Frames generated by compact group actions

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.