Abstract
This paper is devoted to the construction of wavelet (or coherent state) systems arising from the action of certain semidirect products G=Rk⋊H on L2(Rk). For this purpose previous results which guarantee the existence of inversion formulas are applied to the special case where H is Abelian. The questions of systematic construction and conjugacy of such groups are completely resolved by setting up a correspondence to unit groups of commutative associative algebras. As an application the numbers of conjugacy classes of possible Abelian groups are computed for k=2,3,4. For k⩾7, there are uncountably many conjugacy classes. We then compute the admissibility conditions belonging to Abelian groups. The final section contains a characterization of Abelian matrix groups acting ergodically on some subset of Rk. This result ensures that the approach via associative algebras yields all possible groups.
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