Approximation of functions with small mixed smoothness in the uniform norm

Temlyakov, Vladimir; Ullrich, Tino

doi:10.1016/j.jat.2022.105718

Cited by 13 publications

(9 citation statements)

References 44 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%

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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%

“…Although this bound is sometimes weaker than Theorem 3 (see Example 1 in [15]), it has the great advantage that it may also be applied in situations where the Kolmogorov widths in L 2 are not square summable, see, e.g., [35,36]. It would be very interesting to see whether it is possible to unify the two approaches.…”

Section: Remark 5 (Equivalent Widths)mentioning

confidence: 99%

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

Dolbeault,

Krieg,

Ullrich

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…In the recent papers by V.N. Temlyakov and T. Ullrich [36,37], a behavior of some asymptotic characteristics of multivariate functions classes in the uniform norm was studied in the case of "small smoothness" of functions. The crucial point here is the fact, that in a small smoothness setting the corresponding approximation numbers are not square summable.…”

Section: Sampling and Kolmogorov Numbersmentioning

confidence: 99%

“…The crucial point here is the fact, that in a small smoothness setting the corresponding approximation numbers are not square summable. The established estimates for Kolmogorov widths of the Sobolev classes of functions and classes of functions with bounded mixed differences serve as a powerful tool to investigate sampling recovery problem in L 2 and L ∞ (see also Section 7 in [36] and Section 5 in [37]).…”

Section: Sampling and Kolmogorov Numbersmentioning

confidence: 99%

A note on sampling recovery of multivariate functions in the uniform norm

Pozharska¹,

Ullrich²

2021

Preprint

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We study the recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm. Our main interest is to obtain preasymptotic estimates for the corresponding sampling numbers. We obtain results in terms of the decay of related singular numbers of the compact embedding into L 2 (D, ̺ D ) multiplied with the supremum of the Christoffel function of the subspace spanned by the first m singular functions. Here the measure ̺ D is at our disposal. As an application we obtain near optimal upper bounds for the sampling numbers for periodic Sobolev type spaces with general smoothness weight. Those can be bounded in terms of the corresponding benchmark approximation number in the uniform norm, which allows for preasymptotic bounds. By applying a recently introduced sub-sampling technique related to Weaver's conjecture we mostly lose a √ log n and sometimes even less. Finally we point out a relation to the corresponding Kolmogorov numbers.

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“…Temlyakov [27]. Note, that the right-hand side in (1.7) is of particular importance if the linear widths in L 2 are not square-summable [29,28].…”

Section: Introductionmentioning

confidence: 99%

Constructive subsampling of finite frames with applications in optimal function recovery

Bartel¹,

Schäfer²,

Ullrich³

2022

Preprint

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In this paper we present new constructive methods, random and deterministic, for the efficient subsampling of finite frames in C m . Based on a suitable random subsampling strategy, we are able to extract from any given frame with bounds 0 < A ≤ B < ∞ (and condition B/A) a similarly conditioned reweighted subframe consisting of merely O(m log m) elements. Further, utilizing a deterministic subsampling method based on principles developed in [1, Sec. 3], we are able to reduce the number of elements to O(m) (with a constant close to one). By controlling the weights via a preconditioning step, we can, in addition, preserve the lower frame bound in the unweighted case. This allows to derive new quasi-optimal unweighted (left) Marcinkiewicz-Zygmund inequalities for L 2 (D, ν) with constructible node sets of size O(m) for mdimensional subspaces of bounded functions. Those can be applied e.g. for (plain) least-squares sampling reconstruction of functions, where we obtain new quasi-optimal results avoiding the Kadison-Singer theorem. Numerical experiments indicate the applicability of our results.

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Approximation of functions with small mixed smoothness in the uniform norm

Cited by 13 publications

References 44 publications

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

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

A note on sampling recovery of multivariate functions in the uniform norm

Constructive subsampling of finite frames with applications in optimal function recovery

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