Procrustes problems and Parseval quasi-dual frames

Abstract: Parseval frames have particularly useful properties, and in some cases, they can be used to reconstruct signals which were analyzed by a non-Parseval frame. In this paper, we completely describe the degree to which such reconstruction is feasible. Indeed, notice that for fixed frames $\cF$ and $\cX$ with synthesis operators $F$ and $X$, the operator norm of $FX^*-I$ measures the (normalized) worst-case error in the reconstruction of vectors when analyzed with $\cX$ and synthesized with $\cF$. Hence, for any given frame $\cF$, we compute explicitly the infimum of the operator norms of $FX^*-I$, where $\cX$ is any Parseval frame. The $\cX$'s that minimize this quantity are called Parseval quasi-dual frames of $\cF$. Our treatment considers both finite and infinite Parseval quasi-dual frames.