A note on the construction of nonseparable wavelet bases and multiwavelet matrix filters of 𝐿²(ℝⁿ), where 𝕟≥2

Karoui, Abderrazek

doi:10.1090/S1079-6762-03-00109-4

A note on the construction of nonseparable wavelet bases and multiwavelet matrix filters of $L^2(\mathbb {R}^n)$, where $n\geq 2$

Author: Abderrazek Karoui
Journal: Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 32-39
MSC (2000): Primary 39B42, 42C05; Secondary 42C15
DOI: https://doi.org/10.1090/S1079-6762-03-00109-4
Published electronically: April 4, 2003
MathSciNet review: 1988870
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this note, we announce a general method for the construction of nonseparable orthogonal wavelet bases of $L^2(\mathbb {R}^n),$ where $n\geq 2.$ Hence, we prove the existence of such type of wavelet bases for any integer $n\geq 2.$ Moreover, we show that this construction method can be extended to the construction of $n$-D multiwavelet matrix filters.

Antoine Ayache, Construction of non-separable dyadic compactly supported orthonormal wavelet bases for $L^2({\bf R}^2)$ of arbitrarily high regularity, Rev. Mat. Iberoamericana 15 (1999), no. 1, 37–58. MR 1681636, DOI https://doi.org/10.4171/RMI/249

lai W. He and M. Jun Lai, Construction of bivariate nonseparable compactly supported orthonormal multiwavelets with arbitrary high regularity, Preprint.

Hui Ji, Sherman D. Riemenschneider, and Zuowei Shen, Multivariate compactly supported fundamental refinable functions, duals, and biorthogonal wavelets, Stud. Appl. Math. 102 (1999), no. 2, 173–204. MR 1668532, DOI https://doi.org/10.1111/1467-9590.00108
Abderrazek Karoui and Rémi Vaillancourt, Nonseparable biorthogonal wavelet bases of $L^2({\bf R}^n)$, Spline functions and the theory of wavelets (Montreal, PQ, 1996) CRM Proc. Lecture Notes, vol. 18, Amer. Math. Soc., Providence, RI, 1999, pp. 135–151. MR 1676241, DOI https://doi.org/10.1090/crmp/018/13
Abderrazek Karoui, A technique for the construction of compactly supported biorthogonal wavelets of $L^2({\bf R}^n),\ n\geq 2$, J. Math. Anal. Appl. 249 (2000), no. 2, 367–392. MR 1781230, DOI https://doi.org/10.1006/jmaa.2000.6867

karoui2 A. Karoui, A general construction of nonseparable multivariate orthonormal wavelet bases and multidimensional multiwavelet matrix filters, Preprint.

W. Lawton, S. L. Lee, and Zuowei Shen, Stability and orthonormality of multivariate refinable functions, SIAM J. Math. Anal. 28 (1997), no. 4, 999–1014. MR 1453317, DOI https://doi.org/10.1137/S003614109528815X
Yves Meyer, Wavelets and operators, Cambridge Studies in Advanced Mathematics, vol. 37, Cambridge University Press, Cambridge, 1992. Translated from the 1990 French original by D. H. Salinger. MR 1228209
Zuowei Shen, Refinable function vectors, SIAM J. Math. Anal. 29 (1998), no. 1, 235–250. MR 1617183, DOI https://doi.org/10.1137/S0036141096302688