Perturbations of Banach Frames and Atomic Decompositions
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Banach frames and atomic decompositions are sequences which have basis-like properties but which need not be bases. In particular, they allow elements of a Banach space to be written as combinations of the frame or atomic decomposition elements in a stable manner. However, these representations need not be unique. Such exibility is important in many applications. In this paper, we prove that frames and atomic decompositions in Banach spaces are stable under small perturbations. Our results are strongly related to classic results on perturbations of Paley/Wiener and Kato. We also consider duality properties for atomic decompositions, and discuss the consequences for Hilbert frames.