Expansion of Bessel and g-Bessel sequences to dual frames and dual g-frames

Subject Areas : History and biography

M. S. Asgari ¹ (Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran)
G. Kavian ² (Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran)

Received: 2013-01-01 Accepted : 2013-01-01 Published : 2013-03-01

Keywords: G-frames, dual frames, dual g-frames,

Abstract :

In this paper we study the duality of Bessel and g-Bessel sequences in Hilbertspaces. We show that a Bessel sequence is an inner summand of a frame and the sum of anyBessel sequence with Bessel bound less than one with a Parseval frame is a frame. Next wedevelop this results to the g-frame situation.

References:

[1] O. Christensen, An Introduction to Frames and Riesz Bases. Birkhauser, Boston (2003)
[2] O. Christensions, H. O. Kim, R. Y. Kim, Extensions of Bessel sequences to dual pairs of frames. Appl. Comput. Harmon. Anal. 34, (2013), 224-233.
[3] I. Daubechies, A. Grossmann, Y. Meyer, Painless nonorthogonal expansions. J. Math. Phys. 27, (1986), 1271-1283.
[4] R. J. Duffin, A. C. Schaeer, A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 72, (2), (1952), 341-366.
[5] D. F. Li, W. Sun, Expansion of frames to tight frames. Acta Math. Sin. (Engl. Ser.) 25 (2), (2009), 287-292.
[6] W. Sun, G-frames and G-Riesz bases. J. Math. Anal. Appl. 322, (2006), 437-452.