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- W4308739957 abstract "Given an N-dimensional subspace XN of Lp(Ω), we consider the problem of choosing M-sampling points which may be used to discretely approximate the Lp norm on the subspace. We are particularly interested in knowing when the number of sampling points M can be chosen on the order of the dimension N. For the case p=2 it is known that M may always be chosen on the order of N as long as the subspace XN satisfies a natural L∞ bound, and for the case p=∞ there are examples where M may not be chosen on the order of N. We show for all 1≤p<2 that there exist classes of subspaces of Lp([0,1]) which satisfy the L∞ bound, but where the number of sampling points M cannot be chosen on the order of N. We show as well that the problem of discretizing the Lp norm of subspaces is directly connected with frame theory. In particular, we prove that discretizing a continuous frame to obtain a discrete frame which does stable phase retrieval requires discretizing both the L2 norm and the L1 norm on the range of the analysis operator of the continuous frame." @default.
- W4308739957 created "2022-11-15" @default.
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- W4308739957 date "2023-03-01" @default.
- W4308739957 modified "2023-10-07" @default.
- W4308739957 title "Discretizing L norms and frame theory" @default.
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- W4308739957 doi "https://doi.org/10.1016/j.jmaa.2022.126846" @default.
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