Subspaces of L2(G) invariant under translation by an abelian subgroup

By Joseph W. Iverson. 2014

47 citations recorded by the source corpus.

Abstract For a second countable locally compact group G and a closed abelian subgroup H , we give a range function classification of closed subspaces in L 2 ( G ) invariant under left translation by H . For a family A ⊆ L 2 ( G ) , this classification ties with a set of conditions under which the translations of A by H form a continuous frame or a Riesz sequence. When G is abelian, our work relies on a fiberization map; for the more general case, we introduce an analogue of the Zak transform. Both transformations intertwine translation with modulation, and both rely on a new group-theoretic tool: for a closed subgroup Γ ⊆ G , we produce a measure on the space Γ \ G of right cosets that gives a measure space isomorphism G ≅ Γ × Γ \ G . Outside of the group setting, we consider a more general problem: for a measure space X and a Hilbert space H , we investigate conditions under which a family of functions in L 2 ( X ; H ) multiplies with a basis-like system in L 2 ( X ) to produce a continuous frame or a Riesz sequence in L 2 ( X ; H ) . Finally, we explore connections with dual integrable representations of LCA groups, as introduced by Hernandez et al. in [25] .

Subfields: Applied Mathematics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/6e6bcf8369c4728dccc7cec470f241cc6ca1a6ea

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In Applied Mathematics

  1. E. Hern'andez
  2. Davide Barbieri
  3. Victoria Paternostro

In Radiology, Nuclear Medicine and Imaging

  1. E. Hern'andez
  2. Davide Barbieri
  3. Victoria Paternostro

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