Some properties of generalized and approximately dual frames in Hilbert spaces

In the present paper, some sufficient and necessary conditions for two frames $\mathrm{\Phi }=({\phi }_n{)}_n$ and $\mathrm{\Psi }=({\psi }_n{)}_n$ 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 $\mathrm{\Phi }=({\phi }_n{)}_n$ and $\mathrm{\Psi }=({\psi }_n{)}_n$ are close to each other, then we can find approximately dual frames ${\mathrm{\Phi }}^{ad}=({\phi }_n^{ad}{)}_n$ and ${\mathrm{\Psi }}^{ad}=({\psi }_n^{ad}{)}_n$ 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\mathrm{.}$U_{\Phi^{ad}}$and$U_{\Psi^{ad}}$) are the analysis operators (resp. synthesis operators) of the frames $\mathrm{\Phi }$ and $\mathrm{\Psi }$ (resp. ${\mathrm{\Phi }}^{ad}$ and ${\mathrm{\Psi }}^{ad}$), 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.