Frames generated by compact group actions
Joseph W. Iverson

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 be a compact group, and let be a representation of on a Hilbert space . We classify invariant subspaces of in terms of range functions, and investigate frames of the form . This is done first in the setting of translation invariance, where is contained in a larger group and is left translation on . 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,…
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