Document Type : Research Paper
Author
Young Researchers and Elite Club Kahnooj Branch, Islamic Azad University, Kerman, Iran.
Abstract
In this paper, we define super Hilbert module and investigate frames in this space. Super Hilbert modules are generalization of super Hilbert spaces in Hilbert C*-module setting. Also, we define frames in a super Hilbert module and characterize them by using of the concept of g-frames in a Hilbert C*-module. Finally, disjoint frames in Hilbert C*-modules are introduced and investigated.
Keywords
Main Subjects
[1] A. Abdollahi and E. Rahimi, Generalized frames on super Hilbert spaces, Bulletin of the Malaysian Mathematical Sciences Society, 35(3) (2012) 807-818.
[2] A. Arefijamaal and E. Zekaee, Image processing by alternate dual Gabor frames, Bull. Iranian Math. Soc. to appear (2016).
[3] A. Arefijamaal and E. Zekaee, Signal processing by alternate dual Gabor frames, Appl. Comput. Harmon. Anal. 35 (2013) 535-540.
[4] R. Balan, Density and Redundancy of the Noncoherent Weyl-Heisenberg Superframes, Contemp. Math., 247 1999.
[5] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser, Boston,} 2003.
[6] I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986) 1271-1283.
[7] B. DeWitt, Supermanifolds, Cambridge Monographs on Mathematical Physics, Cambridge University Press, New York, NY, USA, 2nd edition, 1992.
[8] R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952) 341-366.
[9] A.M. El Gradechi and L.M. Nieto, Supercoherent states, super Kaler geometry and geometric quantization, Communications in Mathematical Physics, 175(3) (1996) 521-563.
[10] M. Frank and D.R. Larson, Frames in Hilbert $C^*$-modules and $C^*$-algebras, J. Operator Theory 48 (2002) 273-314.
[11] M. Frank and D.R. Larson, A module frame concept for Hilbert $C^*$-modules, Functional and Harmonic Analysis of Wavelets, San Antonio, TX, January 1999, Contemp. Math. 247, Amer. Math. Soc., Providence, RI 2000, 207-233.
[12] A. Ghaani Farashahi, Abstract harmonic analysis of wave-packet transforms over locally compact abelian groups, Banach Journal of Mathematical Analysis, 11(1) (2017) 50-71.
[13] A. Ghaani Farashahi, Continuous partial Gabor transform for semi-direct product of locally compact groups, Bulletin of the Malaysian Mathematical Sciences Society, 38(2) (2015) 779-803.
[14] A. Ghaani Farashahi and R. Kamyabi-Gol, Continuous Gabor transform for a class of non-Abelian groups, Bulletin of the Belgian Mathematical Society-Simon Stevin, 19(4) (2012) 683-701.
[15] Q. Gu and D. Han, Super-wavelets and decomposable wavelet frames,The Journal of Fourier Analysis and Applications, 11(6) (2005) 683-696.
[16] D. Han, W. Jing, D. Larson, and R. Mohapatra, Riesz bases and their dual modular frames in Hilbert $C^*$-modules, J. Math. Anal. Appl. 343 (2008) 246-256.
[17] D. Han, W. Jing, and R. Mohapatra, Perturbation of frames and Riesz bases in Hilbert $C^*$-modules, Linear Algebra Appl. 431 (2009) 746-759.
[18] D. Han and D. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc. 147 (697) 2000.
[19] D. Hua and Y. Huang, The characterization and stability of g-Riesz frames for super Hilbert space, J. Function Spaces,
2015 (2015) 1-9.
[20] W. Jing, Frames in Hilbert $C^*$-modules, Ph. D. thesis, University of Central Florida Orlando, Florida, 2006.
[21] A. Khosravi and B. Khosravi, g-frames and modular Riesz bases in Hilbert $C^*$-modules, Int. J. Wavelets Multiresolut. Inf. Process., 10(2) (2012) 1-12.
[22] A. Khosravi and B. Khosravi, Frames and bases in tensor products of Hilbert spaces and Hilbert $C^*$-modules, Proc. Indian Acad. Sci. Math. Sci. 117 (2007) 1-12.
[23] A. Khrennikov and A. Yu, The Hilbert super space, Soviet Physics-Doklady, 36 (1991) 759-760.
[24] E.C. Lance, Hilbert $C^*$-Modules: A Toolkit for Operator Algebraists, London Math. Soc. Lecture Note Ser. 210, Cambridge Univ. Press, 1995.
[25] R. Oliver, Super Hilbert spaces, Communications in Mathematical Physics, 214(2) (2000) 449-467.
[26] M. Rashidi-Kouchi and A. Nazari, Equivalent continuous g-frames in Hilbert $C^*$-modules, Bull. Math. Anal. Appl.,, 4(4) (2012) 91-98.
[27] M. Rashidi-Kouchi and A. Nazari, Continuous g-frame in Hilbert $C^*$-modules, Abst. Appl. Anal., 2011 (2011) 1-20.
[28] M. Rashidi-Kouchi, A. Nazai, and M. Amini, On stability of g-frames and g-Riesz bases in Hilbert $C^*$-modules, Int. J.Wavelets Multiresolut. Inf. Process, 12(6) (2014) 1-16.
