Sampling Discretization of Integral Norms

Feng, Dagan; Shadrin, Alexei; Temlyakov, Vladimir; Tikhonov, Sergey

doi:10.1007/s00365-021-09539-0

Cited by 20 publications

(4 citation statements)

References 36 publications

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“…Because of this, modern papers on the subject often refer to this class of problems as Marcinkiewics-type discretization problems. It has been recently proven that for all 1 ≤ p < ∞ there are certain entropy conditions on X N ⊆ L p (Ω) which guarantee that the L p -norm on X N can be discretized to satisfy (1.2) using M on the order of N(log(N)) 2 for 1 ≤ p ≤ 2 and N(log(N)) p for p ≥ 2 sampling points [DPTT19], [DPSTT21], [K21], [T18]. These entropy conditions can be fairly technical, but they imply in particular that there exists β > 0 such that (1.3)…”

Section: Introductionmentioning

confidence: 99%

Discretizing $L_p$ norms and frame theory

Freeman¹,

Ghoreishi²

2021

Preprint

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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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Section: Introductionmentioning

confidence: 99%

Discretizing $L_p$ norms and frame theory

Freeman¹,

Ghoreishi²

2021

Preprint

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“…However, it does not lead us to better sample size estimation. In this paper, we apply the idea of Marcinkiewicz type discretization result introduced in [10] to solve random sampling problem in localizable reproducing kernel space. We show that if a sampling set is "good" discretization to the integral norm on Ω for the class of simple functions, then it is a stable sampling set for δ-concentrated functions on Ω.…”

Section: Introductionmentioning

confidence: 99%

“…We pursue the approach of [10,33] with a mild condition on generators. Note that the result in [33] based on strong decay condition of generators…”

Section: Introductionmentioning

confidence: 99%

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Random sampling of signals concentrated on compact set in localized reproducing kernel subspace of $L^p({\mathbb R}^n)$

Patel,

Sivananthan

2021

Preprint

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The paper is devoted to studying the stability of random sampling in a localized reproducing kernel space. We show that if the sampling set on Ω (compact) discretizes the integral norm of simple functions up to a given error, then the sampling set is stable for the set of functions concentrated on Ω. Moreover, we prove with an overwhelming probability that O(µ(Ω)(log µ(Ω)) 3 ) many random points uniformly distributed over Ω yield a stable set of sampling for functions concentrated on Ω.

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