6 research outputs found
Plancherel transform criteria for Weyl-Heisenberg frames with integer oversampling
We investigate the relevance of admissibility criteria based on Plancherel
measure for the characterization of tight Weyl-Heisenberg frames with integer
oversampling. For this purpose we observe that functions giving rise to such
Weyl-Heisenberg frames are admissible with respect to the action of suitably
defined type-I discrete group G. This allows to relate the construction of
Weyl-Heisenberg frames to the Plancherel measure of G, which provides an
alternative proof and a new interpretation of the well-known Zak transform
based criterion for tight Weyl-Heisenberg frames with integer oversampling.Comment: 13 page
Strictly positive definite functions on compact abelian groups
We study the Fourier characterisation of strictly positive definite functions
on compact abelian groups. Our main result settles the case , with and finite. The characterisation
obtained for these groups does not extend to arbitrary compact abelian groups;
it fails in particular for all torsion-free groups.Comment: 9 pages; submitted to Proc. AM
Plancherel Inversion as Unified Approach to Wavelet Transforms and Wigner functions
We demonstrate that the Plancherel transform for Type-I groups provides one
with a natural, unified perspective for the generalized continuous wavelet
transform, on the one hand, and for a class of Wigner functions, on the other.
The wavelet transform of a signal is an -function on an appropriately
chosen group, while the Wigner function is defined on a coadjoint orbit of the
group and serves as an alternative characterization of the signal, which is
often used in practical applications. The Plancherel transform maps
-functions on a group unitarily to fields of Hilbert-Schmidt operators,
indexed by unitary irreducible representations of the group. The wavelet
transform can essentiallly be looked upon as restricted inverse Plancherel
transform, while Wigner functions are modified Fourier transforms of inverse
Plancherel transforms, usually restricted to a subset of the unitary dual of
the group. Some known results both on Wigner functions and wavelet transforms,
appearing in the literature from very different perspectives, are naturally
unified within our approach. Explicit computations on a number of groups
illustrate the theory.Comment: 41 page