Pseudodifferential operators on locally compact abelian groups and Sjstrand's symbol class

Journal fur die Reine und Angewandte Mathematik

Volumn , Issue 613, 2007, Pages 121-146

  Gröchenig, Karlheinz ^a,b   Strohmer, Thomas ^a,c  

^a Institut Fü R Mathematik ^*

^b Institut Fü R Mathematik   (Austria)

^c UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 38949122435     PISSN: 00754102     EISSN: 00754102     Source Type: Journal    
DOI: 10.1515/CRELLE.2007.094     Document Type: Article

References (44)

1. A. G. Baskakov, Wiener's theorem and asymptotic estimates for elements of inverse matrices, Funktsional. Anal. Prilozhen. 24(3) (1990), 64-65.
2. A. G. Baskakov, Estimates for the elements of inverse matrices, and the spectral analysis of linear operators, Izv. Ross. Akad. Nauk Ser. Mat. 61(6) (1997), 3-26.
3. A. Benyi, K. Gröchenig, C. Heil, and K. Okoudjou, Modulation spaces and a class of bounded multilinear pseudodifferential operators, J. Oper. Th. 54 (2005), 389-401.
4. S. Bochner and R. S. Phillips, Absolutely convergent Fourier expansions for non-commutative normed rings, Ann. Math. (2) 43 (1942), 409-418.
5. 2 estimates for Weyl quantization, J. Funct. Anal. 165(1) (1999), 173-204.
6. N. Dunford and J. T. Schwartz, Linear operators, Part I, General theory, with the assistance of William G. Bade and Robert G. Bartle, reprint of the 1958 original, Wiley Classics Library, John Wiley & Sons Inc., New York 1988.
7. H. G. Feichtinger, Generalized amalgams, with applications to Fourier transform, Canad. J. Math. 42(3) (1990), 395-409.
8. H. G. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, II, Monatsh. Math. 108(2-3) (1989), 129-148.
9. H. G. Feichtinger and W. Kozek, Quantization of TF-lattice invariant operators on elementary LCA groups, in: H. G. Feichtinger and T. Strohmer, eds., Gabor Analysis and Algorithms: Theory and Applications, chapter 7, Birkhäuser, Boston (1998), 233-266.
10. G. B. Folland, Harmonic Analysis in Phase Space, Ann. Math. Stud. Princeton Univ. Press, Princeton (NJ) 1989.
11. M. Fornasier and K. Gröchenig, Intrinsic localization of frames, Constr. Approx. 22 (2005), 395-415.
12. q, Bull. Amer. Math. Soc. (N.S.) 13(1) (1985), 1-21.
13. E. Galperin, Uncertainty principles as embeddings of modulation spaces, PhD thesis, Univ. of Connecticut, 2000.
14. I. Gelfand, D. Raikov, and G. Shilov, Commutative normed rings, Chelsea Publishing Co., New York 1964, translated from the Russian.
15. I. Gohberg, M. A. Kaashoek, and H. J. Woerdeman, The band method for positive and strictly contractive extension problems: an alternative version and new applications, Integr. Equ. Oper. Th. 12(3) (1989), 343-382.
16. K. Gröchenig, Aspects of Gabor analysis on locally compact abelian groups, in: H. G. Feichtinger and T. Strohmer, eds., Gabor Analysis and Algorithms: Theory and Applications, chapter 6, Birkhäuser, Boston (1998), 211-231.
17. K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, Boston 2001.
18. K. Gröchenig, A pedestrian's approach to pseudodifferential operators, in: C. Heil, ed., Harmonic Analysis and Applications, in Honor of John J. Benedetto, Birkhäuser, Boston (2006), 139-170.
19. K. Gröchenig, Time-frequency analysis of Sjöstrand's class, Rev. Mat. Iberoam. 22 (2) (2006), 703-724.
20. K. Gröchenig and M. Leinert, Wiener's lemma for twisted convolution and Gabor frames, J. Amer. Math. Soc. 17 (2004), 1-18.
