Integrable Wavelet Transforms with Abelian Dilation Groups

Bradley N. Currey¹ ; Hartmut Führ² ; Keith F. Taylor³

¹Department of Mathematics and Computer Science, St. Louis University, 220 N. Grand Blvd., St.Louis, MO 63103, U.S.A.
²Lehrstuhl A für Mathematik, RWTH Aachen, 52056 Aachen, Germany
³Mathematics and Statistics Dept., Dalhousie University, PO BOX 15000, Halifax, NS, B3H 4R2, Canada

Journal of Lie Theory, Tome 26 (2016) no. 2, pp. 567-595

Résumé

\def\sdir#1{\hbox{$\mathrel\times{\hskip -4.3pt {\vrule height 4.0 pt depth 0 pt}}\hskip 2pt_{#1}$}} \def\R{{\Bbb R}} We consider a class of semidirect products $G = \R^n\sdir{}H$, with $H$ a suitably chosen abelian matrix group. The choice of $H$ ensures that there is a wavelet inversion formula, and we are looking for criteria to decide under which conditions there exists a wavelet such that the associated reproducing kernel is integrable. It is well-known that the existence of integrable wavelet coefficients is related to the question whether the unitary dual of $G$ contains open compact sets. Our main general result reduces the latter problem to that of identifying compact open sets in the quotient space of all orbits of maximal dimension under the dual action of $H$ on $\R^n$. This result is applied to study integrability for certain families of dilation groups; in particular, we give a characterization valid for connected abelian matrix groups acting in dimension three.

Détail

Classification : 42C40, 22E25
Mots-clés : Wavelet transforms, Calderon condition, integrable representations, dual topology, coadjoint orbits