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.

No reviews yet.

No comments yet.