ANALYSIS OF CONTINUOUS FRAMES BY TRANSFERRING TO DISCRITE FRAMES

Subject Areas : Analyze

Reihaneh Raisi Tousi ¹ , Rajabali Kamyabi gol ² , Hoseyn AVAZZADEH ³ , Atefe Razghandi ⁴

1 - Ferdowsi university of mashhad
2 - Department of Mathematics, Ferdowsi University of Mashhad
3 - Department of Mathematics, Ferdowsi University of Mashhad
4 - Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran

Received: 2022-04-11 Accepted : 2023-10-09 Published : 2024-01-18

Keywords: قاب پیوسته, پایه متعامد یکه پیوسته, پایه ریس پیوسته, پایه ریس گسسته,

Abstract :

.In this paper we characterize continuous Riesz bases and give some of their relations with continuous orthonormal bases that are well known in the discrete frame setting. We also give necessary conditions for a continuous frame to be a continuous Riesz basis or a continuous orthonormal basis. Especially, we show that every continuous Riesz basis is the (weak) image of a continuous orthonormal basis via a bounded invertible operator.We show that continuous frames act very similar to discrete frames, In fact there is a correspondence between continuous and discrete frames. As an application, one can solve several problems in continuous frames with the aid of discrete frames. We give some relations between continuous and discrete frames. In particular, we define and then characterize exact continuous frames via continuous Riesz bases and show that continuous exact frames are exactly continuous Riesz bases. Finally, we characterize alternate dual of continuous frames by transferring the problem to discrete frames and using discrete dual frames

References:


[1] S. T. Ali, J. -P. Antoine, J. -P. Gazeau, Continuous frames in Hilbert spaces. Ann. Physics, 222 (1993).

[2] A. A. Arefijamaal and E. Zekaee, Signal processing by alternate dual Gabor frames, Appl. Comput. Harmon. Anal, 35 (2013).

[3] A. A. Arefijamaal, R. A. Kamyabi Gol, R. Raisi Tousi, N. Tavallayi, A new approach to continuous Riesz bases, J. Sci. I. R. Iran, 24 (2013).

[4] A. A. Arefijamaal, A. Razghandi, Characterization of alternate duals of continuous frames and representation frames, Results Math, (2019).

[5] S. H. Avazzadeh, R. A. Kamyabi-Gol and R. Raisi Tousi, Continuous frames and g-frames, Bull. Iranian Math. Soc, 40 (2014).

[6] O. Christensen, An Introduction to Frames and Riesz bases, Birkh¨auser, Boston, (2003).

[7] O. Christensen and K. J. Torben, An Introduction to the Theory of Bases, Frames, and Wavelets, Citeseer, (1999).

[8] M. A. Dehghan and M. A. Hasankhani Fard, G-continuous frames and coorbit spaces. Acta. Math. Acad. Paedagog Nyh´azi (N.S.), 24 (2008).

[9] J. P. Gabardo and D. Han, Frames associated with measurable spaces, Adv. Comput. Math, 18 (2003)

[10] A. Razghandi, R. Raisi Tousi, Using tensor product dual frames for phase retrieval problems, J.Pseudo_Differ.oper. Appl, 12 (2021).