Document Type : Research Paper


1 Department of Mathematics Indraprastha college for Women, University of Delhi, Delhi 110054, India.

2 Department of Mathematics and Statistics, University college of Science, M.L.S. University, Udaipur, Rajasthan, India.


In this note, the notion of generalized continuous K- frame in a Hilbert space is defined. Examples have been given to exhibit the existence of generalized continuous $K$-frames. A necessary and sufficient condition for the existence of a generalized continuous $K$-frame in terms of its frame operator is obtained and a characterization of a generalized continuous $K$-frame for $ \mathcal{H} $ with respect to $ \mu $ is given. Also, a sufficient condition for a generalized continuous $K$-frame is given.  Further, among other results, we prove that generalized continuous $K$-frames are invariant under a linear  homeomorphism. Finally, keeping in mind the importance of perturbation theory in various branches of applied mathematics, we study perturbation of $K$-frames and obtain conditions for the stability of generalized continuous $K$-frames.


[1] A. Aldroubi, C. Cabrelli, and U. Molter, Wavelets on irregular grids with arbitrary dilation matrices and frame atoms for $L^2({Bbb R}^d)$, Appl. Compt. Harmon. Anal., 17 (2004), pp. 119-140.
[2] S. T. Ali, J.-P. Antoine and J.-P. Gazeau, Continuous frames in Hilbert space, Ann. Phy., 222 (1993), pp. 1--37.
[3] E. Alizadeh, A. Rahimi, E. Osgooei, and M. Rahmani, Continuous K-G-frames in Hilbert spaces, Bull. Iran. Math. Soc., 45 (2019), pp. 1091-1104.
[4] M.S. Asgari, Characterizations of fusion frames (frames of subspaces), Proc. Indian Acad. Sci. (Math. Sci.), 119 (2009), pp. 369-382.
[5] P. Balazs and D. Bayer, and A. Rahimi, Multipliers for continuous frames in Hilbert spaces, J. Physics A, 45 (2012), 2240023 (20 p).
[6] B. A. Barnes, Majorization, range inclusion, and factorization for bounded linear operators, Proc. Amer. Math. Soc., 133 (2005), pp. 155-162.
[7] P. G. Casazza, G. Kutyniok, Frames of subspaces, Wavelets, frame and operator theory, 87-113, Comptemp. Math., 345, Amer. Math. Soc., Providence, Rl, 2004.
[8] O. Christensen, An introduction to frames and Riesz bases, Birkhauser, 2016.
[9] O. Christensen and Y.C. Eldar, Oblique dual frames with shift-invariant spaces, Appl. Compt. Harm. Anal., 17 (2004), pp. 48-68.
[10] I. Daubechies, A.Grossman, and Y.Meyer, Painless non-orthogonal expansions, J. Math. Physics, 27 (1986), 1271-1283.
[11] D.X. Ding, Generalized continuous frames constructed by using an iterated function system, J. Geom. Phys., 61 (2011), pp. 1045-1050.
[12] R. J. Duffin and A. C. Schaeffer, A class of non-harmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), pp. 341-366.
[13] Y.C. Eldar, Sampling with arbitrary sampling and reconstruction spaces and oblique dual frame vectors, J. Four. Anal. Appl., 9 (2003), pp. 77-96.
[14] M. Fornasier, Quasi-orthogonal decompositions of structured frames, J. Math. Anal. Appl., 289 (2004), 180-199.
[15] J. P. Gabardo and D. Han, Frame associated with measurable spaces, Adv. Comp. Math., 18 (2003), pp. 127-147.
[16] L. Gavruta, Frames for operators, Appl. Comp. Harm. Anal., 32 (2012), pp. 139-144.
[17] L. Gavruta, New results on frame for operators, Analele Universitatii Oradea Fasc. Matematica, Tom XIX (2012), pp. 55-61.
[18] G. Kaiser, A friendly guide to wavelets, Birkhauser, Boston, MA, 1994.
[19] S. K. Kaushik, L. K. Vashisht, and S. K. Sharma, Some results concerning frames associated with measurable spaces, TWMS J. Pure Appl. Math., 4 (2013), pp. 52-60.
[20] A. Khosravi and B. Khosravi, Fusion frames and G-frames in Banach spaces, Proc. Indian Acad. Sci. (Math. Sci.), 121 (2011), pp. 155-164.
[21] S. Li and H. Ogawa, Pseudo frames for subspaces with applications, J. Fourier Anal. Appl., 10 (2004), pp. 409-431.
[22] A. Rahimi, Multipliers of generalized frames in Hilbert spaces, Bull. Iranian Math. Soc., 37 (2011), pp. 63-80.
[23] A. Rahimi, and P. Balazs, Multipliers for p-Bessel sequences in Banach spaces, Integral Equations Oper. Theory, 68 (2010), pp. 193-205.
[24] X.C. Xiao, Y.C. Zhu, L. Gavruta, Some properties of $K$-frames in Hilbert spaces, Results Math., 63 (2013), pp. 1243–1255.
[25] W. Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl., 322 (2006), pp. 437-452.
[26] C.T. Shieh and V.A. Yurko, Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl., 347 (2008), pp. 266-272.
[27] G. Teschl, Mathematical Methods in Quantum Mechanics; With Applications to Schrodinger Operators, Graduate Studies in Mathematics, Amer. Math. Soc., Rhode Island, 2009.
[28] J. Li, M. Yasuda, and J. Song, Regularity properties of null-additive fuzzy measure on metric space, in: Proc. 2nd Internatinal Conference on Modeling Decisions for Artificial Intelligencer, Tsukuba, Japan, 2005, pp. 59-66.