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It is known in Hilbert space frame theory that a Bessel sequence can beexpanded to a frame. Contrary to Hilbert space situation, using a result ofCasazza and Christensen, we show that there are Banach spaces and approximateBessel sequences which can not be expanded to approximate Schauder frames. Wecharacterize Banach spaces in which one can expand approximate Bessel sequencesto approximate Schauder frames.

Expansion of approximate Bessel sequences to approximate Schauder frames for Banach spaces