We obtain necessary and sufficient conditions for the admissible vectors of a new unitary non irreducible representation U . The group G is an arbitrary semidirect product whose normal factor A is abelian and whose homogeneous factor H is a locally compact second countable group acting on a Riemannian manifold M . The key ingredient in the construction of U is a C 1 intertwining map between the actions of H on the dual group ˆ A and on M . The representation U generalizes the restriction of the metaplectic representation to triangular subgroups of S p(d, R), whence the name “mock metaplectic”. For simplicity, we content ourselves with the case where A = Rn and M = Rd . The main technical point is the decomposition of U as direct integral of its irreducible components. This theory is motivated by some recent developments in signal analysis, notably shearlets. Many related examples are discussed.
An introduction to mocklets
DE MARI CASARETO DAL VERME, FILIPPO;DE VITO, ERNESTO
2010-01-01
Abstract
We obtain necessary and sufficient conditions for the admissible vectors of a new unitary non irreducible representation U . The group G is an arbitrary semidirect product whose normal factor A is abelian and whose homogeneous factor H is a locally compact second countable group acting on a Riemannian manifold M . The key ingredient in the construction of U is a C 1 intertwining map between the actions of H on the dual group ˆ A and on M . The representation U generalizes the restriction of the metaplectic representation to triangular subgroups of S p(d, R), whence the name “mock metaplectic”. For simplicity, we content ourselves with the case where A = Rn and M = Rd . The main technical point is the decomposition of U as direct integral of its irreducible components. This theory is motivated by some recent developments in signal analysis, notably shearlets. Many related examples are discussed.
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