Advances in Computational Mathematics

, Volume 34, Issue 2, pp 185–200

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

Article

DOI: 10.1007/s10444-010-9149-0

Cite this article as:
Han, D., Li, P. & Tang, WS. Adv Comput Math (2011) 34: 185. doi:10.1007/s10444-010-9149-0
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Abstract

Given a frame F = {fj} 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 ℓp-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 ℓp-norm.

Keywords

FramesRiesz basesReconstructionDilation

Mathematics Subject Classifications (2010)

46C9947B9946B15

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Central FloridaOrlandoUSA
  2. 2.Department of MathematicsNanjing University of Aeronautics and AstronauticsNanjingPeople’s Republic of China
  3. 3.Department of MathematicsNational University of SingaporeSingaporeRepublic of Singapore