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A stability criterion for biorthogonal wavelet bases and their related subband coding scheme. (English) Zbl 0784.42022

It is shown that in the consideration of bi-orthogonal (and non- orthogonal) wavelets, a pair of dual filters does not necessarily lead to a pair of stable bi-orthogonal wavelet bases. The objective of this paper is to derive a stability criterion in terms of the action under a so- called transition operator and its dual. The criterion is that the spectra of both operators lie inside the unit circle.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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[1] A. Cohen and I. Daubechies, Nonseparable bidimensional wavelet bases , preprint, Bell AT&T Laboratories, Rev. Mat. Iberoamericana, · Zbl 0792.42021
[2] A. Cohen, I. Daubechies, and J.-C. Feauveau, Biorthogonal bases of compactly supported wavelets , Comm. Pure Appl. Math. 45 (1992), no. 5, 485-560. · Zbl 0776.42020
[3] A. S. Cavaretta, W. Dahmen, and C. A. Micchelli, Stationary subdivision , Mem. Amer. Math. Soc. 93 (1991), no. 453, vi+186. · Zbl 0741.41009
[4] A. Cohen, Ondelettes, analyses multirésolutions et filtres miroirs en quadrature , Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), no. 5, 439-459. · Zbl 0736.42021
[5] A. Cohen, Biorthogonal wavelets , Wavelets ed. C. K. Chiu, Wavelet Anal. Appl., vol. 2, Academic Press, Boston, MA, 1992, pp. 123-152. · Zbl 0760.42018
[6] J.P. Conze and A. Raugi, Fonctions harmoniques pour un operateur de transition et applications , preprint, Dept. de Math., Univ. de Rennes, 1990. · Zbl 0725.60026
[7] I. Daubechies, Orthonormal bases of compactly supported wavelets , Comm. Pure Appl. Math. 41 (1988), no. 7, 909-996. · Zbl 0644.42026
[8] I. Daubechies, Ten lectures on wavelets , CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. · Zbl 0776.42018
[9] T. Eirola, Sobolev characterization of solutions of dilation equations , SIAM J. Math. Anal. 23 (1992), no. 4, 1015-1030. · Zbl 0761.42014
[10] W. M. Lawton, Necessary and sufficient conditions for constructing orthonormal wavelet bases , J. Math. Phys. 32 (1991), no. 1, 57-61. · Zbl 0757.46012
[11] Y. Meyer, Ondelettes et opérateurs. I , Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990. · Zbl 0694.41037
[12] M.J.T. Smith and T.P. Barnwell, Exact reconstruction techniques for tree-structured subband coders , IEEE Trans. Acoust. Speech Signal Process 34 (1986), 434-441.
[13] M. Vetterli, Filter banks allowing perfect reconstruction , Signal Process. 10 (1986), no. 3, 219-244.
[14] L. Villemoes, Energy moments in time and frequency for two-scale difference equation solutions and wavelets , preprint, Math. Institute, Univ. of Denmark, 1991. · Zbl 0759.39005
[15] M. Vetterli and C. Herley, Wavelets and filter banks: theory and design , to appear in IEEE Trans. Acoust. Speech Signal Process. · Zbl 0825.94059
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