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Quasi-Monte Carlo tractability of integration problem in function spaces defined over products of balls

Jie Zhang

http://orcid.org/0000-0002-7954-8954

Department of Mathematics, Beijing Normal University, Beijing 100875, P. R. China

E-mail Address: zhangjie91528@163.com

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and

Yongping Liu

Department of Mathematics, Beijing Normal University, Beijing 100875, P. R. China

E-mail Address: ypliu@bnu.edu.cn

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https://doi.org/10.1142/S0219691319500437Cited by:3 (Source: Crossref)

Abstract

Recently, quasi-Monte Carlo (QMC) rules for multivariate integration in some weighted Sobolev spaces of functions defined over unit cubes ${[0, 1]}^{m}$ , products of $m$ copies of the simplex $𝕋^{d} \subset ℝ^{d}$ and products of $m$ copies of the unit sphere $𝕊^{d} \subset ℝ^{d + 1}$ have been well-studied in the literature. In this paper, we consider QMC tractability of integrals of functions defined over product of $m$ copies of the ball $𝔹^{d} \subset ℝ^{d}$ . The space is a tensor product of $m$ reproducing kernel Hilbert spaces defined by positive and uniformly bounded weights $γ_{m, j}$ for $j = 1, 2, \dots, m$ . We obtain matching necessary and sufficient conditions in terms of this weights for various notions of QMC tractability, including strong polynomial tractability, polynomial tractability, quasi-polynomial tractability, uniformly weak tractability and $(s, t)$ -weak tractability.

Keywords:

AMSC: 41A63, 65Y20, 68Q25

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