Sampling discretization and related problems

Kashin, Boris Sergeevich; Kosov, Egor D.; Лимонова, Ирина Викторовна; Temlyakov, Vladimir

doi:10.1016/j.jco.2022.101653

Cited by 30 publications

(27 citation statements)

References 141 publications

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“…The quest for a systematic comparison has attracted much attention recently, see [3,4,7,8,10,11,12,13,14,15,18,22,23,33,35,36,37], which all appeared in the past five years. It is often expressed in terms of sampling numbers and Kolmogorov (or approximation) numbers, as we summarize below.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A sharp upper bound for sampling numbers in $L_{2}$

Dolbeault,

Krieg,

Ullrich

2022

Preprint

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For a class F of complex-valued functions on a set D, we denote by g n (F ) its sampling numbers, i.e., the minimal worstcase error on F , measured in L 2 , that can be achieved with a recovery algorithm based on n function evaluations. We prove that there is a universal constant c > 0 such that, if F is the unit ball of a separable reproducing kernel Hilbert space, thenwhere d k (F ) are the Kolmogorov widths (or approximation numbers) of F in L 2 . We also obtain similar upper bounds for more general classes F , including all compact subsets of the space of continuous functions on a bounded domain D ⊂ R d , and show that these bounds are sharp by providing examples where the converse inequality holds up to a constant.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

A sharp upper bound for sampling numbers in $L_{2}$

Dolbeault,

Krieg,

Ullrich

2022

Preprint

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“…There are two types of conditions used in the literature: (I) Conditions on the entropy numbers and (II) Conditions in terms of the Nikol'skii-type inequalities. The reader can find a detailed discussion of known results in the very recent survey [14]. In this paper we only prove some discretization results under conditions (II).…”

Section: Introductionmentioning

confidence: 91%

“…Comment 1.2. Historical discussions on sampling discretization for the cases p = 2 and 1 ≤ p < 2 can be found in Subsections D.15 and D.16 of [14] respectively. Here we only mention that the best previously known results are the following.…”

Section: Introductionmentioning

confidence: 99%

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Sampling Discretization of Integral Norms

et al. 2021

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The paper addresses a problem of sampling discretization of integral norms of elements of finite-dimensional subspaces satisfying some conditions. We prove sampling discretization results under two standard kinds of assumptions -conditions on the entropy numbers and conditions in terms of the Nikol'skii-type inequalities. We prove some upper bounds on the number of sample points sufficient for good discretization and show that these upper bounds are sharp in a certain sense. Then we apply our general conditional results to subspaces with special structure, namely, subspaces with the tensor product structure. We demonstrate that applications of results based on the Nikol'skiitype inequalities provide somewhat better results than applications of results based on the entropy numbers conditions. Finally, we apply discretization results to the problem of sampling recovery.

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“…In the recent advances in complexity theory and data analysis Marcinkiewicz-Zygmund families are implicit in the discretization of norms. For a nice survey see [15].…”

Section: Introductionmentioning

confidence: 99%

Marcinkiewicz-Zygmund inequalities for polynomials in Fock space

Gröchenig

Ortega-Cerdà

2022

Math. Z.

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We study the relation between Marcinkiewicz-Zygmund families for polynomials in a weighted $$L^2$$ L 2 -space and sampling theorems for entire functions in the Fock space and the dual relation between uniform interpolating families for polynomials and interpolating sequences. As a consequence we obtain a description of signal subspaces spanned by Hermite functions by means of Gabor frames.

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Sampling discretization and related problems

Cited by 30 publications

References 141 publications

A sharp upper bound for sampling numbers in $L_{2}$

A sharp upper bound for sampling numbers in $L_{2}$

Sampling Discretization of Integral Norms

Marcinkiewicz-Zygmund inequalities for polynomials in Fock space

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