2013/09/30 by Corach, G., Massey, P., Ruiz, M.
#15A60 #42C15 #65F35 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1309.7914
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.