38 research outputs found
Decompositions of Rational Gabor Representations
Let be a group of unitary operators where is a translation
operator and is a modulation operator acting on Assuming that is a non-singular rational matrix of
order with at least one rational non-integral entry, we obtain a direct
integral irreducible decomposition of the Gabor representation which is defined
by the isomorphism where We also show that the left regular representation
of \left( \mathbb{Z}_{m}\times B\mathbb{Z}% ^{d}\right) \rtimes\mathbb{Z}^{d}
which is identified with via is unitarily equivalent to a direct
sum of many disjoint
subrepresentations: It is shown that for any the
subrepresentation of the left regular representation is disjoint from the
Gabor representation. Furthermore, we prove that there is a subrepresentation
of the left regular representation of which has a
subrepresentation equivalent to if and only if Using a central decomposition of the representation
and a direct integral decomposition of the left regular representation, we
derive some important results of Gabor theory. More precisely, a new proof for
the density condition for the rational case is obtained. We also derive
characteristics of vectors in such that
is a Parseval frame in $L^{2}(\mathbb{R})^{d}.
Dihedral Group Frames which are Maximally Robust to Erasures
Let be a natural number larger than two. Let be the Dihedral group, and an
-dimensional unitary representation of acting in as
follows. and
for For any representation which is unitarily equivalent to
we prove that when is prime there exists a Zariski open subset
of such that for any vector any subset of
cardinality of the orbit of under the action of this representation is
a basis for However, when is even there is no vector in
which satisfies this property. As a result, we derive that if
is prime, for almost every (with respect to Lebesgue measure) vector in
the -orbit of is a frame which is maximally
robust to erasures. We also consider the case where is equivalent to an
irreducible unitary representation of the Dihedral group acting in a vector
space and we
provide conditions under which it is possible to find a vector
such that has the Haar
property
Admissibility For Monomial Representations of Exponential Lie Groups
Let be a simply connected exponential solvable Lie group, a closed
connected subgroup, and let be a representation of induced from a
unitary character of . The spectrum of corresponds via the
orbit method to the set of coadjoint orbits that meet the
spectral variety A_\tau = f + \h^\perp. We prove that the spectral measure of
is absolutely continuous with respect to the Plancherel measure if and
only if acts freely on some point of . As a corollary we show that
if is nonunimodular, then has admissible vectors if and only if the
preceding orbital condition holds
Groups with frames of translates
Let be a locally compact group with left regular representation
We say that admits a frame of translates if there exist a
countable set and such that
is a frame for The present
work aims to characterize locally compact groups having frames of translates,
and to this end, we derive necessary and/or sufficient conditions for the
existence of such frames. Additionally, we exhibit surprisingly large classes
of Lie groups admitting frames of translates