Discretizing L_p norms and frame theory
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Given an N-dimensional subspace X of L_p([0,1]), we consider the problem of choosing M-sampling points which may be used to discretely approximate the L_p 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 X 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 L_p([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 L_p 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 L_2 norm and the L_1 norm on the range of the analysis operator of the continuous frame.
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