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

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Abstract

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.

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  • [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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