An operator theory approach to the approximate duality of Hilbert space frames

Abstract

The purpose of this paper is to state some properties and applications of approximate duals constructed by bounded operators. It is shown that there are close relationships between this kind of approximate duals and some concepts in frame theory such as the closeness of two frames, nearly Parseval and scalable frames. We get some equivalent conditions for a frame to be close to another frame in terms of approximate duals. Also, using approximate duals, some conditions for a frame to be nearly Parseval or scalable are derived. Moreover, some other properties of approximate duals are obtained; in particular, some characterizations of them are stated and some of their applications for Gabor frames and erasures are given, especially the excesses of them are considered.