Abstract
Improving and extending the concept of dual for frames, fusion frames and continuous frames, the notion of dual for continuous fusion frames in Hilbert spaces will be studied. It will be shown that generally the dual of c-fusion frames may not be defined. To overcome this problem, the new concept namely Q-dual for c-fusion frames will be defined and some of its properties will be investigated.
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Ali, S.T., Antoine, J.P., Gazeau, J.P.: Continuous frames in Hilbert spaces. Ann. Phys. 222, 1–37 (1993)
Antoine, J.-P., Balazs, P.: Frames and semi-frames. J. Phys. A: Math. Theor. 44, 479–501 (2011)
Arefijamaal, A., Zekaee, E.: Signal processing by alternate dual Gabor frames. Appl. Comput. Harmon. Anal. 35(3), 535–540 (2013)
Askari-Hemmat, A., Dehghan, M.A., Radjabalipour, M.: Generalized frames and their redundancy. Proc. Am. Math. Soc. 129, 1143–1147 (2001)
Balazs, P., Bayer, D., Rahimi, A.: Multipliers for continuous frames in Hilbert spaces. J. Phys. A: Math. Theor. 45, 244023 (2012) (20 pp)
Casazza, P.G., Kutyniok, G.: Frames of subspaces. In: Wavelets, frames and operator theory, College Park, MD, 2003. Contemp. Math. 345, 87–113. American Mathematical Society, Providence (2004)
Christensen, O.: Introduction to frames and Riesz bases. Birkhauser, Boston (2003)
Daubenchies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27, 1271–1283 (1986)
Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)
Faroughi, M.H., Ahmadi, R.: C-fusion frames. J. Appl. Sci. 8(16), 2881–2887 (2008)
Faroughi, M.H., Ahmadi, R., Afsar, Z.: Some properties of c-frame of subspaces. J. Nonlinear Sci. Appl. 1(3), 155–168 (2008)
Fereydooni, A., Safapour, A., Rahimi, A.: Adjoint of pair frames. U. P. B. Sci. Bull. Ser. A 74(4), 131–140 (2012)
Fornasier, M., Rauhut, H.: Continuous frames, function spaces and the discretization problem. J. Fourier Anal. Appl. 11, 245–287 (2005)
Gabardo, J.P., Han, D.: Frame associated with measurable space. Adv. Comp. Math. 18(3), 127–147 (2003)
Găvruţa, P.: On the duality of fusion frames. J. Math. Anal. Appl. 333, 871–879 (2007)
Ghaani Farashahi, A.: Abstract harmonic analysis of wave packet transforms over locally compact Abelian groups. Banach J. Math. Anal. (2016). http://projecteuclid.org/accepted/euclid.bjma
Heineken, S.B., Morillas, P.M., Benavente, A.M., Zakowicz, M.I.: Dual fusion frame. Arch. Math. 103, 355–365 (2014)
Janssen, A.J.E.M.: The duality condition for Weyl-Heisenberg frames. In: Feichtinger, H.G., Strohmer, T. (eds.) Gabor analysis: theory and applications. Birkhauser, Boston (1998)
Kaiser, G.: A friendly guide to wavelets. Birkhauser, Boston (1994)
Laugesen, R.S.: Gabor dual spline windows. Appl. Comput. Harmon. Anal. 27, 180–194 (2009)
Najati, A., Rahimi, A., Faroughi, M.H.: Continuous and discrete frames of subspaces in Hilbert spaces. Southeast Asian Bull. Math. 32, 305–324 (2008)
Rahimi, A., Najati, A., Dehghan, Y.N.: Continuous frames in Hilbert spaces. Methods Funct. Anal. Topol. 12, 170–182 (2006)
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Rahimi, A., Darvishi, Z. & Daraby, B. On the duality of c-fusion frames in Hilbert spaces. Anal.Math.Phys. 7, 335–348 (2017). https://doi.org/10.1007/s13324-016-0146-4
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DOI: https://doi.org/10.1007/s13324-016-0146-4