Generalized Calderón conditions and regular orbit spaces

• Autorzy: Hartmut Führ
• Języki publikacji: EN

Abstrakty

EN

The construction of generalized continuous wavelet transforms on locally compact abelian groups A from quasi-regular representations of a semidirect product group G = A ⋊ H acting on L²(A) requires the existence of a square-integrable function whose Plancherel transform satisfies a Calderón-type resolution of the identity. The question then arises under what conditions such square-integrable functions exist. The existing literature on this subject leaves a gap between sufficient and necessary criteria. In this paper, we give a characterization in terms of the natural action of the dilation group H on the character group of A. We first prove that a Calderón-type resolution of the identity gives rise to a decomposition of Plancherel measure of A into measures on the dual orbits, and then show that the latter property is equivalent to regularity conditions on the orbit space of the dual action. Thus we obtain, for the first time, sharp necessary and sufficient criteria for the existence of a wavelet inversion formula associated to a quasi-regular representation. As a byproduct and special case of our results we deduce that discrete series subrepresentations of the quasi-regular representation correspond precisely to dual orbits with positive Plancherel measure and associated compact stabilizers. Only sufficiency of the conditions was previously known.

Kategorie tematyczne

• 28D15: General groups of measure-preserving transformations
• 42C40: Wavelets and other special systems
• 22D10: Unitary representations of locally compact groups

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2010
• Tom: 120
• Numer: 1
• Strony: 103-126

Daty

wydano

2010

Twórcy

autor

Hartmut Führ

• Lehrstuhl A für Mathematik, RWTH Aachen, 52056 Aachen, Germany

Identyfikatory

DOI

10.4064/cm120-1-8