, Volume 34, Issue 2, pp 185-200
Date: 01 Jun 2010

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

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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 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 \(H^{p}_{F}\) -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 \(H_{F}^{p}\) converges in both the Hilbert space norm and the ||·|| F, p -norm which is induced by the ℓ p -norm.

Communicated by Qiyu Sun.
Pengtong Li’s work was partially supported by National Natural Science Foundation of China (No. 10771101).