Ali Akbar AREFİJAMAAL, Fahimeh ARABYANI NEYSHABURI, Mitra SHAMSABADİ

On the duality of frames and fusion frames

Optimal frame bounds play a key role in many applications of frame theory, such as filter banks. In this paper, we study the relation between the bounds of a frame and its alternate dual and then present some approach to construct a family of Parseval frames. Also, we survey some problems on duals of fusion frames. In particular, we discuss on some essential differences between duals of ordinary frames and fusion frames. Finally, we characterize duals of some fusion frames.

Keywords:

Frame bounds dual frames, fusion frames, dual fusion frames,

PDF

___

Ali, S. T. Antoine, J. P. and Gazeau, J. P. Continuous frames in Hilbert spaces. Ann. Physics. 222, 1-37, 1993.
Arefijamaal, A. and Zekaee, E. Signal processing by alternate dual Gabor frames, Appl. Comput. Harmon. Anal. 35, 535-540, 2013.
Benedetto,J. Powell, A. and Yilmaz, O. Sigm-Delta quantization and finite frames. IEEE Trans. Inform. Theory. 52, 1990-2005, 2006.
Bodmannand,B. G. and Paulsen, V. I. Frames, graphs anderasures, Linear. Algebra. Appl. 404, 118-146, 2005.
Bolcskel, H., Hlawatsch, F. and Feichtinger, H. G. Frame-theoretic analysis of oversampled filter banks. IEEE Trans. Signal Process. 46, 3256-3268, 1998.
Candes, E. J. and Donoho, D. L. New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities. Commun. Pure Appl. Anal. 56, 216-266, 2004.
Casazza, P. G. and Kutyniok, G. Frames of subspaces, Contemp. Math. 345, 87-114, 2004.
Casazza, P. G., Kutyniok, G. and Li, S. Fusion frames and distributed processing, Appl. Comput. Harmon. Anal. 25 (1), 114-132, 2008.
Casazza, P. G., Kutyniok, G., Li, S. and Rozell, c. J. Modeling Sensor Networks with Fusion Frames,Wavelets XII (San Diego, CA, 2007), 67011M-1–67011M-11, SPIE Proc. 6701, SPIE, Bellingham, WA.
Casazza, P. G., Kutyniok, G. and Lammers, M. C. Duality principles in frame theory, J. Fourier Anal. Appl. 10(4), 383–408, 2004.
Christensen, O. Frames and Bases: An Introductory Course, Birkhäuser, Boston. 2008.
Duffin, R. and Schaeffer, A. A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72, 341-366, 1952.
Gavruµa, P. On the duality of fusion frames, J. Math. Anal. Appl. 333 (2), 871-879, 2007.
Kaftal, V., Larson, D. R. and Zhang, Sh. Operator-valued frames, Trans. Amer. Math. Soc. 361, 6349–6385, 2009.
Leng, J., Gue, Q. and Huang, T. The duals of fusion frames for experimental data transmission coding of energy physics, Hindawi. Publ. Corp. 2013, 1-9, 2013.
Mller, V. Spectral theory of linear operators and spectral systems in Banach algebras, Operator Theory: Advances and Applications, 139. Birkhuser Verlag, Basel, 2003.
Ruiz, M. A. and Stojanoff, D. Some properties of frames of subspaces obtained by operator theory methods, J. Math. Anal. Appl. 343, 366-378, 2008.
Sadeghi, Gh. and Arefijamaal, A. von Neumann-Schatten frames, Medi. J. Math. 9 (3), 525-535, 2012.
Sun, W. G-frames and G-Riesz bases. J. Math. Anal. Appl. 322, 437-452, 2006.