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.
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