Advances in Computational Mathematics

, Volume 34, Issue 2, pp 185–200 | Cite as

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 \(H^{p}_{F}\) 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.


Frames Riesz bases Reconstruction Dilation 

Mathematics Subject Classifications (2010)

46C99 47B99 46B15 


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© 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

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