Reproducing formulae related to the metaplectic representation.
by
Filippo De Mari
Università di Genova
Coauthors: Elena Cordero (Università di Torino). Krzysztof Nowak (Drexel University). Anita Tabacco (Universita' di Torino)

Consider the following problem: describe the subgroups H of the symplectic group G for which there exist a "window" f Î L²(R^d) such that the restriction to H of the metaplectic representation m of G gives rise to a reproducing formula

f= ó õ H  áf, m(h)fñm(h)f  dh

that holds (weakly) for all f Î L²(R^d). One looks for invariants that decide whether a group H enjoys the property or not. Further, in the affirmative case, one seeks conditions that single out the "good" windows, namely those for which the formula holds.

Examples of such subgroups in dimension d=2 include the affine group of the plane with one-dimensional dilations (the wavelet transform inversion formula) and the group generated by translations dilations and sheering that arises in the construction of contourlet frames.

We discuss several structural properties related to this multi-faceted issue, showing that a rich interplay between analytic, geometric and algebraic features comes naturally into the picture. Although some of our results are of a general nature, we mostly concentrate on the 10-dimensional group G=Sp(2, R).

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Date received: May 13, 2005

Copyright © 2005 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caqw-32.