Discretizing $L_p$ norms and frame theory

Abstract

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.