Banach algebras of pseudodifferential operators and their almost diagonalization
By Karlheinz Gröchenig, Ziemowit Rzeszotnik. 2007
90 citations recorded by the source corpus.
We define new symbol classes for pseudodifferntial operators and investigate their pseudodifferential calculus. The symbol classes are parametrized by commutative convolution algebras. To every solid convolution algebra over a lattice we associate a symbol class. Then every operator with such a symbol is almost diagonal with respect to special wave packets (coherent states or Gabor frames), and the rate of almost diagonalization is described precisely by the underlying convolution algebra. Furthermore, the corresponding class of pseudodifferential operators is a Banach algebra of bounded operators on $L^2 $. If a version of Wiener's lemma holds for the underlying convolution algebra, then the algebra of pseudodifferential operators is closed under inversion. The theory contains as a special case the fundamental results about Sj\"ostrand's class and yields a new proof of a theorem of Beals about the H\"ormander class of order 0.
Subfields: Applied Mathematics, Mathematical Physics, Signal Processing
View Paper: https://www.semanticscholar.org/paper/8e5fa8bdd477e6da7246fcf3d1d4528f31e483a4
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In Applied Mathematics
- Karlheinz Gröchenig
- J. Toft
- Ziemowit Rzeszotnik
- Elena Cordero
- Fabio Di Nicola
- L. Rodino
- Qiyu Sun
- Thomas Strohmer
In Mathematical Physics
- J. Toft
- Karlheinz Gröchenig
- Ziemowit Rzeszotnik
- L. Rodino
- Fabio Di Nicola
- Elena Cordero
- Thomas Strohmer
- Hans G. Feichtinger
In Signal Processing
- Thomas Strohmer
- Ziemowit Rzeszotnik
- J. Toft
- Karlheinz Gröchenig
- L. Rodino
- Elena Cordero
- Christopher Heil
- Qiyu Sun
Citations in the corpus
- Convergence Analysis of the Finite Section Method and Banach Algebras of Matrices (2010, background): “More general conditions were obtained in [15, 29].”
- Wiener algebras of Fourier integral operators (2013, background): “The Hörmander symbol class S0 0,0(R 2d) can be characterized by means of modulation spaces as follows, see for example [23]: S0 0,0 = ⋂”
- Isomorphism properties of Toeplitz operators and pseudo-differential operators between modulation spaces (2009, background): “5] or [21] for the general case of (2), and to [22, 23] for further generalizations.”