Some properties of generalized and approximately dual frames in Hilbert spaces

TL;DR

This paper investigates properties of generalized and approximately dual frames in Hilbert spaces, providing new characterizations, conditions, and constructions, especially for Gabor frames, based on analysis and synthesis operator properties.

Contribution

It introduces new criteria and characterizations for approximately and generalized dual frames, including their stability and construction methods, with applications to Gabor frames.

Findings

Characterization of dual frames via analysis and synthesis operators.

Existence of approximately dual frames close to given frames.

Construction methods for approximately dual Gabor frames.

Abstract

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.