Pseudodifferential operators on locally compact abelian groups and Sjöstrand's symbol class
By Karlheinz Gröchenig, Thomas Strohmer. 2006
73 citations recorded by the source corpus.
Abstract We investigate pseudodifferential operators on arbitrary locally compact abelian groups. As symbol classes for the Kohn-Nirenberg calculus we introduce a version of Sjöstrand's class. Pseudodifferential operators with such symbols form a Banach algebra that is closed under inversion. Since “hard analysis” techniques are not available on locally compact abelian groups, a new time-frequency approach is used with the emphasis on modulation spaces, Gabor frames, and Banach algebras of matrices. Sjöstrand's original results are thus understood as a phenomenon of abstract harmonic analysis rather than “hard analysis” and are proved in their natural context and generality.
Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Signal Processing
View Paper: https://www.semanticscholar.org/paper/f3f5923e5b7ca773c94372968bd78ab6cdd390b1
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In Applied Mathematics
In Computer Vision and Pattern Recognition
In Signal Processing
Citations in the corpus
- History and Evolution of the Density Theorem for Gabor Frames (2007, result): “We also refer to the new results in [92, 99, 172] for a deeper understanding of the role of Wiener’s lemmas in time-frequency analysis.”
- Wiener’s Lemma for Infinite Matrices II (2007, background): “The lw-stability is one of the basic assumptions for infinite matrices arising in many fields of mathematics (see [2, 4, 24, 28, 29, 39, 42, 43] for a sample of papers), but little is known about practical criteria for the lw-stability of an infinite matrix, see [46] for the diagonalblocks-dominated”
- Stability of Localized Operators (2008, background): “commutative matrices arises in the study of spline approximation ([26,28]), wavelet and affine wavelets ([20,42]), Gabor frame ([6,35,36]), non-uniform sampling ([61]), and pseudo-differential operators ([13,14,29,34, 38 ,40,56,57,67,68,69]), and the l p -stability for those matrices is one of few b”
- Reproducing formulas for generalized translation invariant systems on locally compact abelian groups (2014, unknown): “Finally, as an alternative generalization of uniform lattices, we mention the idea of so-called quasi-lattices, see [28,29].”