G-frames and G-Riesz Bases

By Wenchang Sun. 2005

551 citations recorded by the source corpus.

G-frames are generalized frames which include ordinary frames, bounded invertible linear operators, as well as many recent generalizations of frames, e.g., bounded quasi-projectors and frames of subspaces. G-frames are natural generalizations of frames and provide more choices on analyzing functions from frame expansion coefficients. We give characterizations of g-frames and prove that g-frames share many useful properties with frames. We also give generalized version of Riesz bases and orthonormal bases. As an application, we get atomic resolutions for bounded linear operators.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/285beed79a3a9c70fbf01bd6b5119cc060fbef66

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In Applied Mathematics

Citations in the corpus

Reproducing formulas for generalized translation invariant systems on locally compact abelian groups (2014, methodology): “fk2 ≤ X j∈J Z Mj hf,f j,ki 2 dµ Mj (k) ≤ Bkfk 2 for all f∈ H. (2.7) The above deﬁnition and statements about continuous frames carry over to continuous g-frames; we refer to the original paper by Sun [47] for a detailed account of g-frames. Lemma 2.4 is also easily transferred to this new setup. We ”
G-FRAMES AND STABILITY OF G-FRAMES IN HILBERT SPACES (2008, methodology): “The mean of g-frames has been presented by W. Sun in [ 15 ].”
On the duality of fusion frames (2007, unknown): “Generalizations of frame theory were give in [1,5,6,10,11,16,19].”
Fusion frames and distributed processing (2006, unknown): “Some further results on the theory from [11] can be found in [3], and an extension was derived by Sun [27, 28], however without any possibility of equipping the subspaces with an underlying structure.”
Characterizations of g-frames and g-Riesz bases in Hilbert spaces (2008, methodology): “1 ([9]) A sequence {Λj ∈ L (U ,V j)}j∈J is called a generalized frame, or simply a g-frame, for U with respect to {Vj}j∈J if there are two positive constants A and B such that A‖f‖2 ≤ ∑”
Fusion frames and g-frames (2008, background): “1 in [12] we have ∑ i∈I ‖Γif ‖2 = ∑ i∈I ‖Λ̃if ‖2 + ∑ i∈I ‖Γif − Λ̃if ‖2.”