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Orthogonal frames of translates. (English) Zbl 1042.42038
Summary: Two Bessel sequences are orthogonal if the composition of the synthesis operator of one sequence with the analysis operator of the other sequence is the 0 operator. We characterize when two Bessel sequences are orthogonal when the Bessel sequences have the form of translates of a finite number of functions in \(L^2(\mathbb R^d)\). The characterizations are applied to Bessel sequences which have an affine structure, and a quasi-affine structure. These also lead to characterizations of superframes. Moreover, we characterize perfect reconstruction, i.e., duality, of subspace frames for translation invariant (bandlimited) subspaces of \(L^2(\mathbb R^d)\).

MSC:
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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[1] Aldroubi, A, Non-uniform weighted average sampling and reconstruction in shift-invariant and wavelet spaces, Appl. comput. harmon. anal., 13, 2, 151-161, (2002) · Zbl 1016.42022
[2] Aldroubi, A, A portrait of frames, Proc. amer. math. soc., 123, 6, 1661-1668, (1995) · Zbl 0851.42030
[3] A. Aldroubi, C. Cabrelli, U. Molter, Wavelets on irregular grids with arbitrary dilation matrices and frame atoms for \(L\^{}\{2\}(R\^{}\{d\})\), Preprint, 2003 · Zbl 1060.42025
[4] A. Aldroubi, D. Larson, W.S. Tang, E. Weber, The geometry of frame representations of Abelian groups, 2002, submitted for publication · Zbl 1054.43008
[5] R. Balan, Weyl-Heisenberg super frames, Preprint, 1999
[6] Balan, R, Multiplexing of signals using superframes, (), 118-130
[7] Balan, R; Daubechies, I; Vaishampayan, V, The analysis and design of windowed Fourier frame based multiple description source coding schemes, IEEE trans. inform. theory, 46, 2491-2536, (2000) · Zbl 0998.94011
[8] R. Balan, Z. Landau, Topologies of Weyl-Heisenberg sets, Preprint, 2002
[9] Benedetto, J; Li, S, The theory of multiresolution analysis frames and applications to filter banks, Appl. comput. harmon. anal., 5, 4, 389-427, (1998) · Zbl 0915.42029
[10] Bownik, M, A characterization of affine dual frames in \(L\^{}\{2\}(R\^{}\{n\})\), Appl. comput. harmon. anal., 8, 2, 203-221, (2000) · Zbl 0961.42018
[11] M. Bownik, E. Weber, Affine frames, GMRA’s, and the canonical dual, Studia Math. (2003), in press · Zbl 1063.42023
[12] Chui, C; Czaja, W; Maggioni, M; Weiss, G, Characterization of general tight wavelet frames with matrix dilations and tightness preserving oversampling, J. Fourier anal. appl., 8, 2, 173-200, (2002) · Zbl 1005.42020
[13] Daubechies, I; Grossmann, A; Meyer, Y, Painless nonorthogonal expansions, J. math. phys., 27, 5, 1271-1283, (1986) · Zbl 0608.46014
[14] Daubechies, I; Han, B, The canonical dual frame of a wavelet frame, Appl. comput. harmon. anal., 12, 3, 269-285, (2002) · Zbl 1013.42023
[15] Duffin, R; Schaeffer, A, A class of nonharmonic Fourier series, Trans. amer. math. soc., 72, 341-366, (1952) · Zbl 0049.32401
[16] Frazier, M; Jawerth, B, A discrete transform and decomposition of distribution spaces, J. funct. anal., 93, 34-170, (1990) · Zbl 0716.46031
[17] Han, D; Larson, D, Frames, bases and group representations, (), No. 697 · Zbl 0971.42023
[18] Hernandez, E; Labate, D; Weiss, G, A unified characterization of reproducing systems generated by a finite family II, J. geom. anal., 12, 4, 615-662, (2002) · Zbl 1039.42032
[19] Labate, D, A unified characterization of reproducing systems generated by a finite family, J. geom. anal., 12, 3, 469-491, (2002) · Zbl 1029.42026
[20] Li, S; Ogawa, H, Pseudo-duals of frames with applications, Appl. comput. harmon. anal., 11, 2, 289-304, (2001) · Zbl 0984.42024
[21] Ron, A; Shen, Z, Affine systems in \(L\^{}\{2\}(R\^{}\{d\})\) II: dual systems, J. Fourier anal. appl., 3, 5, 617-637, (1997)
[22] Ron, A; Shen, Z, Affine systems in \(L\^{}\{2\}(R\^{}\{d\})\): the analysis of the analysis operator, J. funct. anal., 148, 2, 408-447, (1997) · Zbl 0891.42018
[23] E. Weber, The geometry of sampling on unions of lattices, Proc. Amer. Math. Soc. (2002), in press
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