Weaving \(K\)-frames in Hilbert spaces. (English) Zbl 1407.42022
Frames in a Hilbert space were introduced by R. J. Duffin and A. C. Schaeffer [Trans. Am. Math. Soc. 72, 341–366 (1952; Zbl 0049.32401)]. Motivated by examples arising from sampling theory in [“Atomic system for subspaces”, in: Proceedings of the international conference on sampling theory and applications, SampTA 2001. 163–165 (2001); https://www.researchgate.net/profile/Hans_Feichtinger/publication/2857251_Atomic_Systems_for_Subspaces/links/0912f50b0749dad8ac000000.pdf], H. G. Feichtinger and T. Werther introduced a family of analysis and synthesis systems with frame like properties for closed subspaces of Hilbert spaces. These systems were termed as atomic systems or local atoms. In [Appl. Comput. Harmon. Anal. 32, No. 1, 139–144 (2012; Zbl 1230.42038)], L. Găvruţa studied atomic systems with reference to a bounded linear operator \(K\) on a Hilbert space \(\mathcal{H}\). These are called \(K\)-frames. In [Oper. Matrices 10, No. 4, 1093–1116 (2016; Zbl 1358.42025)], T. Bemrose et al. introduced the notion of weaving frames motivated by a problem in distributed signal processing.
In the paper under review, the authors introduce and study weaving \(K\)-frames. The authors give some necessary and sufficient conditions for weaving \(K\)-frames. One of the main results of the paper states that woven \(K\)-frames and weakly woven \(K\)-frames are equivalent.
In the paper under review, the authors introduce and study weaving \(K\)-frames. The authors give some necessary and sufficient conditions for weaving \(K\)-frames. One of the main results of the paper states that woven \(K\)-frames and weakly woven \(K\)-frames are equivalent.
Reviewer: N. Shravan Kumar (New Delhi)
MSC:
| 42C15 | General harmonic expansions, frames |
| 42C30 | Completeness of sets of functions in nontrigonometric harmonic analysis |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
References:
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