Frames generated by compact group actions

TL;DR

This paper classifies invariant subspaces and analyzes frames generated by compact group actions on Hilbert spaces, introducing new tools like an operator-valued Zak transform and a bracket calculus.

Contribution

It develops a comprehensive framework for understanding frames from compact group representations, including a new operator-valued Zak transform and a duality theorem for multi-generator frames.

Findings

Classification of invariant subspaces via range functions

Introduction of an operator-valued Zak transform for translation-invariant frames

A duality theorem for frames with multiple generators

Abstract

Let $K$ be a compact group, and let $\rho$ be a representation of $K$ on a Hilbert space $\mathcal{H}_\rho$. We classify invariant subspaces of $\mathcal{H}_\rho$ in terms of range functions, and investigate frames of the form $\{\rho(\xi) f_i\}_{\xi \in K, i \in I}$. This is done first in the setting of translation invariance, where $K$ is contained in a larger group $G$ and $\rho$ is left translation on $\mathcal{H}_\rho = L^2(G)$. 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.