Procrustes Problems and Parseval Quasi-Dual Frames

GUSTAVO CORACH; PEDRO MASSEY; MARIANO RUIZ

ACTA APPLICANDAE MATHEMATICAE

SPRINGER

Lugar: Berlin; Año: 2014 vol. 131 p. 179 - 195

0167-8019

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 norm 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.