Some properties of generalized and approximately dual frames in Hilbert spaces

In the present paper, some sufficient and necessary conditions for two frames $Φ=(φ_n)_n$ and $Ψ=(ψ_n)_n$ 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 $Φ=(φ_n)_n$ and $Ψ=(ψ_n)_n$ are close to each other, then we can find approximately dual frames $Φ^{ad}=(φ^{ad}_n)_n$ and $Ψ^{ad}=(ψ^{ad}_n)_n$ of them which are close to each other and $T_ΦU_{Φ^{ad}}=T_ΨU_{Ψ^{ad}}$, where $T_Φ$ and $T_Ψ$ (resp. $U_{Φ^{ad}}$ and $U_{Ψ^{ad}}$) are the analysis operators (resp. synthesis operators) of the frames $Φ$ and $Ψ$ (resp. $Φ^{ad}$ and $Ψ^{ad}$), 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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