Some properties of controlled K-frames in Hilbert spaces

Document Type : Research Paper


1 Department of Mathematics, Damghan University, Damghan and Department of Basic Sciences, Faculty of Valiasr, Tehran Branch, Technical and Vocational University (TVU), Tehran, Islamic Republic of Iran.

2 Department of Basic Sciences, Faculty of Enghelab-e Eslami, Tehran Branch, Technical and Vocational University (TVU), Tehran, Islamic Republic of Iran.

3 Department of Mathematics, Damghan University, P.O.Box 36716-41167, Damghan, Islamic Republic of Iran.



    In this paper, we reintroduce the concept of controlled K-frames and then, we show that this definition is equivalent
with the concept that has been recently introduced in \cite{Nouri}. Meanwhile, we correct
one of the results which was obtained in the mentioned paper. In the sequel, we construct some new controlled K-frames
by some operator theory tools. Finally, we provide some conditions under which the sum of two controlled K-frames remains
a controlled K-frame.


[1] S.T. Ali, J.P. Antoine and J.P. Gazeau, Continuous frames in Hilbert space, Ann. Phys., 222 (1993), 1-37.
[2] P. Balazs, B. Laback, G. Eckel and W.A. Deutsch, Time-frequency sparsity by removing perceptually irrelevant components using a simple model of simultaneous masking, The IEEE/ACM Transactions on Audio, Speech, and Language Processing, 18 (2010), 34-49.
[3] P. Balazs, G.P. Antoine and A. Grybo{'s}, Weighted and controlled frames: Mutual relationship and first numerical properties, Int. J. Wavelets Multiresolut. Inf. Process., 8(1) (2010), 109-132.
[4] P. Balazs, M. D"orfler, N. Holighaus, F. Jaillet and G. Velasco, Theory, implementation and applications of nonstationary Gabor frames, J. Comput. Appl. Math., 236 (2011), 1481-1496.
[5] J.J. Benedetto and S. Li, The theory of multiresolution analysis frames and applications to filter banks, Appl. Comput. Harmon. Anal., 5 (1998), 389-427.
[6] H. B{"o}lcskei, F. Hlawatsch and H.G. Feichtinger, Frame-theoretic analysis of oversampled filter banks, IEEE Trans. Signal Process., 46 (1998), 3256-3268.
[7] P.G. Cassaza, G. Kutyniok and S. Li, Fusion frames and distributed processing, Appl. Comput. Harmon. Anal, 25(1) (2008), 114-132.
[8] N. Cotfas and J.P. Gazeau, Finite tight frames and some applications, J. Phys. A, Math. Theor., 43 (2010), 193001 (26pp).
[9] S. Dahlke, M. Fornasier and T. Raasch, Adaptive frame methods for elliptic operator equations, Adv. Comput. Math., 27 (2007), 27-63.
[10] I. Daubechies, A. Grossmann and Y. Meyer, Ainless nonorthogonal expansions, J. Math. Phys., 27 (1986), 1271-1283.
[11] R.G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Am. Math. Soc., 17 (1966), 413-415.
[12] R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Am. Math. Soc., 72 (1952), 341-366.
[13] H.G. Feichtinger and T. Werther, Atomic systems for subspaces, In: Zayed, L. (ed.) Proceedings SampTA 2001, Orlando, (2001), 163-165.
[14] D. Gabor, Theory of communication. Part 1: The analysis of information, Journal of the Institution of Electrical Engineers-Part III: Radio and Communication Engineering, 93(26) (1946),  429-441.
[15] L. G{u{a}}vru{c{t}}a, Frames for operators, Appl. Comput. Harmon. Anal., 32 (2012), 139-144.
[16] D. Hua and Y. Huang, Controlled K-g-frames in Hilbert spaces, Result. Math., 72 (2017), 1227-1283.
[17] L. Jacques, Ondelettes, Rep{'e}res et Couronne Solaire, Th{'e}se de Doctorat, Univ. Cath. Louvain, Louvain-la-Neuve, (2004).
[18] A. Khosravi and K. Musazadeh, Controlled fusion frames, Methods Funct. Anal. Topol., 18(3) (2012), 256-265.
[19] D. Li and J. Leng, Generalized frames and controlled operators in Hilbert spaces, Ann. Funct. Anal., 10(4) (2017), 537-552.
[20] P. Majdak, P. Balazs, W. Kreuzer and M. D"orfler, newblock A time-frequency method for increasing the signal-to-noise ratio in system identification with exponential sweeps, In Proceedings of the 36th International Conference on Acoustics, Speech and Signal Processing, ICASSP 2011, Prag, (2011).
[21] M. Nouri, A. Rahimi and Sh Najafzadeh, Controlled K-frames in Hilbert Spaces, J. Ramanujan Soc. Math. Math. Sci., 4(2) (2015), 39-50.
[22] A. Rahimi and A. Fereydooni, Controlled G-frames and their G-multipliers in Hilbert spaces, An. Stiint. Univ. “Ovidius” Constanta, Ser. Mat., 21(2) (2013), 223-236.
[23] G. Ramu and P.S. Johnson, Frame operators of K-frames, SeMA J.}, 73(2) (2016), 171-181.
[24] R. Stevenson, Adaptive solution of operator equations using wavelet frames, SIAM J. Numer. Anal., 41 (2003), 1074-1100.[25] W. Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl., 322 (2006), 437-452.
[26] X. Xiao, Y. ZhuL and L. G{u{a}}vru{c{t}}a, Some properties of K-frames in Hilbert spaces, Result. Math., 63 (2013), 1243-1255.