Elsevier

Journal of Complexity

Volume 67, December 2021, 101573
Journal of Complexity

A note on EC-tractability of multivariate approximation in weighted Korobov spaces for the standard information class,☆☆

https://doi.org/10.1016/j.jco.2021.101573Get rights and content

Abstract

In this paper we study exponential tractability of multivariate approximation problem for weighted Korobov spaces in the worst case setting. The considered algorithms are constructive and use the class Λstd of only function evaluations. We give matching necessary and sufficient conditions for notions of EC-quasi-polynomial tractability and EC-uniform weak tractability in terms of two weight parameters of the problem.

Section snippets

Introduction and main results

Recently, research in information-based complexity theory has focused on tractability of multivariate continuous problems. Studies on this area started in 1994 (see [18], [19]) and since then a huge number of results emerged on this topic. Nowadays tractability of multivariate problems is an active and popular research field (see [9], [10], [11] and the references therein). It has been studied in different settings including the worst case, the probabilistic case and the average case setting,

Preliminaries

For a fixed ω(0,1), let Kd,a,b be the analytic Korobov kernel given by (1.5) with a,b satisfying (1.3). By (1.4) we know that the reproducing kernel Hilbert space H(Kd,a,b) is a tensor product of the univariate reproducing kernel Hilbert spaces H(K1,aj,bj),j=1,,d with reproducing kernels K1,aj,bj, i.e.,H(Kd,a,b)=H(K1,a1,b1)H(K1,a2,b2)H(K1,ad,bd).

Proof of part 1

For the notion of EC-QPT, observe that for dN and ε(0,1),exp(t(1+lnd)(1+ln(1+lnε1)))=(ed)t(1+ln(1+lnε1))=(e(1+lnε1))t(1+lnd), where e=exp(1). In the following proof we will sometimes use these equivalent formulations to establish EC-QPT.

Proof

We first prove the “only if” part. Assume that APP is EC-QPT for the class Λstd. Then by (1.8) we know that APP is also EC-QPT for the class Λall and this implies (1.9). Furthermore, it follows from the proof of Theorem 8 in [4] that the exponent t

Acknowledgements

The author would like to thank F. Pillichshammer and two anonymous referees for helpful suggestions and comments regarding an improved presentation of the results.

Cited by (4)

Supported by the National Natural Science Foundation of China (Grant No. 11871006) and Doctoral Foundation Project of Shandong Jianzhu University (X19090Z).

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Communicated by F. Pillichshammer.

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