Abstract
In this paper, we first determine the relations among the best bounds A and B of the g-frame, the g-frame operator S and the pre-frame operator Q and give a necessary and sufficient condition for a g-frame with bounds A and B in a complex Hilbert space. We also introduce the definition of a g-frame sequence and obtain a necessary and sufficient condition for a g-frame sequence with bounds A and B in a complex Hilbert space. Lastly, we consider the stability of a g-frame sequence for a complex Hilbert space under perturbation.
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Duffin, R. J., Schaeffer, A. C.: A class of nonharmonic Fourier series. Trans. Amer. Math. Soc., 72, 341–366 (1952)
Casazza, P. G.: The art of frame theory. Taiwanese J. Math., 4(2), 129–201 (2000)
Christensen, O.: Frames Riesz bases and discrete Gabor/wavelet expansions. Bull. Amer. Math. Soc., 38(3), 273–291 (2001)
Christensen, O.: An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2003
Mallat, S.: A Wavelet Tour of Signal Processing, Academic Press, San Diego, 1999
Walnut, D. F.: An Introduction to Wavelet Analysis, Birkhäuser, Boston, 2002
Casazza, P. G., Christensen, O.: Perturbation of operator and applications to frame theory. J. Fourier Anal. Appl., 3, 543–557 (1997)
Christensen, O.: Operators with closed range pseudo-inverses and perturbation of frames for a subspace. Canad. Math. Bull., 42(1), 37–45 (1999)
Christensen, O.: Frames and pseudo-inverses. J. Math. Anal. Appl., 195, 401–414 (1995)
Zhu, Y. C.: q-Besselian frames in Banach spaces. Acta Mathematica Sinica, English Series, 23(9), 1707–1718 (2007)
Sun, W.: G-frame and g-Riesz base. J. Math. Anal. Appl., 322(1), 437–452 (2006)
Sun, W.: Stability of g-frames. J. Math. Anal. Appl., 326(2), 858–868 (2007)
Casazza, P. G., Kutyniok, G.: Frames of subspaces, in: Wavelets, Frames and Operator Theory, Contemp. Math., Vol. 345, Providence, RI, 2004, 87–113
Asgari, M. S., Khosravi, A.: Frames and bases of subspaces in Hilbert spaces. J. Math. Anal. Appl., 308, 541–553 (2005)
Fornasier, M.: Quasi-orthogonal decompositions of structured frames. J. Math. Anal. Appl., 289, 180–199 (2004)
Zhu, Y. C.: Characterizations of g-frames and g-Riesz bases in Hilbert spaces. Acta Mathematica Sinica, English Series, 24(10), 1727–1736 (2008)
Rudin, W.: Functional Analysis, Second Edition, McGraw-Hill, New York, 1991
Taylor, A. E., Lay, D. C.: Introduction to Functional Analysis, Wiley, New York, 1980
Kato, T.: Perturbation Theory for Linear Operator, Springer-Verlag, New York, 1984
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Partly supported by Natural Science Foundation of Fujian Province of China (No. 2009J01007) and Education Commission Foundation of Fujian Province of China (No. JA08013)
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Wang, Y.J., Zhu, Y.C. G-frames and g-frame sequences in Hilbert spaces. Acta. Math. Sin.-English Ser. 25, 2093–2106 (2009). https://doi.org/10.1007/s10114-009-7615-8
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DOI: https://doi.org/10.1007/s10114-009-7615-8