Controlled $g$-dual frames and their approximates in Hilbert spaces

Abstract

In this paper, we introduced and characterized the controlled $g$-duals of a frame in a separable Hilbert space $\mathcal{H}$ . Afterwards, we obtained new $C$-controlled $g$-dual frames from the given $C$-controlled $g$-dual frames. In addition, the approximation for controlled $g$-dual frames was defined and some of their properties were investigated. Finally, we characterized the relationship between approximately $C$-controlled dual and $C$-controlled $g$-dual.

Keywords

• frames
• controlled frames
• g-dual frame
• approximate g-dual

References

1. [1] P. Balazs, J.P. Antoine and A. Grybos, Weighted and Controlled Frames, Int. J. Wavelets Multiresolut. Inf. Process. 8 (1), 109-132, 2010.
2. [2] I. Bogdanova, P. Vandergheynst, J.-P. Antoine, L. Jacques and M. Morvidone, Stere- ographic wavelet frames on the sphere, Applied Comput. Harmon. Anal. 19, 223-252, 2005.
3. [3] P. Casazza and O. Christensen, Perturbation of operators and applications to frame theory, J. Fourier Anal. Appl. 3, 543-557, 1997.
4. [4] O. Christensen, A Paley-Wiener theorem for frames, Proc. Amer. Math. Soc. 123, 2199-2202, 1995.
5. [5] O. Christensen, An introduction to Frame and Riesz Bases, Birkhäuser, Boston, 2003.
6. [6] O. Christensen and R.S. Laugesen, Approximately dual frames in Hilbert spaces and applications to Gabor frames, Sampl. Theory Sig. Image Process. 9 (3), 7789, 2011.
7. [7] M.A. Dehgan and M.A. Hasankhani, g-dual frames in Hilbert spaces, U.P.B. Sci. Bull. 75, 129-140, 2013.
8. [8] R. Duffin and A. Schaeffer, A class of non-harmonic Fourier series, Trans. Amer. Math. Soc. 72, 341-366, 1952.

9. [9] S.M. Ramezani, G-duals of continuous frames and their perturbations in Hilbert spaces, U.P.B. Sci. Bull. 82, 75-82, 2020.
10. [10] S.M. Ramezani and A. Nazari, g-orthonormal bases, g-Riesz bases and g-dual of g- frames, U.P.B. Sci. Bull. 78, 91-98, 2016.
11. [11] M. Rashidi-Kouchi, A. Rahimi and F.A. Shah, Duals and multipliers of controlled frames in Hilbert spaces, Int. J. Wavelets, Multiresolut. Inf. Process. 16 (5), 1-13, 2018.
12. [12] W. Sun, Stability of g-frames, J. Math. Anal. Appl. 326 (2), 858-868, 2007.