Intrinsic Localization of Frames

By Massimo Fornasier, Karlheinz Gröchenig. 2005

133 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Signal Processing

View Paper: https://www.semanticscholar.org/paper/25c38677c7b48d96b81858ceb97ac655303239f2

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In Applied Mathematics

In Computer Vision and Pattern Recognition

In Signal Processing

Citations in the corpus

Continuous Frames, Function Spaces, and the Discretization Problem (2004, background): “[22, 23, 24, 33, 35, 27] and their recent generalizations [12, 13, 47, 48], we show in this section how classes of abstract Banach spaces called (generalized) coorbit spaces can be associated to a given continuous frame.”
Density, overcompleteness, and localization of frames (2006, methodology): “to a reference system E. An analogous polynomial or exponential “intrinsic localization” was independently introduced by Gr¨ ochenig in [Gr¨ o03]; see also [For03], [ GF05 ].”
Banach frames for α-modulation spaces (2007, background): “We refer to [30,37] for further information where a characterization of a large class of algebras of this type is presented.”
Deformation of Gabor systems (2013, methodology): “It combines methods from the theory of localized frames [22, 25], the stability of operators on `-spaces [1, 36] and weak limit techniques in the style of Beurling [7].”
Time-Frequency Analysis of Sj\"ostrand's Class (2004, background): “is invertible on R and kerA = R⊥. The unique operator A† that satisﬁes A†Ah = AA†h = h for h ∈ R and kerA† = R⊥ is called the (Moore-Penrose) pseudo-inverse of A. The following lemma is borrowed from [19]. Lemma 4.5 (Pseudoinverses). If A ∈ Cv has a (Moore-Penrose) pseudoinverse A†, then A† ∈ C v. T”
Pseudodifferential operators on locally compact abelian groups and Sjöstrand's symbol class (2006, background): “10 ([11]) Let A = [Ai,j ]i,j∈D be a matrix in Cv where v is an admissible weight.”
Basic definition and properties of Bessel multipliers (2005, background): “ors might be worthwhile. The combination of two multipliers are connected to the Gram matrix, see Lemma 5.3. It will be interesting to apply the results for the decay properties of the Gram matrix in [9] to this topic. An interesting step away from the unstructured frame will be to investigate frame”
Wiener’s Lemma for Infinite Matrices II (2007, methodology): “We remark that the above generalization from invertibility to pseudo-invertibility in the Wiener’s lemma holds for any Banach algebra, see [ 18 ] for the holomorphic calculus approach used in the proof, and also for another approach from Banach algebra.”
Modulation Spaces: Looking Back and Ahead (2006, methodology): “, the paper [100] by Fornasier and Gröchenig, and this also was developed in [6].”
Banach frames for multivariate α-modulation spaces (2006, methodology): “In particular, we use an important recent result by Fornasier and Gr¨ochenig [ 10 ] that a so-called self-localized frame has a self-localized dual frame.”
Time-Frequency Analysis of Sjöstrand's Class (2004, unknown): “The following lemma is borrowed from [19].”
Density, Overcompleteness, and Localization of Frames. II. Gabor Systems (2005, background): “An analogous polynomial or exponential “intrinsic localization” was independently introduced by Gröchenig in [36]; see also [33, 38].”
Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces (2008, background): “In a more general setting Gröchenig introduced in [27, 22] the localization theory for families of Banach spaces, see also [2] for an approach to localization theory not based on the spectral invariance of Banach algebras.”