Frames and their associated $emph{H}_{{kern-2pt}emph{F}}^{emph{p}}$-subspaces

Given a frame F = {f _j } for a separable Hilbert space H, we introduce the linear subspace H^p_FH^{p}_{F} of H consisting of elements whose frame coefficient sequences belong to the ℓ ^p -space, where 1 ≤ p < 2. Our focus is on the general theory of these spaces, and we investigate different aspects of these spaces in relation to reconstructions, p-frames, realizations and dilations. In particular we show that for closed linear subspaces of H, only finite dimensional ones can be realized as H^p_FH^{p}_{F}-spaces for some frame F. We also prove that with a mild decay condition on the frame F the frame expansion of any element in H_F^pH_{F}^{p} converges in both the Hilbert space norm and the ||·|| _{F, p} -norm which is induced by the ℓ ^p -norm.