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 F and X 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 X and synthesized with F . Hence, for any given frame F , we compute explicitly the infimum of the operator norm of FX∗−I, where X is any Parseval frame. The X ’s that minimize this quantity are called Parseval quasi-dual frames of F . Our treatment considers both finite and infinite Parseval quasi-dual frames.