Weighted and Controlled Continuous g-Frames and their Multipliers in Hilbert Spaces

Abstract

A generalization of weighted, multiplier, controlled from frame and Bessel sequences to continuous g-frames and continuous g-Bessel sequences in Hilbert spaces is presented in this study. Moreover, we find a dual of a continuous g-frame in the case that the multiplier operator is invertible. Finally, it is demonstrated that a controlled continuous g-frame is equivalent to a continuous g-frame.

Keywords

• Continuous g-frame,continuous g-multiplier,weighted continuous g-frame

References

1. [1] M. R. Abdollahpour, M. H. Faroughi, ”Continuous g-Frames in Hilbert Spaces”, South-east Asian. Bull. Math., 32, (2008), 1-19.
2. [2] S. T. Ali, J. P. Antoine, J. P. Gazeau, ”Continuous Frames in Hilbert Spaces”, Annals of Physics, 222, (1993), 137.
3. [3] P. Balazs, ”Basic definition and properties of Bessel multipliers”, J. Math. Anal. Appl., 325(1), (2007) 571-585.
4. [4] P. Balazs, J. P. Antoine, A. Grybos, ”Weighted and Controlled Frames”, Int. J. Wavelets Multiresolut. Inf. Process., 8(1), (2010), 109-132.
5. [5] P. Balazs, D. Bayer, A. Rahimi, ”Multipliers for continuous frames in Hilbert spaces”, J. Phys. A:Math. Theor., 45, (2012) 1-24.
6. [6] 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.
7. [7] O. Christensen, ”An introduction to Frame and Riesz Bases”, Birkh¨auser, Boston, 2003.
8. [8] I. Daubechies, A. Grossmann, Y. Meyer, ”Painless nonorthogonal expansions”, J. Math. Phys., 27, (1986), 1271- 1283.

9. [9] I. Daubechies, ”Ten Lectures on Wavelets”, (1992).
10. [10] M. Dorfler, ”Gabor analysis for a class of signals called music”, Ph.D. Thesis, University of Vienna, (2003).
11. [11] R. Duffin, A. Schaeffer, ”A class of nonharmonic Fourier series”, Trans. Amer. Math. Soc., 72, (1952), 341-366.
12. [12] H. G. Feichtinger and K. Nowak, ”A first survey of Gabor multipliers”, Advances in Gabor Analysis, 99-128, Appl. Numer. Harmon. Anal., Birkh¨auser, Boston, MA, (2003).
13. [13] J. P. Gabardo, D. Han, ”Frames Associated with Measurable Space”, Adv. Comp. Math., 18, (2003), 127-147.
14. [14] H. Heuser, ”Functional analysis”, John Wiley, New York, (1982).
15. [15] G. Kaiser, ”A Friendly Guide to Wavelets”, Birkh¨auser, (1994).
16. [16] G. Matz and F. Hlawatsch, ”Linear Time-Frequency Filters”, One-line Algorithms and Applicatioins, eds. A. Papandredreou-Suppappola, Boca Raton (FL): CRC Press, Ch., 6, (2002), 205-271.
17. [17] A. Najati, A. Rahimi, ”Generalized Frames in Hilbert spaces”, Bull. Iranian Math. Soc., 35(1), (2009), 97-109.
18. [18] A. Rahimi, ”Multipliers of Genralized frames in Hilbert spaces”, Bull. Iranian Math. Soc., 37(1), (2011), 63-88.
19. [19] A. Rahimi and P. Balazs, ”Multipliers for p-Frames in Banach spaces”, Integral Equations and Operator Theory, 68, (2010), 193-205.
20. [20] A. Rahimi and A. Fereydooni, ”Controlled g-Frames and Their g-Multipliers in Hilbert spaces”, An. St. Univ. Ovidius Constanta, 21 (2013), 223-236.
21. [21] W. Sun, ”g-Frame and g-Riesz bases”, J. Math. Anal. Appl., 322, (2006), 437-452.