tailieunhanh - On characterization and stability of alternate dual of g-frames

In this decomposition, dual frames have a key role. G-frames, introduced by Sun, cover many other recent generalizations of frames. In this paper, we give some characterizations of dual g-frames. Moreover, we prove that if two g-frames are close to each other, then we can find duals of them which are close to each other. | Turkish Journal of Mathematics Research Article Turk J Math (2013) 37: 71 – 79 ¨ ITAK ˙ c TUB doi: On characterization and stability of alternate dual of g-frames Ali Akbar AREFIJAMAAL1, ∗, Soheila GHASEMI2 Department of Mathematics and Computer Sciences, Sabzevar Tarbiat Moallem University, Sabzevar, Iran 2 Department of Mathematics and Computer Sciences, Sabzevar Tarbiat Moallem University, Sabzevar, Iran 1 Received: • Accepted: • Published Online: • Printed: Abstract: One of the essential applications of frames is that they lead to expansions of vectors in the underlying Hilbert space in terms of the frame elements. In this decomposition, dual frames have a key role. G-frames, introduced by Sun, cover many other recent generalizations of frames. In this paper, we give some characterizations of dual g-frames. Moreover, we prove that if two g-frames are close to each other, then we can find duals of them which are close to each other. Key words: Frame, dual frame, g-Riesz sequence, alternate dual, g-frame 1. Introduction and preliminaries Frames were first introduced by Duffin and Schaeffer [11]. Today they are a very useful tool in wavelet theory, signal processing and many other fields [4, 5, 6, 15]. The main feature of a frame is to represent every element of underlying Hilbert space as a linear combination of the frame elements. Specifically, if H is a separable Hilbert space and {fi }∞ i=1 is a frame for H , then any f ∈ H can be expressed as f = ∞ f, hi fi , i=1 ∞ 2 for some dual frame {hi }∞ i=1 of {fi }i=1 . A dual frame in which the coefficients f, hi has minimal l -norm for all f ∈ H is called the canonical dual. Unfortunately, computing the canonical dual is highly non-trivial in general. Moreover, the frame {fi }∞ i=1 might have a certain structure which is not shared by the canonical dual. This complication appears, for example, if {fi