Controlled G-Frames and Their G-Multipliers in Hilbert spaces

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Multipliers have been recently introduced by P. Balazs as operators for Bessel sequences and frames in Hilbert spaces. These are opera- tors that combine (frame-like) analysis, a multiplication with a fixed sequence ( called the symbol) and synthesis. One of the last extensions of frames is weighted and controlled frames that introduced by P.Balazs, J-P. Antoine and A. Grybos to improve the numerical efficiency of iterative algorithms for inverting the frame operator. Also g-frames are the most popular generalization of frames that include almost all of the frame extensions. In this manuscript the concept of the controlled g- frames will be defined and we will show that controlled g-frames are equivalent to g-frames and so the controlled operators C and C' can be used as preconditions in applications. Also the multiplier operator for this family of operators will be introduced and some of its properties will be shown.

[1] A. Aldroubi, C. Cabrelli, U. Molter, Wavelets on irregular grids with arbitrary dilation matrices and frame atomics for L2 (Rd), Appl. Comput. Harmon. Anal. 17 (2004) 119-140.

[2] M. L. Arias, M. Pacheco, Bessel fusion multipliers, J. Math. Anal. Appl. 348(2) (2008), 581-588.

[3] P. Balazs, Basic definition and properties of Bessel multipliers, J. Math. Anal. Appl. 325(1) (2007) 571-585.

[4] P. Balazs, J.P. Antoine, A. Grybos, Wighted and Controlled Frames, Int. J. Wavelets Multiresolut. Inf. Process. 8(1) (2010) 109-132.

[5] P. Balazs, W.A. Deutsch, A. Noll, J. Rennison, J. White, STx Pro­grammer Guide, Version: 3.6.2. Acoustics Research Institute, Austrian Academy of Sciences, 2005.

[6] P. Balazs, B. Laback, G. Eckel, W.A. Deutsch, Introducing time-frequency sparsity by removing perceptually irrelevant components using a simple model of simultaneous masking. IEEE Transactions on Audio, Speech and Language Processing, forthcoming:-, 2009.

[7] I. Bogdanova, P. Vandergheynst, J.P. Antoine, L. Jacques, M. Morvidone, Stereographic wavelet frames on the sphere, Applied Comput. Harmon. Anal. (19) (2005) 223-252.

[8] P.G. Casazza, G. Kutyniok, Frames of Subspaces, Wavelets, Frames and Operator Theory, Amer. Math. Soc. 345 (2004) 87-113.

[9] O. Christensen, Y.C. Eldar, Oblique dual frames and shift-invariant spaces, Appl. Comput. Harmon. Anal. 17 (2004) 48-68.

[10] Ph. Depalle, R. Kronland-Martinet, B. Torresani, Time-frequency multi­pliers for sound synthesis. In Proceedings of the Wavelet XII conference, SPIE annual Symposium, San Diego, August 2007.

[11] M. Dörfler, B. Torresani, Representation of operators in the time- frequency domain and generalized gabor multipliers. J. Fourier Anal. Appl. (2009).

[12] M. Dorfler, Gabor analysis for a class of signals called music, Ph.D. thesis, University of Vienna, 2003.

[13] Y.C. Eldar, Sampling with arbitrary sampling and reconstruction spaces and oblique dual frame vectors, J. Fourier Anal. Appl. 9 (2003) 77-96.

[14] H.G. Feichtinger, K. Nowak, A first survey of Gabor multipliers, Birkhauser, Boston, 2003, Chapter 5, 99-128.

[15] A. Fereydooni, A. Safapour, Banach pair frames.

[16] A. Fereydooni, A. Safapour, Pair frames, arXiv:1109.3766v2.

[17] A. Fereydooni, A. Safapour, A. Rahimi, Aadjoint of pair frames, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys 2013.

[18] M. Fornasier, Decomposition of Hilbert spaces: local construction of global frames, Proc. Int. Conf. on Constructive function theory, Varna, B. Bo- janov Ed., DARBA, Sofia, 2003 275-281.

[19] M. Fornasier, Quasi-orthogonal decompsition of structured frames, J. Math. Anal. Appl. 289 (2004) p 180-199.

[20] H. Heuser, Functional analysis, John Wiley, New York, 1982.

[21] S. Li, H. Ogawa, Pseudoframes for subspaces with application, J. Fourier Anal. Appl. 10 (2004) 409-431.

[22] G. Matz, F. Hlawatsch, Linear Time-Frequency Filters: On-line Algo­rithms and Applications, chapter 6 in ’Application in Time-Frequency Signal Processing’, pp. 205-271. eds. A. Papandreou-Suppappola, Boca Raton (FL): CRC Press, 2002.

[23] G. Matz, D. Schafhuber, K. Grochenig, M. Hartmann, F. Hlawatsch, Analysis, Optimization, and Implementation of Low-Interference Wire­less Multicarrier Systems. IEEE Trans. Wireless Comm. 6(4) (2007) 1­11.

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

[25] A. Najati, A. Rahimi, Generalized Frames in Hilbert spaces, Bull. Iranian Math. Soc. 35(1) (2009) 97-109.

[26] P. Oswald, Multilevel Finite Element Approximation: Theory and Appli­cation, Teubner Skripten zur Numerik, Teubner, Stuttgart, 1994.

[27] A. Rahimi, Frames and Their Generalizations in Hilbert and Banach Spaces, LAP Lambert Academic Publishing, 2011.

[28] A. Rahimi, Multipliers of Genralized frames in Hilbert spaces, Bull. Ira­nian Math. Soc. 37(1) (2011) 63-88.

[29] A. Rahimi, P. Balazs, Multipliers of p-Bessel sequences in Banach spaces, Integral Equations Operator Theory, 68(2) (2010) 193-205.

[30] R. Schatten, Norm Ideals of Completely Continious Operators, Springer, Berlin, 1960.

[31] W. Sun, G-frames and G-Riesz bases, J. Math. Anal. Appl. 322(1) (2006) 437-452.

[32] K. Zhuo, Operator Theory in Function Spaces, Marcel Dekker, Inc, 1990.

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