21. K. Gröchenig and G. Zimmermann, Hardy's theorem and the short-time Fourier transform of Schwartz functions, J. London Math. Soc. 63 (2001), 205-214.
22. S. Haran, Quantizations and symbolic calculus over the p-adic numbers, Ann. Inst. Fourier (Grenoble) 43(4) (1993), 997-1053.
23. E. Hewitt and K. Ross, Abstract Harmonic Analysis, Vol. 1 and 2, Grundl. Math. Wiss. 152, Springer, Berlin-Heidelberg-New York 1963.
24. L. Hörmander, The analysis of linear partial differential operators, I, Distribution theory and Fourier analysis, second edition, Springer-Verlag, Berlin 1990.
25. L. Hörmander, The analysis of linear partial differential operators, III, Pseudo-differential operators, corrected reprint of the 1985 original, Grundl. Math. Wiss. 274, Springer-Verlag, Berlin 1994.
26. E. Kaniuth and G. Kutyniok, Zeros of the Zak transform on locally compact abelian groups, Proc. Amer. Math. Soc. 126(12) (1998), 3561-3569.
27. V. G. Kurbatov, Linear Differential and Difference Equations, Voronezh Univ. Press, Voronezh (Russia) 1990 (Russian).
28. V. G. Kurbatov, Functional Differential Operators and Equations, Math. Appl. 473, Kluwer Academic Publishers, Dordrecht-Boston-London 1999.
29. N. Lerner, Fefferman-Phong inequality and an algebra of pseudodifferential operators, preprint 2005.
30. Y. I. Lyubarskii, Frames in the Bargmann space of entire functions, Adv. Sov. Math. 429 (1992), 107-113.
31. V. Maslov, Méthodes opératorielles, translated from the Russian by Djilali Embarek, Éditions Mir, Moscow 1987.
32. M. S. Osborne, On the Schwartz-Bruhat space and the Paley-Wiener theorem for locally compact abelian groups, J. Funct. Anal. 19 (1975), 40-49.
33. H. Reiter and J. D. Stegeman, Classical harmonic analysis and locally compact groups, second edition, London Math. Soc. Monogr. (N.S.) 22, The Clarendon Press Oxford University Press, New York 2000.
34. M. A. Rieffel, Projective modules over higher-dimensional noncommutative tori, Canad. J. Math. 40(2) (1988), 257-338.
35. P. Ruelle, E. Thiran, D. Versiegen, and J. Weyers, Quantum mechanics on p-adic fields, J. Math. Phys. 30(12) (1989), 2854-2874.
36. J. Saranen and G. Vainikko, Periodic integral and pseudodifferential equations with numerical approximation, Monogr. Math., Springer-Verlag, Berlin 2002.
37. K. Seip and R. Wallsten, Density theorems for sampling and interpolation in the Bargmann-Fock space II, J. reine angew. Math. 429 (1992), 107-113.
38. J. Sjöstrand, An algebra of pseudodifferential operators, Math. Res. Lett. 1(2) (1994), 185-192.
39. J. Sjöstrand, Wiener type algebras of pseudodifferential operators, in: Séminaire sur les Équations aux Dérivées Partielles, 1994-1995, Exp. No. IV, 21, École Polytech., Palaiseau 1995.
40. E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, with the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III, Princeton Univ. Press, Princeton (NJ) 1993.
41. T. Strohmer, Pseudodifferential operators and Banach algebras in mobile communications, Appl. Comput. Harm. Anal. 20(2) (2006), 237-249.
42. J. Toft, Subalgebras to a Wiener type algebra of pseudo-differential operators, Ann. Inst. Fourier (Grenoble) 51(5) (2001), 1347-1383.
43. V. S. Vladimirov, On the spectrum of some pseudodifferential operators over the field of p-adic numbers, Alg. Anal. 2(6) (1990), 107-124.
44. N. Wiener, Tauberian theorems, Ann. Math. (2) 33(1) (1932), 1-100.