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

Abstract

Given a frame F = { f j } for a separable Hilbert space H, we introduce the linear subspace H p F of H consisting of elements whose frame coefficient sequences belong to the 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 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 p F converges in both the Hilbert space norm and the || · || F, pnorm which is induced by the p -norm.