arXiv subjects

Joseph W. Iverson

Publications and source records attributed to Joseph W. Iverson.

24 records · Page 2Linked to original sources

arXiv2019

Multiplication-invariant operators and the classification of LCA group frames

In this paper we study the properties of multiplication invariant (MI) operators acting on subspaces of the vector-valued space $L^2(X;\mathcal H)$. We characterize such operators in terms of range functions by showing that there is an isomorphism between the category of MI spaces (with MI operators as morphisms) and the category of measurable range functions whose morphisms are measurable range operators. We investigate how global properties of an MI operator are reflected by local pointwise properties of its corresponding range operator. We also establish several results about frames generated by multiplications in $L^2(X;\mathcal H)$. This includes the classification of frames of multiplications with respect to unitary equivalence by measurable fields of Gramians. Finally, we show applications of our results in the study of abelian group frames and translation-invariant (TI) operators acting on subspaces of $L^2(\mathcal G)$, where $\mathcal G$ is a locally compact group.

math.FA↗

arXiv2019

The Zak transform and representations induced from characters of an abelian subgroup

We consider a variant of the Zak transform for a finite group $G$ with respect to a fixed abelian subgroup $H$, and demonstrate a relationship with representations of $G$ induced from characters of $H$. We also show how the Zak transform can be used to study right translations by $H$ in $L^2(G)$, and give some examples of applications for equiangular tight frames.

math.FA↗

arXiv2019

Optimal line packings from nonabelian groups

We use group schemes to construct optimal packings of lines through the origin. In this setting, optimal line packings are naturally characterized using representation theory, which in turn leads to a necessary integrality condition for the existence of equiangular central group frames. We conclude with an infinite family of optimal line packings using the group schemes associated with certain Suzuki 2-groups, specifically, extensions of Heisenberg groups. Notably, this is the first known infinite family of equiangular tight frames generated by representations of nonabelian groups.

math.FA↗

arXiv2017

Optimal line packings from finite group actions

We provide a general program for finding nice arrangements of points in real or complex projective space from transitive actions of finite groups. In many cases, these arrangements are optimal in the sense of maximizing the minimum distance. We introduce our program in terms of general Schurian association schemes before focusing on the special case of Gelfand pairs. Notably, our program unifies a variety of existing packings with heretofore disparate constructions. In addition, we leverage our program to construct the first known infinite family of equiangular lines with Heisenberg symmetry.

math.FA↗

arXiv2015

Frames generated by compact group actions

Let $K$ be a compact group, and let $ρ$ be a representation of $K$ on a Hilbert space $\mathcal{H}_ρ$. We classify invariant subspaces of $\mathcal{H}_ρ$ in terms of range functions, and investigate frames of the form $\{ρ(ξ) f_i\}_{ξ\in K, i \in I}$. This is done first in the setting of translation invariance, where $K$ is contained in a larger group $G$ and $ρ$ is left translation on $\mathcal{H}_ρ= L^2(G)$. For this case, our analysis relies on a new, operator-valued version of the Zak transform. For more general representations, we develop a calculational system known as a "bracket" to analyze representation structures and frames with a single generator. Several applications are explored. Then we turn our attention to frames with multiple generators, giving a duality theorem that encapsulates much of the existing research on frames generated by finite groups, as well as classical duality of frames and Riesz sequences.

math.CA↗

arXiv2015

Subspaces of $L^2(G)$ invariant under translation by an abelian subgroup

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 $\mathscr{A} \subset L^2(G)$, this classification ties with a set of conditions under which the translations of $\mathscr{A}$ 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 $Γ\subset G$, we produce a measure on the space $Γ\backslash G$ of right cosets that gives a measure space isomorphism $G \cong Γ\times Γ\backslash G$. Outside of the group setting, we consider a more general problem: for a measure space $X$ and a Hilbert space $\mathcal{H}$, we investigate conditions under which a family of functions in $L^2(X;\mathcal{H})$ multiplies with a basis-like system in $L^2(X)$ to produce a continuous frame or a Riesz sequence in $L^2(X;\mathcal{H})$. Finally, we explore connections with dual integrable representations of LCA groups, as introduced by Hern{á}ndez et al.

math.CA↗