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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 \(L^2R^d\); A. Aldroubi, C. Cabrelli, U. Molter, Wavelets on irregular grids with arbitrary dilation matrices and frame atoms for \(L^2R^d\) · 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; 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; R. Balan, Weyl-Heisenberg super frames, Preprint, 1999
[6] Balan, R., Multiplexing of signals using superframes, (Aldroubi, A.; Laine, A., Wavelets and Applications in Signal and Image Processing, vol. VIII. Wavelets and Applications in Signal and Image Processing, vol. VIII, SPIE Proceedings, vol. 4119 (2000)), 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; 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; 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, (Mem. Amer. Math. Soc., vol. 147 (September 2000), AMS: AMS Providence, RI), 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) · Zbl 0904.42025
[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; E. Weber, The geometry of sampling on unions of lattices, Proc. Amer. Math. Soc. (2002), in press
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