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Approximately dual frames as a generalization of duality notion in Hilbertspaces have applications in Gabor systems, wavelets, coorbit theory and sensormodeling. In recent years, the computing of the associated deviations of thecanonical and alternate dual frames from the original ones has been consideredby some authors. However, the quantitative measurement of the associateddeviations of the alternate and approximately dual frames from the originalones has not been satisfactorily answered. In this paper, among other things,it is proved that if the sequence $\Psi=(\psi_n)_n$ is sufficiently close tothe frame $\Phi=(\varphi_n)_n$, then $\Psi$ is a frame for $\mathcal H$ andapproximately dual frames $\Phi^{ad}=(\varphi^{ad}_n)_n$ and$\Psi^{ad}=(\psi^{ad}_n)_n$ can be found which are close to each other andparticularly, we estimate the deviation from perfect reconstruction in terms ofthe operator ${\mathcal A}_1:=T_\Phi U_{\Phi^{ad}}$ and ${\mathcal A}_2:=T_\PsiU_{\Psi^{ad}}$ and their approximation rates, where $T_X$ and $U_X$ denote thesynthesis and analysis operators of the frame $X$, respectively. Finally, wedemonstrate how our results apply in the practical case of Gabor systems. It isworth mentioning that some of our perturbation conditions are quite differentfrom those used in the previous literatures on this topic.

The effect of perturbations of frames on their alternate and approximately dual frames