Dihedral Group Frames which are Maximally Robust to Erasures

Let $n$ be a natural number larger than two. Let $D_{2n}=\langle r,s : r^{n}=s^{2}=e, srs=r^{n-1} \rangle$ be the Dihedral group, and $\kappa$ an $n$-dimensional unitary representation of $D_{2n}$ acting in $\mathbb{C}^n$ as follows. $(\kappa (r)v)(j)=v((j-1)\mod n)$ and $(\kappa(s)v)(j)=v((n-j)\mod n)$ for $v\in\mathbb{C}^n.$ For any representation which is unitarily equivalent to $\kappa,$ we prove that when $n$ is prime there exists a Zariski open subset $E$ of $\mathbb{C}^{n}$ such that for any vector $v\in E,$ any subset of cardinality $n$ of the orbit of $v$ under the action of this representation is a basis for $\mathbb{C}^{n}.$ However, when $n$ is even there is no vector in $\mathbb{C}^{n}$ which satisfies this property. As a result, we derive that if $n$ is prime, for almost every (with respect to Lebesgue measure) vector $v$ in $\mathbb{C}^{n}$ the $\Gamma$-orbit of $v$ is a frame which is maximally robust to erasures. We also consider the case where $\tau$ is equivalent to an irreducible unitary representation of the Dihedral group acting in a vector space $\mathbf{H}_{\tau}\in\left\{\mathbb{C},\mathbb{C}^2\right\}$ and we provide conditions under which it is possible to find a vector $v\in\mathbf{H}_{\tau}$ such that $\tau\left( \Gamma\right) v$ has the Haar property

Decompositions of Rational Gabor Representations

Let $\Gamma=\langle T_{k},M_{l}:k\in\mathbb{Z}^{d},l\in B\mathbb{Z}^{d}\rangle$ be a group of unitary operators where $T_{k}$ is a translation operator and $M_{l}$ is a modulation operator acting on $L^{2}\left( \mathbb{R}^{d}\right) .$ Assuming that $B$ is a non-singular rational matrix of order $d,$ 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 $\pi:\left( \mathbb{Z}_{m}\times B\mathbb{Z}^{d}\right) \rtimes\mathbb{Z}^{d}\rightarrow\Gamma$ where $\pi\left( \theta,l,k\right) =e^{2\pi i\theta}M_{l}T_{k}.$ 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 $\Gamma$ via $\pi$ is unitarily equivalent to a direct sum of $\mathrm{card}\left( \left[ \Gamma,\Gamma\right] \right)$ many disjoint subrepresentations: $L_{0},L_{1},\cdots,L_{\mathrm{card}\left( \left[ \Gamma,\Gamma\right] \right) -1}.$ It is shown that for any $k\neq 1$ the subrepresentation $L_k$ of the left regular representation is disjoint from the Gabor representation. Furthermore, we prove that there is a subrepresentation $L_{1}$ of the left regular representation of $\Gamma$ which has a subrepresentation equivalent to $\pi$ if and only if $\left\vert \det B\right\vert \leq1.$ Using a central decomposition of the representation $\pi$ 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 $f$ in $L^{2}(\mathbb{R})^{d}$ such that $\pi(\Gamma)f$ is a Parseval frame in $L^{2}(\mathbb{R})^{d}.

Admissibility For Monomial Representations of Exponential Lie Groups

Let $G$ be a simply connected exponential solvable Lie group, $H$ a closed connected subgroup, and let $\tau$ be a representation of $G$ induced from a unitary character $\chi_f$ of $H$. The spectrum of $\tau$ corresponds via the orbit method to the set $G\cdot A_\tau / G$ of coadjoint orbits that meet the spectral variety A_\tau = f + \h^\perp. We prove that the spectral measure of $\tau$ is absolutely continuous with respect to the Plancherel measure if and only if $H$ acts freely on some point of $A_\tau$. As a corollary we show that if $G$ is nonunimodular, then $\tau$ has admissible vectors if and only if the preceding orbital condition holds

Groups with frames of translates

Let $G$ be a locally compact group with left regular representation $\lambda_{G}.$ We say that $G$ admits a frame of translates if there exist a countable set $\Gamma\subset G$ and $\varphi\in L^{2}(G)$ such that $(\lambda_{G}(x) \varphi)_{x \in\Gamma}$ is a frame for $L^{2}(G).$ 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

Dihedral Group Frames with the Haar Property

We consider a unitary representation of the Dihedral group $D_{2n}=\mathbb{Z}_{n}\rtimes\mathbb{Z}_{2}$ obtained by inducing the trivial character from the co-normal subgroup $\left\{0\right\}\rtimes\mathbb{Z}_{2}.$ This representation is naturally realized as acting on the vector space $\mathbb{C}^{n}.$ We prove that the orbit of almost every vector in $\mathbb{C}^{n}$ with respect to the Lebesgue measure has the Haar property (every subset of cardinality $n$ of the orbit is a basis for $\mathbb{C}^{n}$) if $n$ is an odd integer. Moreover, we provide explicit sufficient conditions for vectors in $\mathbb{C}^{n}$ whose orbits have the Haar property. Finally, we derive that the orbit of almost every vector in $\mathbb{C}^{n}$ under the action of the representation has the Haar property if and only if $n$ is odd. This completely settles a problem which was only partially answered in \cite{Oussa}