Dual pairs of frames yield a procedure for obtaining perfect reconstruction of elements in the underlying Hilbert space in terms of superpositions of the frame elements. However, practical constraints often force us to apply sequences that do not exactly form dual frames. In this article, we consider the important case of generalized shift-invariant systems and provide various ways of estimating the deviation from perfect reconstruction that occur when the systems do not form dual frames. The deviation from being dual frames will be measured either in terms of a perturbation condition or in terms of the deviation from equality in the duality conditions.
Benavente, A., Christensen, O., & Zakowicz, M. I. (2017). Generalized shift-invariant systems and approximately dual frames. Annals of Functional Analysis, 8(2), 177-189. https://doi.org/10.1215/20088752-3784315