[29] W. Sun, g-Frames and g-Riesz bases, J. Math. Anal. Appl. 322 (2006) 437-452.
[30] L. Zang, W. Sun, and D. Chen, Excess of a class of g-frames, Journal of Mathematical Analysis and Applications, 352(2) (2009) 711-717.
[2] A. Arefijamaal and E. Zekaee, Image processing by alternate dual Gabor frames, Bull. Iranian Math. Soc. to appear (2016).
[3] A. Arefijamaal and E. Zekaee, Signal processing by alternate dual Gabor frames, Appl. Comput. Harmon. Anal. 35 (2013) 535-540.
[4] R. Balan, Density and Redundancy of the Noncoherent Weyl-Heisenberg Superframes, Contemp. Math., 247 1999.
[5] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser, Boston,} 2003.
[6] I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986) 1271-1283.
[7] B. DeWitt, Supermanifolds, Cambridge Monographs on Mathematical Physics, Cambridge University Press, New York, NY, USA, 2nd edition, 1992.
[8] R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952) 341-366.
[9] A.M. El Gradechi and L.M. Nieto, Supercoherent states, super Kaler geometry and geometric quantization, Communications in Mathematical Physics, 175(3) (1996) 521-563.
[10] M. Frank and D.R. Larson, Frames in Hilbert $C^*$-modules and $C^*$-algebras, J. Operator Theory 48 (2002) 273-314.
[11] M. Frank and D.R. Larson, A module frame concept for Hilbert $C^*$-modules, Functional and Harmonic Analysis of Wavelets, San Antonio, TX, January 1999, Contemp. Math. 247, Amer. Math. Soc., Providence, RI 2000, 207-233.
[12] A. Ghaani Farashahi, Abstract harmonic analysis of wave-packet transforms over locally compact abelian groups, Banach Journal of Mathematical Analysis, 11(1) (2017) 50-71.
[13] A. Ghaani Farashahi, Continuous partial Gabor transform for semi-direct product of locally compact groups, Bulletin of the Malaysian Mathematical Sciences Society, 38(2) (2015) 779-803.
[14] A. Ghaani Farashahi and R. Kamyabi-Gol, Continuous Gabor transform for a class of non-Abelian groups, Bulletin of the Belgian Mathematical Society-Simon Stevin, 19(4) (2012) 683-701.
[15] Q. Gu and D. Han, Super-wavelets and decomposable wavelet frames,The Journal of Fourier Analysis and Applications, 11(6) (2005) 683-696.
[16] D. Han, W. Jing, D. Larson, and R. Mohapatra, Riesz bases and their dual modular frames in Hilbert $C^*$-modules, J. Math. Anal. Appl. 343 (2008) 246-256.
[17] D. Han, W. Jing, and R. Mohapatra, Perturbation of frames and Riesz bases in Hilbert $C^*$-modules, Linear Algebra Appl. 431 (2009) 746-759.
[18] D. Han and D. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc. 147 (697) 2000.
[19] D. Hua and Y. Huang, The characterization and stability of g-Riesz frames for super Hilbert space, J. Function Spaces,
2015 (2015) 1-9.
[20] W. Jing, Frames in Hilbert $C^*$-modules, Ph. D. thesis, University of Central Florida Orlando, Florida, 2006.
[21] A. Khosravi and B. Khosravi, g-frames and modular Riesz bases in Hilbert $C^*$-modules, Int. J. Wavelets Multiresolut. Inf. Process., 10(2) (2012) 1-12.
[22] A. Khosravi and B. Khosravi, Frames and bases in tensor products of Hilbert spaces and Hilbert $C^*$-modules, Proc. Indian Acad. Sci. Math. Sci. 117 (2007) 1-12.
[23] A. Khrennikov and A. Yu, The Hilbert super space, Soviet Physics-Doklady, 36 (1991) 759-760.
[24] E.C. Lance, Hilbert $C^*$-Modules: A Toolkit for Operator Algebraists, London Math. Soc. Lecture Note Ser. 210, Cambridge Univ. Press, 1995.
[25] R. Oliver, Super Hilbert spaces, Communications in Mathematical Physics, 214(2) (2000) 449-467.
[26] M. Rashidi-Kouchi and A. Nazari, Equivalent continuous g-frames in Hilbert $C^*$-modules, Bull. Math. Anal. Appl.,, 4(4) (2012) 91-98.
[27] M. Rashidi-Kouchi and A. Nazari, Continuous g-frame in Hilbert $C^*$-modules, Abst. Appl. Anal., 2011 (2011) 1-20.
[28] M. Rashidi-Kouchi, A. Nazai, and M. Amini, On stability of g-frames and g-Riesz bases in Hilbert $C^*$-modules, Int. J.Wavelets Multiresolut. Inf. Process, 12(6) (2014) 1-16.
[29] W. Sun, g-Frames and g-Riesz bases, J. Math. Anal. Appl. 322 (2006) 437-452.
[30] L. Zang, W. Sun, and D. Chen, Excess of a class of g-frames, Journal of Mathematical Analysis and Applications, 352(2) (2009) 711-717.
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