Document Type : Research Paper

Authors

1 Department of Mathematics, Shabestar Branch, Islamic Azad University, Shabestar, Iran.

2 Research Institute for Fundamental Sciences, University of Tabriz, Tabriz, Iran.

3 Faculty of Physic, University of Tabriz, Tabriz, Iran.

Abstract

In this manuscript, we study the relation between K-fusion frame and its local components which leads to the definition of a $C$-controlled $K$-fusion frames, also we extend a theory based on K-fusion frames on Hilbert spaces, which prepares exactly the frameworks not only to model new frames on Hilbert spaces but also for deriving robust operators. In particular, we define the analysis, synthesis and frame operator for $C$-controlled $K$-fusion frames, which even yield a reconstruction formula. Also, we define dual of $C$-controlled $K$-fusion frames and study some basic properties and perturbation of them.

Keywords

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[1] Y. Alizadeh and M. Abdollahpour, Controlled Continuous G -Frames and Their Multipliers in Hilbert Spaces, Sahand Commun. Math. Anal., 15(1) (2019), pp. 37-48.

[2] F. Arabyani and A. Arefijamal, Some constructions of K-frames and their duals, Rocky Mountain., 47 (2017), pp. 1749-1764.

[3] P. Balazs, J.-P. Antoine and A. Grybo's, Weighted and Controlled frames: mutual relationship and first numerical properties, Int. J. Wavelets, Multi. Info. Proc., 8(1) (2010), pp. 109-132.

[4] B.G. Bodmann and V.I. Paulsen, Frame paths and error bounds for sigma-delta quantization, Appl Comput Harmon Anal., 22 (2007), pp. 176-197.

[5] I. Bogdanova, P. Vandergheynst, J.P. Antoine, L. Jacques and M. Morvidone, Stereographic wavelet frames on the sphere, Appl. Comput. Harmon. Anal., 16 (2005), pp. 223-252.

[6] O. Christensen, An Introduction to Frames and Riesz Basesو Birkhauser, 2016.

[7] R.G. Douglas, On majorization, Factorization and Range Inclusion of Operators on Hilbert Spaces, Proc Amer. Math. Soc., 17 (1996), pp. 413-415.

[8] R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Am. Math. Soc., 72 (1952), pp. 341-366.

[9] A. Fang and P. Tong Li, K-fusion Frames and the Corresponding Generators for Unitary Systems, Acta Math. Sci., (2018), pp.843-854.

[10] H. Jamali and M. Kolahdouz, Richardson and Chebyshev Iterative Methods by Using G-frames, Sahand Commun. Math. Anal., 13(1) (2019), pp. 129-139.

[11] D. Hua and Y. Huang, Controlled K-G-Frames in Hilbert Spaces, Results. Math., 523 (2016), pp. 152-168.

[12] A. Khosravi and K. Musazadeh, Controlled fusion frames, Meth. Func. Anal. Topol.,18(3), (2012), pp. 256-265.

[13] K. Musazadeh and K. Khandani, Some results on controlled frames in Hilbert space, Acta Math. Sci., 36(3)(2016), pp. 655-665.

[14] A. Rahimi and A. Fereydooni, Controlled G-Frames and Their G-Multipliers in Hilbert spaces, Analele Stiintifice ale Universitatii Ovidius Constanta., 2(12), (2012), pp. 223-236.

[15] A. Rahimi , SH. Najafzadeh and M.Nouri, Controlled K-frames in Hilbert spaces, International journal of Analysis and Applications., 4(2), (2015), pp. 39-50.

[16] G. Rahimlou, R. Ahmadi, M. Jafarizadeh and S. Nami, Continuous K -Frames and their Dual in Hilbert Spaces, Sahand Commun. Math. Anal., 17(3), (2020), pp. 145-160.

[17] M. Rashidi-Kouchi, A. Rahimi and Firdous A. Shah, Duals and multipliers of controlled frames in Hilbert spaces, Int. J. Wavelets Multiresolut. Inf. Process., 16(5), (2018), pp. 1-13.

[18] M. Rashidi-Kouchi, Frames in super Hilbert modules, Sahand Commun. Math. Anal., 9(1), (2018), pp. 129-142.

[19] V. Sadri, R. Ahmadi, M. Jafarizadeh and S. Nami, Continuous K -Fusion Frames in Hilbert Spaces, Sahand Commun. Math. Anal., 17(1), (2020), pp. 39-55.

[20] W. Sun, G-frames and g-Riesz bases, J. Math. Anal., 322 (2006), pp. 437-452.