Square integrable representations, von Neumann algebras and an application to Gabor analysis

Journal of Fourier Analysis and Applications

Volumn 10, Issue 4, 2004, Pages 325-349

  Bekka, Bachir ^a  

^a UNIVERSITÉ DE LORRAINE   (France)

Author keywords

Coupling constant; Discrete series; Discrete subgroups of Lie groups; Gabor wavelets; Heisenberg group; Von Neumann algebras; Von Neumann dimension

Indexed keywords

EID: 4544291752     PISSN: 10695869     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00041-004-3036-3     Document Type: Article

References (40)

1. Atiyah, M. (1976). Elliptic operators, discrete groups and von Neumann algebras, Astérisque, 32, 43-72.
2. Atiyah, M. and Schmid, W. (1977). A geometric construction of the discrete series for semisimple Lie groups, Invent. Math., 42, 1-62.
3. Baggett, L. (1990). Processing a radar signal and representations of the discrete Heisenberg group, Colloq. Math., 60/61, 195-203.
4. Beardon, A.F. (1983). The Geometry of Discrete Groups, Springer.
5. 1-algebras of exponential Lie groups and of motion groups, Math. Z., 204, 515-526.
6. Borel, A. and Harder, G. (1978). Existence of discrete cocompact subgroups of reductive groups over local fields, J. Reine Angew. Math., 298, 53-64.
7. Borel, A. (1972). Représentations de groupes localement compacts, Lecture Notes Math., 276, Springer.
8. Bargmann, V. (1947). Irreducible unitary representations of the Lorentz group, Ann. Math., 48, 568-640.
9. Bargmann, V., Butera, P., Giradello, L., and Klauder, J. (1971). On the completeness of the coherent states, Rep. Math. Phys., 2, 221-228.
10. Corwin, L. and Greenleaf, F.P. (1990). Representations of Nilpotent Lie Groups and their Applications, Part I: Basic Theory and Examples, Cambridge University Press.
11. Daubechies, I. (1992). Ten Lectures on Wavelets, SIAM.
12. Dixmier, J. (1969). Les Algèbres D'opérateurs dans L'espace Hilbertien (Algèbres de von Neumann), Gauthier-Villars.
13. 2-Betti numbers, Israel J. Math., 117, 183-219.
14. Feichtinger, H.G. and Strohmer, T., Eds., (1998). Gabor Analysis and Algorithms, Theory and Applications, Birkhäuser.
15. Folland, G.B. (1989). Harmonic Analysis in Phase Space, Ann. Math. Studies, 122, Princeton University Press.
16. Goodman, F.M., de la Harpe, P., and Jones, V. (1989). Coxeter Graphs and Towers of Algebras, Springer.
17. Grossmann, A., Morlet, J., and Paul, T. (1986). Transforms associated to square integrable group representations II: examples, Ann. Inst. Henri Poincaré, 45, 293-309.
18. Gröchenig, K. (2001). Foundation of Time-Frequency Analysis, Birkhäuser.
19. Jones, V. and Sunder, V.S. (1997). Introduction to Subfactors, Cambridge University Press.
20. Kadison, R. and Ringrose, J. (1997). Fundamentals of the theory of operator algebras, vol. II, Am. Math. Soc.
21. Klauder, J.S. and Streater, R.F. (1994). Wavelets and the Poincaré half-plane, J. Math. Phys., 35, 471-478.
22. 3) to two cocompact subgroups, Rocky Mountain J. Math., 22, 1349-1363.
23. Langlands, R.P. (1966). Dimension of spaces of automorphic forms, in: Algebraic Groups and Discontinuous Groups, Proc. Symp. Pure Math. IX, Am. Math. Soc.
24. 2-Invariants: Theory and Applications to Geometry and K-Theory, Springer.
25. Margulis, G.A. (1991). Discrete Subgroups of Semisimple Lie Groups, Springer.
26. Meyer, Y. (2000). Le traitement du signal et l'analyse mathématique, Ann. Inst. Fourier, 50, 593-632.
27. Moore, C.C. and Wolf, J. (1973). Square integrable representations of nilpotent groups, Trans. Am. Math. Soc., 185, 445-462.
28. Neeb, K.-H. (2000). Holomorphy and Convexity in Lie Theory, de Gruyter.
29. Perelomov, A. (1971). On the completeness of a system of coherent states, Sov. J. Theor. Math. Phys., 6, 156-165.
30. Perelomov, A. (1973). Coherent states for Lobachevskian plane, Funct. Anal. Appl., 7, 215-222.
31. Rǎdulescu, F.G. (1998). The Γ-equivariant form of the Berezin quantization of the upper half plane, Memoirs Am. Math. Soc., 630.
32. Raghunathan, M.S. (1972). Discrete Subgroups of Lie Groups, Springer.
33. Ramanathan, J. and Steger, T. (1995). Incompleteness of sparse coherent states, Appl. Comput. Harm. Anal., 2, 148-153.
34. Rieffel, M.A. (1981). von Neumann algebras associated with pairs of lattices in Lie groups, Math. Ann., 257, 403-418.
35. Rosenberg, J. (1978). Square-integrable factor representations of locally compact groups, Trans. Am. Math. Soc., 237, 1-33.
36. Seip, K. (1992). Density theorems for sampling and interpolation in the Bargmann-Fock space I, J. Reine Angew. Math., 429, 91-106.
37. Sugiura, M. (1990). Unitary Representations and Harmonic Analysis, North-Holland.
38. Takesaki, M. (1979). Theory of Operator Algebras I, Springer.
39. von Neumann, J. (1968). Mathematische Grundlagen der Quantenmechanik, Springer.
40. Zimmer, R.J. (1984). Ergodic Theory and Semisimple Groups, Birkhäuser.