On a New G-Frame and Duality

Document Type: Research Paper


1 Ferdowsi University of Mashhad

2 Department of Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran.



We introduce a new g-frame (singleton g-frame), g-orthonormal basis and g-Riesz basis and study corresponding notions in some other generalizations of frames.
Also, we investigate duality  for some kinds of g-frames. Finally, we illustrate an example which provides a  suitable translation from discrete frames to Sun's g-frames.


[1] S.T. Ali, J.-P. Antoine and J.-P. Gazeau, Continuous frames in Hilbert spaces, Ann. Phys., 222(1) (1993), 1-37.
[2] A.A. Arefijamaal, R.A. Kamyabi Gol, R. Raisi Tousi and N. Tavallayi, A new approach to continuous Riesz bases, J. Sci., Islam. Repub. Iran, 24(1) (2013), 63-69.
[3] S. H. Avazzadeh, R.A. Kamyabi-Gol and R. Raisi Tousi, Continuous frames and g-frames, Bull. Iran. Math. Soc., 40(4) (2014), 1047-1055.
[4] O. Christensen, An Introduction to Frames and Riesz Bases, Birkh"{a}user, Boston, 2003.
[5] O. Christensen and K.J. Torben, An Introduction to the Theory of Bases, Frames, and Wavelets, Citeseer, 1999.
[6] D.-X Ding, Generalized continuous frames by using an iterated function system, J. Geom. Phys., 61(6) (2011), 1045-1050.
[7] M.A. Dehghan and M.A. Hasankhani Fard, G-continuous frames and coorbit spaces, Acta Math. Acad. Paedagog. Nyh'{a}zi. (N.S.), 24(3) (2008), 373-383.
[8] H.G. Feichtinger and T. Strohmer  (Eds.), Gabor Analysis and Algorithms: Theory and Applications, Birkh"user  Inc., Boston, MA, 1998.
[9] J.-P. Gabardo and D. Han, Frames associated with measurable spaces, Adv. Comput. Math., 18(2-4)  (2003), 127-147.
[10] K. Gr"{o}chenig, Foundations of Time-Frequency Analysis, Birkh"{a}user, Boston, 2001.
[11] D. Han and D.R. Larson, Frames, bases  and group representations, Mem. Am. Math. Soc., 147(697) (2000).
[12] C.E. Heil and D.F. Walnut, Continuous and discrete wavelet transforms, SIAM Rev., 31(4) (1989), 628-666.
[13] W. Rudin, Real and Complex Analysis, McGraw-Hill, 2009.
[14] W. Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl., 322(1) (2006), 437-452.