Paley-wiener type perturbations of frames and the deviation from perfect reconstruction

Frame theory is an efficient tool to obtain expansions of elements in separable Hilbert spaces that are similar to the ones obtained via orthonormal bases, however, with considerably more exibility. In this paper we give a survey of known results about frame expansions and perturbation theory, combined with an extension to approximately dual frames. We will show, e.g., that perturbation of a pair of dual frames in the Paley-Wiener sense leads to a deviation from perfect reconstruction that can be controlled in terms of the frame bounds of the involved sequences. The paper contains an Appendix, which motivates the analysis of frames via classical results.