Some properties of generalized and approximately dual frames in Hilbert spaces

In the present paper, some sufficient and necessary conditions for two frames $\Phi=(\varphi_n)_n$ and $\Psi=(\psi_n)_n$ under which they are approximately or generalized dual frames are determined depending on the properties of their analysis and synthesis operators. We also give a new characterization for approximately dual frames associated with a given frame and given operator by using of bounded operators. Among other things, we prove that if two frames $\Phi=(\varphi_n)_n$ and $\Psi=(\psi_n)_n$ are close to each other, then we can find approximately dual frames $\Phi^{ad}=(\varphi^{ad}_n)_n$ and $\Psi^{ad}=(\psi^{ad}_n)_n$ of them which are close to each other and $T_\Phi U_{\Phi^{ad}}=T_\Psi U_{\Psi^{ad}}$, where $T_\Phi$ and $T_\Psi$ (resp. $U_{\Phi^{ad}}$ and $U_{\Psi^{ad}}$) are the analysis operators (resp. synthesis operators) of the frames $\Phi$ and $\Psi$ (resp. $\Phi^{ad}$ and $\Psi^{ad}$), respectively. We then give some consequences on generalized dual frames. Finally, we apply these results to find some construction results for approximately dual frames for a given Gabor frame.

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