PERTURBATIONS OF BANACH FRAMES AND ATOMIC DECOMPOSITIONS

Banach frames and atomic decompositions are sequences that have basis- like properties but which need not be bases. Ir. particular, they allo w elements of a Banach space to be written as linear combinations of t he frame or atomic decomposition elements in a stable manner. In this paper we prove several functional-analytic properties of these decompo sitions, and show how these properties apply to Gabor and wavelet syst ems. We first prove that frames and atomic decompositions are stable u nder small perturbations. This is inspired by corresponding classical perturbation results for bases, including the Paley-Wiener basis stabi lity criteria and the perturbation theorem of Kato. We introduce new a nd weaker conditions which ensure the desired stability. We then prove duality properties of atomic decompositions and consider some consequ ences for Hilbert frames. Finally, we demonstrate how our results appl y in the practical case of Gabor systems in weighted L-2 spaces. Such systems can form atomic decompositions for L-w(2)(IR), but cannot form Hilbert frames for L-w(2)(IR) unless the weight is trivial.