Localization of Frames, Banach Frames, and the Invertibility of the Frame Operator

By Karlheinz Gröchenig. 2004

232 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/1feae3c587e84596bad4a01e083edb6f410e4835

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In Applied Mathematics

Citations in the corpus

Wiener's lemma for infinite matrices with polynomial off-diagonal decay (2005, background): “, des termes llows: rtant and 4], time- In this Note, we give a simple elementary proof to Wiener’s lemma for infinite matrices with polynomial off-diagonal”
Highly sparse representations from dictionaries are unique and independent of the sparseness measure (2007, unknown): “The result is a direct corollary of Gr¨ochenig’s main theorem on localized frames (see [32, Theorem 3.5]), which in particular implies that the dual frame is also ( s + d + (cid:15) )- localized.”
Stability of Localized Operators (2008, methodology): “Jaffard’s result ([42]) for infinite matrices with polynomial decay was used in numerical analysis ([17,58,59]), wavelet analysis ([42]), time-frequency analysis ([31,32,33]) and sampling ([4,24, 33 ,63]); and the Sj¨ostrand’s result ([56]) for infinite matrices was used in the study of pseudo-diffe”
Linear Independence of Finite Gabor Systems (2006, background): “For an interesting recent paper that deals with the issue of extending frames beyond the Hilbert space setting, see [58].”
Weight Functions in Time-Frequency Analysis (2006, background): “Another application of subconvolutive weights is in the definition of time-frequency molecules [16].”
Composition and spectral invariance of pseudodifferential Operators on Modulation Spaces (2006, background): “w . These methods have proven to be extremely useful in a number of completely dierent contexts of analysis [ 16 ,20,26] and hint at a more general principle for the treatment of “weighted Banach algebras”.”
Wiener’s Lemma for Infinite Matrices II (2007, methodology): “... property at infinity for dual generators of a shift-invariant space ([1, 27]); the Wiener’s lemma for matrices associated with twisted convolution was used in the study the decay properties of the dual Gabor frame for L2 ([3, 23, 24]); the Jaard’s result ([26]) for infinite matrices with polynom”
History and Evolution of the Density Theorem for Gabor Frames (2007, background): “This suggests that we can define localization much more abstractly, and indeed, this is exactly what is done in [91, 12, 13].”
Banach frames for multivariate α-modulation spaces (2006, methodology): “The frames constructed are not orthonormal bases, but each system is “locally” orthonormal (in a certain sense that will be explained in Section 4). One of the main tools we use to construct the brushlet frames is the theory of localized frames introduced by Gr¨ochenig [ 13 ].”
Density, Overcompleteness, and Localization of Frames. II. Gabor Systems (2005, background): “As evidence of this fact, we note that Gr¨ochenig has independently introduced a concept of localized frames, for a completely diﬀerent purpose, in [Gr¨o04].”
Weight Functions in Time-Frequency Analysis (2006, background): “Another application of subconvolutive weights is in the definition of time-frequency molecules [16].”