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Analysis of the affine transformations of the time-frequency plane

Published online by Cambridge University Press:  17 April 2009

Filippo De Mari  and
Krzysztof Nowak
Filippo De Mari
Affiliation:
DIMET, Piazzale J. F. Kennedy, Pad. D, 16129 Genova, Italy e-mail: demari@dima.unige.it
Krzysztof Nowak
Affiliation:
Purchase College, 735 Anderson Hill Rd.Purchase, NY 10577–1400United States of America e-mail: knowak@gmx.de
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Abstract

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We consider two aspects of the action of the extended metaplectic representation of the group G of affine, measure and orientation preserving maps of the time-frequency plane on L2 functions on the line. On the one hand, we list, up to equivalence, all possible reproducing formulas that arise by restricting the representation to connected Lie subgroups of G. On the other hand, we describe, in terms of Weyl calculus, the commutative von Neumann algebras generated by restriction to one-parameter subgroups.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 63 , Issue 2 , April 2001 , pp. 195 - 218
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Baraniuk, R., Shear madness: signal-dependent and metaplectic time-frequency representations, (Ph.D. Thesis) (University of Illinois, Urbana-Champaign, 1992).Google Scholar
[2]Daubechies, I., Ten lectures on wavelets (CBMS-NSF Regional Conference Series, no. 6, SIAM, Philadelphia, 1992).Google Scholar
[3]De Mari, F. and Nowak, K., ‘Canonical subgroups of ℍ1 × SL(2,ℝ)’, Boll. Un. Mat. Ital. (to appear).Google Scholar
[4]Feichtinger, H.G. and Gröchenig, K., ‘Gabor wavelets and the Heisenberg group; Gabor expansions and short time Fourier transform from the group theoretical point of view’, in Wavelets - A Tutorial in Theory and Applications, (Chui, C.K., Editor) (Academic Press, Boston, 1992). pp. 359–397.Google Scholar
[5]Feichtinger, H.G. and Strohmner, T., Eds., Gabor analysis and algorithms - Theory and applications, Applied and Numerical Harmonic Analysis (Birkhauser Boston, Boston, MA, 1997).Google Scholar
[6]Folland, G.B., Harmonic analysis in phase space, Annals of Mathematics Studies (Princeton University Press, Princeton, N.J., 1989).Google Scholar
[7]Folland, G.B. and Sitaram, A., ‘The uncertainty principle: A mathematical survey’, J. Fourier Anal. Appl. 3 (1997), 207–237.Google Scholar
[8]Frazier, M., Jawerth, B. and Weiss, G., Littlewood-Plaey theory and the study of function spaces, CBMS Regional Conference Series in Mathematics 79(American Mathematical Society,Providence, R.I.,1991).Google Scholar
[9]Hörmander, L., The analysis of linear partial differential operators III (Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985).Google Scholar
[10]Meyer, Y., Wavelets, algorithms and applications (SIAM, Philadelphia, PA, 1993).Google Scholar
[11]Nowak, K., ‘Certain function spaces related to the metaplectic representation’, Colloq. Math. 76 (1998), 143–152.Google Scholar