Document Type : Research Paper

Authors

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

2 Institute of Fundamental Science, University of Tabriz, Tabriz, Iran.

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

Abstract

The notion of $k$-frames was recently introduced by G\u avru\c ta in Hilbert  spaces to study atomic systems with respect to a bounded linear operator. A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous super positions. In this manuscript, we construct a continuous $k$-frame, so called c$k$-frame along with an atomic system for this version of frames. Also we introduce a new method for obtaining the dual of a c$k$-frame and prove some new results about it.

Keywords

[1] S.T. Ali, J.P. Antoine, and J.P. Gazeau, Continuous frames in Hilbert spaces, Ann.Phys., 222 (1993), pp. 1-37.
[2] S.T. Ali, J.P. Antoine, and J.P. Gazeau, Coherent States, Wavelets and their Generalizations, Springer Graduate Texts in Contemporary Physics, 1999.
[3] F. Arabyani and A.A. Arefijamal, Some constructions of $k$-frames and their duals, Rocky Mountain., 47(6)(2017), pp. 1749-1764.
[4] J. Benedetto, A. Powell, and O. Yilmaz, Sigma-Delta quantization and finite frames, IEEE Trans. Inform.Th., 52(2006), pp. 1990-2005.
[5] H. Bolcskel , F. Hlawatsch, and H.G Feichyinger, Frame-Theoretic analysis of oversampled filter bank, IEEE Trans. Signal Processing.,46(12)(1998), pp. 3256- 3268.
[6] E.J. Candes and D.L. Donoho, New tight frames of curvelets and optimal representation of objects with piecwise $C^2$ singularities, Comm. Pure and App. Math.,56 (2004), pp. 216-266.
[7] P.G. Casazza and G. Kutyniok, Frame of subspaces, Contemp. Math. 345, Amer. Math. Soc., Providence, RI., (2004), pp. 87-113.
[8] P.G. Casazza, G. Kutyniok, and S. Li, Fusion frames and Distributed Processing, Appl. Comput. Harmon. Anal.,25 (2008), pp. 114-132.
[9] P.G. Casazza and J. Kovacevic, Equal-norm tight frames with erasures, Adv. Comput. Math., 18 (2003), pp. 387-430.
[10] O. Christensen, An Introduction to Frames and Riesz Bases, 2nd ed. Birkhauser,Boston, 2016.
[11] I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal Expansions, J. Math. Phys., 27(1986), pp. 1271-1283.
[12] R.G. Douglas, On majorization, factorization and range inclusion of operators on Hilbert spaces, Proc. Amer. Math. Soc., 17(2) (1966), pp. 413-415.
[13] R.J. Duffin and A.C. Schaeffer, A class of nonharmonik Fourier series, Trans. Amer. Math. Soc.,72 (1952), pp. 341-366.
[14] M.H. Faroughi and E. Osgooei, C-Frames and C-Bessel Mappings, Bull. Iranian Math. Soc., 38(1) (2012), pp. 203-222.
[15] M. Fornasier and H. Rauhut, Continuous frames, function spaces, and the discretization problem, J. Fourier Anal. Appl., 11(3) (2005), pp.245-287.
[16] J.P. Gabardo and D. Han, Frames associated with measurable spaces, Adv. Comput. Math., 18(2003), pp.127-147.
[17] L.Gavruta, Frames for operators, Appl. Comput. Harmon. Anal.,32 (2012), pp. 139-144.
[18] B. Hassibi, B. Hochwald, A. Shokrollahi, and W. Sweldens, Representation theory for high-rate multiple-antenna code design, IEEE Trans. Inform.Theory., 47 (2001), pp. 2335-2367.
[19] G. Kaiser, A Friendly Guide to Wavelets, Birkhuser Boston, 2011.
[20] M. Mirzaee, M. Rezaei, and M.A. Jafarizadeh, Quantum tomography with wavelet transform in Banach space on homogeneous space, Eur. Phys. J. B., 60 (2007), pp. 193-201.
[21] A. Rahimi A. Najati, and Y.N. Dehgan, Continuous frame in Hilbert space, Methods Func. Anal. Top., 12 (2006), pp. 170-182.
[22] A. Rahimi, A. Najati, and M H. Faroughi, Continuous and discrete frames of subspaces in Hilbert spaces, Southeast Asian Bull. Math., 32 (2008), pp. 305-324.
[23] M. Rahmani, On some properties of c-frames, J. Math. Research with Appl., 37(4) (2017), pp. 466-476.
[24] W. Rudin, Functional Analysis, New York, Tata Mc Graw-Hill Editions, 1973.
[25] W. Rudin, Real and Complex Analysis, New York, Tata Mc Graw-Hill Editions, 1987.
[26] S. Sakai, $C^*$-Algebras and $W^*$-Algebras, New York, Springer-Verlag, 1998.
[27] X. Xiao, Y. Zhu, and L. Gavruta, Some Properties of $k$-frames in Hilbert Spaces, Results. Math., 63 (2012), pp.1243-1255.