Abstract
In this paper, a new characterization is obtained for approximately dual frames of a given frame. 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 \({T_\Phi U_{\Phi^{ad}}=T_\Psi U_{\Psi^{ad}}}\), where T X and U X denote the synthesis and analysis operators of the frame X, respectively. Finally, the results are applied to Gabor systems to obtain some practical examples.
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Javanshiri, H. Some Properties of Approximately Dual Frames in Hilbert Spaces. Results. Math. 70, 475–485 (2016). https://doi.org/10.1007/s00025-016-0587-y
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DOI: https://doi.org/10.1007/s00025-016-0587-y