Indian Journal of Science and Technology

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Perturbations of Generalized Dual Banach Frames


Affiliations
1 Department of Mathematics, Faculty of Science, Islamic Azad University, Central Tehran Branch, Tehran, Iran, Islamic Republic of
 

The classical duals and generalized duals of frames play a fundamental role in the abstract frame theory. In this paper we first define the concepts of dual, canonical dual, pseudo-dual and approximate dual for Bessel sequences with respect to a BK-space of scalar-valued sequences in Banach spaces, and illuminate their relationship with Banach frames. Then we prove that classical dual and approximate dual of Banach frames are stable under small perturbations so that the results obtained1 is a special case of it. We also apply a new characterization of classical dual Banach frames to discuss a stability problem for them. For approximate dual Banach frames constructed via perturbation theory, we provide a bound on the deviation from perfect reconstruction.

Keywords

Xd-Frames, Banach Frames, Approximate Duals, Pseudo-duals, Perturbations
User

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  • Perturbations of Generalized Dual Banach Frames

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Authors

Seyedeh Sara Karimizad
Department of Mathematics, Faculty of Science, Islamic Azad University, Central Tehran Branch, Tehran, Iran, Islamic Republic of
Mohammad Sadegh Asgari
Department of Mathematics, Faculty of Science, Islamic Azad University, Central Tehran Branch, Tehran, Iran, Islamic Republic of

Abstract


The classical duals and generalized duals of frames play a fundamental role in the abstract frame theory. In this paper we first define the concepts of dual, canonical dual, pseudo-dual and approximate dual for Bessel sequences with respect to a BK-space of scalar-valued sequences in Banach spaces, and illuminate their relationship with Banach frames. Then we prove that classical dual and approximate dual of Banach frames are stable under small perturbations so that the results obtained1 is a special case of it. We also apply a new characterization of classical dual Banach frames to discuss a stability problem for them. For approximate dual Banach frames constructed via perturbation theory, we provide a bound on the deviation from perfect reconstruction.

Keywords


Xd-Frames, Banach Frames, Approximate Duals, Pseudo-duals, Perturbations

References





DOI: https://doi.org/10.17485/ijst%2F2014%2Fv7i1%2F46682