Frames in Banach spaces

The notion of a frame in a Banach space with respect to a model space of sequences is introduced. This notion is different from the notions of an atomic decomposition, Banach frame in the sense of Gröchenig, (unconditional) Schauder frame in the sense of Han and Larson, and other known definitions of frames for Banach spaces. The frames introduced in this paper are shown to play a universal role in the solution of the general problem of representation of functions by series. A projective description of these frames is given. A criterion for the existence of a linear frame expansion algorithm and an analogue of the extremality property for a frame expansion are obtained.