalanced multiwavelets have antisymmetric component, it is impossible for the balanced multiwavelets by the method mentioned above to have symmetry. In this paper, we give an algorithm for constructing a pair of biorthogonal symmetric refinable function vectors from any orthogonal refinable function vector, which has symmetric and antisymmetric components. Then, a general scheme is given for high balanced biorthogonal multiwavelets with symmetry from the constructed pair of biorthogonal refinable function vectors. Moreover, we discuss the approximation orders of the biorthogonal symmetric refinable function vectors. An example is given to illustrate our results.

MSC:

42C40

Nontrigonometric harmonic analysis involving wavelets and other special systems

Citations:

Zbl 1129.42019; Zbl 1193.42100

References:

[1]	Huang, Y.; Zhao, Y., Balanced multiwavelets on the interval \([0,1]\) with ar bitrary integer dilation factor \(a\), International Journal of Wavelets, Multiresolution and Information Processing, 10, 1 (2012) · Zbl 1242.42031 · doi:10.1142/S0219691311004523
[2]	Han, B.; Mo, Q., Multiwavelet frames from refinable function vectors, Advances in Computational Mathematics, 18, 2-4, 211-245 (2003) · Zbl 1059.42030 · doi:10.1023/A:1021360312348
[3]	Jiang, Q.
[4]	Li, B.; Luo, T.; Peng, L., Balanced interpolatory multiwavelets with multiplicity \(r\), International Journal of Wavelets, Multiresolution and Information Processing, 10, 4 (2012) · Zbl 1252.42044 · doi:10.1142/S0219691312500397
[5]	Li, R.; Wu, G., The orthogonal interpolating balanced multiwavelet with rational coefficients, Chaos, Solitons and Fractals, 41, 2, 892-899 (2009) · Zbl 1198.42047 · doi:10.1016/j.chaos.2008.04.019
[6]	Li, Y.; Yang, S.; Yuan, D., Bessel multiwavelet sequences and dual multiframelets in Sobolev spaces, Advances in Computational Mathematics, 38, 3, 491-529 (2013) · Zbl 1263.42023 · doi:10.1007/s10444-011-9246-8
[7]	Li, Y., Sampling approximation by framelets in Sobolev space and its application in modifying interpolating error, Journal of Approximation Theory, 175, 43-63 (2013) · Zbl 1298.42036
[8]	Yang, S.; Peng, L., Construction of high order balanced multiscaling functions via PTST, Science in China Series F: Information Sciences, 49, 4, 504-515 (2006) · Zbl 1129.42019 · doi:10.1007/s11432-006-2010-x
[9]	Yang, S.; Peng, L., Raising approximation order of refinable vector by increasing multiplicity, Science in China Series A, 49, 1, 86-97 (2006) · Zbl 1193.42100 · doi:10.1007/s11425-005-0040-2
[10]	Shouzhi, Y.; Hongyong, W., High-order balanced multiwavelets with dilation factor \(a\), Applied Mathematics and Computation, 181, 1, 362-369 (2006) · Zbl 1148.65321 · doi:10.1016/j.amc.2006.01.041
[11]	Yang, S., Biorthogonal interpolatory multiscaling functions and corresponding multiwavelets, The ANZIAM Journal, 49, 1, 85-97 (2007) · Zbl 1135.42332 · doi:10.1017/S1446181100012694
[12]	Chui, C. K.; Lian, J.-A., A study of orthonormal multi-wavelets, Applied Numerical Mathematics, 20, 3, 273-298 (1996) · Zbl 0877.65098 · doi:10.1016/0168-9274(95)00111-5
[13]	Keinert, F., Wavelets and Multiwavelets. Wavelets and Multiwavelets, Studies in Advanced Mathematics (2003), New York, NY, USA: CRC Press, New York, NY, USA · Zbl 1058.65150
[14]	Strela, V., Multiwavelets: Theory and Application (1996), Cambridge, Mass, USA: Department of Mathematics, Institute of Technology, Cambridge, Mass, USA