Some Properties of Approximately Dual Frames in Hilbert Spaces

In this paper, a new characterization is obtained for approximately dual frames of a given frame. Among other things, it is proved that if the sequence Psi = (psi(n))(n) is sufficiently close to the frame Phi = (phi(n))(n), then Psi is a frame for H and approximately dual frames Phi(ad) = (phi(ad)(n))(n) and Psi(ad) = (psi(ad)(n))(n) can be found which are close to each other and T Phi U Phi ad = T Psi U Psi ad, where T-X and U-X denote the synthesis and analysis operators of the frame X, respectively. Finally, the results are applied to Gabor systems to obtain some practical examples.