Geodesic
Matrix-valued Gabor frames over LCA groups for operators
Filomat, Tome 37 (2023) no. 28, p. 9543

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DOI

Găvruta studied atomic systems in terms of frames for range of operators (that is, for subspaces), namely Θ-frames, where the lower frame condition is controlled by the Hilbert-adjoint of a bounded linear operatorΘ. For a locally compact abelian groupGand a positive integer n, westudy frames of matrix-valued Gabor systems in the matrix-valued Lebesgue space L2(G,Cn×n) , where a bounded linear operator Θ on L2(G,Cn×n) controls not only lower but also the upper frame condition. We term such frames matrix-valued (Θ,Θ∗)-Gabor frames. Firstly, we discuss frame preserving mapping in terms of hyponormal operators. Secondly, we give necessary and sufficient conditions for the existence of matrix-valued (Θ,Θ∗)- Gabor frames in terms of hyponormal operators. It is shown that if Θ is adjointable hyponormal operator, then L2(G,Cn×n) admits a λ-tight (Θ,Θ∗)-Gabor frame for every positive real number λ. A characterization of matrix-valued (Θ,Θ∗)-Gabor frames is given. Finally, we show that matrix-valued (Θ,Θ∗)-Gabor frames are stable under small perturbation of window functions. Several examples are given to support our study.
DOI : 10.2298/FIL2328543J
Classification : 42C15, 42C30, 42C40
Keywords: Frames, Gabor frames, hyponormal operator, locally compact abelian group 
Jyoti ; Lalit Kumar Vashisht; Uttam Kumar Sinha. Matrix-valued Gabor frames over LCA groups for operators. Filomat, Tome 37 (2023) no. 28, p. 9543 . doi: 10.2298/FIL2328543J
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     title = {Matrix-valued {Gabor} frames over {LCA} groups for operators},
     journal = {Filomat},
     pages = {9543 },
     year = {2023},
     volume = {37},
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     doi = {10.2298/FIL2328543J},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2328543J/}
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