^{}Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, P.O. Box 1159-91775, Iran.

Abstract

$K$-frames as a generalization of frames were introduced by L. G\u{a}vru\c{t}a to study atomic systems on Hilbert spaces which allows, in a stable way, to reconstruct elements from the range of the bounded linear operator $K$ in a Hilbert space. Recently some generalizations of this concept are introduced and some of its difference with ordinary frames are studied. In this paper, we give a new generalization of $K$-frames. After proving some characterizations of generalized $K$-frames, new results are investigated and some new perturbation results are established. Finally, we give several characterizations of $K$-duals.

[1] A. Abdollahi and E. Rahimi, Some results on g-frames in Hilbert spaces, Turk. J. Math., 35 (2011) 695-704.

[2] M.R. Abdollahpour, M.H. Faroughi, and A. Rahimi, Pg-frames in Banach spaces, Methods Funct. Anal. Topology, 13 (2007) no. 3, 201-210.

[3] A.A. Arefijamaal and S. Ghasemi, On characterization and stability of alternate dual of g-frames, Turk. J. Math., 37 (2013) 71-79.

[4] M.S. Asgari and H. Rahimi, Generalized frames for operators in Hilbert spaces, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 17 (2014) no. 2, 1450013 (20 pages).

[5] H. Bolcskei, F. Hlawatsch and H.G. Feichtinger, Frame-theoretic analyssis of oversampled filter banks, IEEE Trans. Signal Process., 46 (1998) 3256-3268.

[6] O. Christensen, An Introduction to Frame and Riesz Bases, Birkhäuser, 2002.

[7] B. Dastourian and M. Janfada, *-frames for operators on Hilbert modules, Wavelets and Linear algebras., 3 (2016) 27-43.

[8] B. Dastourian and M. Janfada, Frames for operators in Banach spaces via semi-inner products, Int. J. Wavelets Multiresult. Inf. Process., 14 (2016) no. 3, 1650011 (17 pages).

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

[10] R.G. Douglas, On majorization, factorization and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc., 17 (1966) no. 2, 413-415.

[11] N.E. Dudey Ward and J.R. Partington, A construction of rational wavelets and frames in Hardy-Sobolev space with applications to system modelling, SIAM J. Control Optim., 36 (1998) 654-679.

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

[13] Y.C. Eldar, Sampling with arbitrary sampling and reconstruction spaces and oblique dual frame vectors, J. Fourier. Anal. Appl., 9 (2003) no. 1, 77-96.

[14] Y.C. Eldar and T. Werther, General framework for consistent sampling in Hilbert spaces, Int. J. Wavelets Multi. Inf. Process., 3 (2005) no. 3, 347-359.

[15] H.G. Feichtinger and T. Werther, Atomic systems for subspaces, in: L. Zayed (Ed.), Proceedings SampTA 2001, Orlando, FL, (2001) 163-165.

[16] P.J.S.G. Ferreira, Mathematics for multimedia signal processing II: Discrete finite frames and signal reconstruction, In: Byrnes, J.S. (ed.) Signal processing for multimedia, IOS Press, Amsterdam (1999) 35-54.

[17] L. Gávruta, Frames for operators, Appl. Comput. Harmon. Anal., 32 (2012) 139-144.

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

[19] A. Najati, M.H. Faroughi, and A. Rahimi, G-frames and stability of g-frames in Hilbert space, Methods Funct. Anal. Topology, 14 (2008) 271-286.

[20] F.A. Neyshaburi and A.A. Arefijamaal, Some constructions of K-frames and their duals, To appear in Rocky Mountain J. Math.

[21] S. Obeidat, S. Samarah, P.G. Casazza, and J. C. Tremain, Sums of Hilbert space frames, J. Math. Anal. Appl., 351 (2009) 579-585.

[22] T. Strohmer and R. Jr. Heath, Grassmanian frames with applications to coding and communications, Appl. Comput. Harmon. Anal., 14 (2003) 257-275.

[23] W. Sun, g-frames and g-Riesz bases, J. Math. Anal. Appl., 322 (2006) 437–452.

[24] Y.J. Wang and Y.C. Zhu, G-Frames and g-Frame sequences in Hilbert spaces, Acta Mathematica Sinica, 25 (2009) no. 12, 2093-2106.

[25] X.C. Xiao and X.M. Zeng, Some properties of g-frames in Hilbert C^{*}-modules, J. Math. Anal. Appl., 363 (2010) 399-408.

[26] X. Xiao, Y. Zhu, and L. Gávruta, Some properties of $K$-frames in Hilbert spaces, Results. Math., 63 (2013) no. 3-4, 1243-1255.

[27] X. Xiao, Y. Zhu, Z. Shu, and M. Ding, G-frames with bounded linear operators, Rocky Mountain J. Math., 45 (2015) no. 2, 675-693.