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 Ψ = ( ψ n ) n is sufficiently close to the frame Φ = ( φ n ) n , then Ψ is a frame for H and approximately dual frames Φ a d = ( φ n a d ) n and Ψ a d = ( ψ n a d ) n can be found which are close to each other and T Φ U Φ a d = T Ψ U Ψ a d , 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.