Document Type : Research Paper
Authors
- Vahid Sadri ^{1}
- Reza Ahmadi ^{} ^{2}
- Mohammad Jafarizadeh ^{3}
- Susan Nami ^{3}
^{1} Department of Mathematics, Shabestar Branch, Islamic Azad University Shabestar, Iran.
^{2} Institute of Fundamental Sciences, University of Tabriz, Tabriz, Iran.
^{3} Faculty of Physic, University of Tabriz, Tabriz, Iran.
Abstract
The study of the c$k$-fusions frames shows that the emphasis on the measure spaces introduces a new idea, although some similar properties with the discrete case are raised. Moreover, due to the nature of measure spaces, we have to use new techniques for new results. Especially, the topic of the dual of frames which is important for frame applications, have been specified completely for the continuous frames. After improving and extending the concept of fusion frames and continuous frames, in this paper we introduce continuous $k$-fusion frames in Hilbert spaces. Similarly to the c-fusion frames, dual of continuous $k$-fusion frames may not be defined, we however define the $Q$-dual of continuous $k$-fusion frames. Also some new results and the perturbation of continuous $k$-fusion frames will be presented.
Keywords
[1] F. Arabyani Neyshaburi and A.A. Arefijamaal, Characterization and Construction of $k$-Fusion Frames and Their Duals in Hilbert Spaces, Results. Math., (2018) to appear.
[2] H. Blocsli, H.F. Hlawatsch, and H.G. Fichtinger, Frame-Theoretic analysis of oversampled filter bank, IEEE Trans. Signal Processing., 46 (1998), pp. 3256- 3268.
[3] E.J. Candes and D.L. Donoho, New tight frames of curvelets and optimal representation of objects with piecwise $C^2$ singularities, Comm. Pure and App. Math., 57 (2004), pp. 219-266.
[4] P.G. Casazza and O. Christensen, Perturbation of Operators and Application to Frame Theory, J. Fourier Anal. Appl., 3 (1997), pp. 543-557.
[5] P.G. Casazza and G. Kutyniok, Frames of Subspaces, Contemp. Math., 345 (2004), pp. 87-114.
[6] P.G. Casazza, G. Kutyniok, and S. Li, Fusion Frames and distributed processing, Appl. comput. Harmon. Anal., 25 (2008), pp. 114--132.
[7] P.G. Casazza and J.C. Tremain, Consequences of The Marcus/Spielman/Srivastava Solution of The Kadison-Singer Problem, New Trends in Appl. Harm. Anal., (2016), pp. 191-213.
[8] O. Christensen, Frames and Bases: An Introductory Course (Applied and Numerical Harmonic Analysis), Birkhauser, 2008.
[9] R.G. Douglas, On majorization, Factorization and Range Inclusion of Operators on Hilbert Spaces, Proc Amer. Math. Soc., 17 (1996), pp. 413-415.
[10] R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), pp. 341-366.
[11] M.H. Faroughi and R. Ahmadi, C-Fusion Frame, J. of Appl. Sci., 8 (2008), pp. 2881-2887.
[12] M.H. Faroughi and R. Ahmadi, Fusion Integral, Math. Nachr., 284 (2011), pp. 681-693.
[13] M.H. Faroughi and R. Ahmadi, Some Properties of C-Frames of Subspaces, J. Nonlinear Sci. Appl., 1 (2008), pp. 155-168.
[14] M.H. Faroughi and E. Osgooei, C-Frames and C-Bessel Mappings, BIMS., 38 (2012), pp. 203-222.
[15] H.G. Feichtinger and T. Werther, Atomic Systems for Subspaces, Proceedings SampTA. Orlando, FL, (2001), pp. 163-165.
[16] L. Gavruta, Frames for Operators, Appl. Comp. Harm. Annal., 32 (2012), pp. 139-144.
[17] B. Hassibi, B. Hochwald, B. Shokrollahi, and W. Sweldens, Representation theory for high-rate multiple-antenna code design, IEEE Trans. Inform. Theory., 47 (2001), pp. 2335-2367.
[18] R. Kadison and I. Singer, Extensions of pure states, American Journal of Math., 81 (1959), pp. 383-400.
[19] M. Khayyami and A. Nazari, Construction of Continuous $g$-Frames and Continuous Fusion Frames, Sahand Commun. Math. Anal., 4 (2016), pp. 43-55.
[20] A. Najati, A. Rahimi, and M.H. Faroughi, Continuous and Discrete Frames of Subspaces in Hilbert Space, South. Asian Bull. Math., 32 (2008), pp. 305-324.
[21] A. Rahimi, Z. Darvishi, and B. Daraby, On The Duality of C-Fusion Frames in Hilbert Spaces, Anal. and Math. Phys., 7 (2016), pp. 335-348.
[22] W. Sun, G-Frames and G-Riesz bases, J. Math. Anal. Appl., 326 (2006), pp. 437-452.
[23] X. Xiao, Y. Zhu, and L. Gavruta, Some Properties of $k$-Frames in Hilbert Spaces, Results. Math., 63 (2013), pp. 1243-1255.