×

zbMATH — the first resource for mathematics

A pair of orthogonal frames. (English) Zbl 1116.42009
Summary: We start with a characterization of a pair of frames to be orthogonal in a shift-invariant space and then give a simple construction of a pair of orthogonal shift-invariant frames. This is applied to obtain a construction of a pair of Gabor orthogonal frames as an example. This is also developed further to obtain constructions of a pair of orthogonal wavelet frames.

MSC:
42C15 General harmonic expansions, frames
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] C. Bhatt, B. D. Johnson, E. Weber, Orthogonal wavelet frames and vector valued wavelet transforms, preprint. · Zbl 1136.42026
[2] de Boor, C.; DeVore, R.; Ron, A., The structure of finitely generated shift-invariant spaces in \(L_2(\mathbb{R}^d)\), J. funct. anal., 119, 37-78, (1994) · Zbl 0806.46030
[3] Bownik, M., The structure of shift-invariant subspaces of \(L^2(\mathbb{R}^n)\), J. funct. anal., 177, 283-309, (2000) · Zbl 0986.46018
[4] Christensen, O., Frames and pseudo-inverses, J. math. anal. appl., 195, 401-414, (1995) · Zbl 0845.47002
[5] Daubechies, I.; Han, B.; Ron, A.; Shen, Z., Framelets: MRA-based constructions of wavelet frames, Appl. comput. harmon. anal., 14, 1-46, (2003) · Zbl 1035.42031
[6] Goh, S.S.; Lim, Z.Y.; Shen, Z., Symmetric and antisymmetric tight wavelet frames, Appl. comput. harmon. anal., 20, 411-421, (2006) · Zbl 1106.42027
[7] Helson, H., Lectures on invariant subspaces, (1964), Academic Press New York · Zbl 0119.11303
[8] Holub, J.R., Pre-frame operators, Besselian frames and near-Riesz bases in Hilbert spaces, Proc. amer. math. soc., 122, 779-785, (1994) · Zbl 0821.46008
[9] Ron, A.; Shen, Z., Frames and stable bases for shift-invariant subspaces of \(L_2(\mathbb{R}^d)\), Canad. J. math., 47, 1051-1094, (1995) · Zbl 0838.42016
[10] Ron, A.; Shen, Z., Affine systems in \(L_2(\mathbb{R}^d)\): the analysis of the analysis operator, J. funct. anal., 148, 408-447, (1997) · Zbl 0891.42018
[11] Ron, A.; Shen, Z., Affine systems in \(L_2(\mathbb{R}^d)\) II: dual systems, J. Fourier anal. appl., 3, 617-637, (1997) · Zbl 0904.42025
[12] Ron, A.; Shen, Z., Weyl – heisenberg frames and Riesz bases in \(L_2(\mathbb{R}^d)\), Duke math. J., 89, 237-282, (1997) · Zbl 0892.42017
[13] Weber, E., Orthogonal frames of translates, Appl. comput. harmon. anal., 17, 69-90, (2004) · Zbl 1042.42038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.