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

Abstract

Approximately dual frames as a generalization of duality notion in Hilbert spaces have applications in Gabor systems, wavelets, coorbit theory and sensor modeling. In recent years, the computing of the associated deviations of the canonical and alternate dual frames from the original ones has been considered by some authors. However, the quantitative measurement of the associated deviations of the alternate and approximately dual frames from the original ones 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 to the frame $\Phi=(\varphi_n)_n$, then $\Psi$ is a frame for $\mathcal H$ and approximately dual frames $\Phi^{ad}=(\varphi^{ad}_n)_n$ and $\Psi^{ad}=(\psi^{ad}_n)_n$ can be found which are close to each other and particularly, we estimate the deviation from perfect reconstruction in terms of the operator ${\mathcal A}_1:=T_\Phi U_{\Phi^{ad}}$ and ${\mathcal A}_2:=T_\Psi U_{\Psi^{ad}}$ and their approximation rates, where $T_X$ and $U_X$ denote the synthesis and analysis operators of the frame $X$, respectively. Finally, we demonstrate how our results apply in the practical case of Gabor systems. It is worth mentioning that some of our perturbation conditions are quite different from those used in the previous literatures on this topic.