Remez-Type and Nikol’skii-Type Inequalities: General Relations and the Hyperbolic Cross Polynomials

Temlyakov, Vladimir; Tikhonov, Sergey

doi:10.1007/s00365-017-9370-x

Cited by 14 publications

(12 citation statements)

References 23 publications

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“…We refer to [29] for further information, including results in higher dimensions, as well as to [47] for results on multivariate Remez inequalities for hyperbolic cross index sets.…”

Section: Accuracy and Conditioningmentioning

confidence: 99%

Approximating Smooth, Multivariate Functions on Irregular Domains

Adcock

Huybrechs

2020

Forum of Mathematics, Sigma

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In this paper, we introduce a method known as polynomial frame approximation for approximating smooth, multivariate functions defined on irregular domains in $d$ dimensions, where $d$ can be arbitrary. This method is simple, and relies only on orthogonal polynomials on a bounding tensor-product domain. In particular, the domain of the function need not be known in advance. When restricted to a subdomain, an orthonormal basis is no longer a basis, but a frame. Numerical computations with frames present potential difficulties, due to the near-linear dependence of the truncated approximation system. Nevertheless, well-conditioned approximations can be obtained via regularization, for instance, truncated singular value decompositions. We comprehensively analyze such approximations in this paper, providing error estimates for functions with both classical and mixed Sobolev regularity, with the latter being particularly suitable for higher-dimensional problems. We also analyze the sample complexity of the approximation for sample points chosen randomly according to a probability measure, providing estimates in terms of the corresponding Nikolskii inequality for the domain. In particular, we show that the sample complexity for points drawn from the uniform measure is quadratic (up to a log factor) in the dimension of the polynomial space, independently of $d$ , for a large class of nontrivial domains. This extends a well-known result for polynomial approximation in hypercubes.

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“…We refer to [29] for further information, including results in higher dimensions, as well as to [47] for results on multivariate Remez inequalities for hyperbolic cross index sets.…”

Section: Accuracy and Conditioningmentioning

confidence: 99%

Approximating Smooth, Multivariate Functions on Irregular Domains

Adcock

Huybrechs

2020

Forum of Mathematics, Sigma

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“…We now discuss some results from [20] about relation between discretization and the Remez-type inequality. (5.4)…”

Section: Discussionmentioning

confidence: 99%

Sampling discretization of the uniform norm

Kashin¹,

Konyagin²,

Temlyakov³

2021

Preprint

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Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. We pay special attention to the case of trigonometric polynomials with frequencies from an arbitrary finite set with fixed cardinality. We give two different proofs of the fact that for any N -dimensional subspace of the space of continuous functions it is sufficient to use e CN sample points for an accurate upper bound for the uniform norm. Previous known results show that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. Also, we prove a general result, which connects the upper bound on the number of sampling points in the discretization theorem for the uniform norm with the best m-term bilinear approximation of the Dirichlet kernel associated with the given subspace. We illustrate application of our technique on the example of trigonometric polynomials.

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“…has been recently initiated in [43]. It turns out that for such polynomials the problem to obtain the optimal Remez inequalities has different solutions when p < ∞ and p = ∞.…”

Section: Some Historical Remarks and An Application To Remez Inequalimentioning

confidence: 99%

“…It is worth mentioning that this result is sharp with respect to the logarithmic factor. This is because the following statement is false (see [43]).…”

Section: Some Historical Remarks and An Application To Remez Inequalimentioning

confidence: 99%

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Integral norm discretization and related problems

Feng¹,

Prymak²,

Temlyakov³

et al. 2019

Russ. Math. Surv.

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The problem of replacing an integral norm with respect to a given probability measure by the corresponding integral norm with respect to a discrete measure is discussed in the paper. The above problem is studied for elements of finite dimensional spaces. Also, discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. We pay special attention to the case of the multivariate trigonometric polynomials with frequencies from a finite set with fixed cardinality. Both new results and a survey of known results are presented.

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Remez-Type and Nikol’skii-Type Inequalities: General Relations and the Hyperbolic Cross Polynomials

Abstract: In this paper we study the Remez-type inequalities for trigonometric polynomials with harmonics from hyperbolic crosses. The interrelation between the Remez and Nikolskii inequalities for individual functions and its applications are discussed.

Cited by 14 publications

References 23 publications

Approximating Smooth, Multivariate Functions on Irregular Domains

Approximating Smooth, Multivariate Functions on Irregular Domains

Sampling discretization of the uniform norm

Integral norm discretization and related problems

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