Some results on g-frames in Hilbert spaces

In this paper we show that every g-frame for a Hilbert space H can be represented as a linear combination of two g-orthonormal bases if and only if it is a g-Riesz basis. We also show that every g-frame can be written as a sum of two tight g-frames with g-frame bounds one or a sum of a g-orthonormal basis and a g-Riesz basis for H. We further give necessary and sufficient conditions on g-Bessel sequences {Li \in L (H,Hi) : i \in J} and {Gi \in L(H,Hi): i \in J} and operators L1, L2 on H so that {LiL1+GiL2: i \in J} is a g-frame for H. We next show that a g-frame can be added to any of its canonical dual g-frame to yield a new g-frame.

Anahtar Kelimeler:

Frame, g-frame, g-orthonormal basis, tight g-frame, g-Bessel sequence

Some results on g-frames in Hilbert spaces

Keywords:

Frame, g-frame, g-orthonormal basis, tight g-frame, g-Bessel sequence,

PDF

___

Asgari, M.S. and Khosravi, A.: Frames and bases of subspaces in Hilbert spaces, J. Math. Anal. Appl., 308, 553 (2005).
Casazza, P.G.: Every frame is a sum of three (but not two) orthonormal bases and other frame representations, J. Fourier Anal. Appl. 4 (1998), no. 6, 727-732.
Casazza, P.G. and Kutynoik, G.: Frames of subspaces,Wavelets, frames and operator theory, Contemp. Math., Amer. Math. Soc., Providence, R. I., 345, 87-113 (2004).
Daubechies, I., Grossmann, A. and Meyer, Y.: Painless nonorthogonal expansions, J. Math. Phys., 27, 1271-1283 (1986).
Duﬃn, R.J. and Schaeﬀer, A.C.: A class of non-harmonic Fourier series, Trans. Amer. Math. Soc., 72, 341-366 (1952).
Fornasier, M.: Decompositions of Hilbert space: local construction of global frames, Proc. Int. Conf. Constrictive Function Theory, Varna 2002, B. Bojanov Ed., DARBA, SoŞa, 275-281 (2003).
Kaushik, S.K.: A Note on Frame Sequences in Hilbert Spaces, Int. J. Contemp. Math. Sciences, 3, 791 - 799 (2008).
Najati, A. and Rahimi, A.: Generalized Frames in Hilbert Spaces, Bulletin of the Iranian Mathematical Society 35, 109 (2009).
Najati, A., Faroughi, M.H. and Rahimi, A.: G-frames and stability of g-frames in Hilbert space, Methods of Functional Analysis and Topology, 14, 271-286 (2008).
Obeidat, S., Samarah, S., Casazza, P.G. and Tremain, J.C.: Sums of Hilbert Space Frames, J. Math. Anal. Appl., , 579-585 (2009).
Sun, W.: G-frames and g-Riesz bases, J. Math. Anal. Appl. 322, 437-452 (2006).
Zhu, Y.C.: Characterizations of g-Frames and g-Riesz Bases in Hilbert Spaces, Acta Mathematica Sinica, 24, 1736 (2008).
Abdolaziz ABDOLLAHI, Elham RAHIMI Department of Mathematics, College of Sciences, Shiraz University, Shiraz 71454, IRAN e-mail: abdolahi@shirazu.ac.ir, rahimie@shirazu.ac.ir