Document Type : Research Paper

**Authors**

Department of Mathematics, Science and Research Branch, Islamic Azad University, Kerman, Iran.

**Abstract**

A generalization of the known results in fusion frames and $g$-frames theory to continuous fusion frames which defined by M. H. Faroughi and R. Ahmadi, is presented in this study. Continuous resolution of the identity (CRI) is introduced, a new family of CRI is constructed, and a number of reconstruction formulas are obtained. Also, new results are given on the duality of continuous fusion frames in Hilbert spaces.

**Keywords**

**Main Subjects**

[1] M.R. Abdollahpour and M.H. Faroughi, *Continuous $g$-frames in Hilbert spaces*, Southeast Asian Bull. Math., 32, 1-19(2008).

[2] S.T. Ali, J.P. Antoine, and J.P Gazeau, *Continuous frames in Hilbert spaces*, Annals of Physics. 222, 1-37 (1993).

[3] M.S. Asgari and A. Khosravi, *Frames and bases of subspaces in Hilbert spaces*, J. Math. Anal. Appl., 308, 541-553 (2005).

[4] O. Christensen, *An introduction to frame and Riesz bases*, Birkhäuser, Boston, 2003.

[5] I. Daubechies, A. Grossmann, and Y. Meyer, *Painless nonorthogonal expansions*, J. Math. Phys., 27, 1271-1283 (1986).

[6] I. Daubechies, *Ten Lectures on Wavelets*, CBMS-NSF conference series in applied mathematics, 61, SIAM, Philadelphia, 1992.

[7] R. Duffin and A. Schaeffer, *A class of nonharmonic Fourier series*, Trans. Amer. Math. Soc., 72, 341-366 (1952).

[8] M.H. Faroughi and R. Ahmadi, *Fusion integral*, J. Mathematische Nachrichten. 284, No. 5-6, 681-693 (2011).

[9] M.H. Faroughi and R. Ahmadi, *Some properties of $C$-fusion frames*, Turk J Math., 34, 393-415. (2010).

[10] J.P. Gabardo and D. Han, *Frames Associated with Measurable Space*, Adv. Comp. Math., 18, 127-147 (2003).

[11] P. Gavruta, *On the duality of fusion frames in Hilbert spaces*, J. Math. Anal. Appl., 333, 871-879 (2007).

[12] G. Kaiser, *A Friendly Guide to Wavelets*, Birkhäuser, 1994.

[13] A. Khosravi and K. Musazadeh, *Fusion frames and $g$-frames*, J. Math. Anal. Appl., 342, 1068-1083 (2008).

[14] W. Sun, *$g$-frame and $g$- Riesz bases*, J. Math. Anal. Appl., 322, 437-452 (2006).