# High-discrepancy sequences

Shu Tezuka

2 引用 (Scopus)

### 抄録

First, it is pointed out that the uniform distribution of points in [0, 1]d is not always a necessary condition for every function in a proper subset of the class of all Riemann integrable functions to have the arithmetic mean of function values at the points converging to its integral over [0, 1]d as the number of points goes to infinity. We introduce a formal definition of the d-dimensional high-discrepancy sequences, which are not uniformly distributed in [0, 1]d, and present motivation for the application of these sequences to high-dimensional numerical integration. Then, we prove that there exist non-uniform (∞, d)-sequences which provide the convergence rate O(N-1) for the integration of a certain class of d-dimensional Walsh function series, where N is the number of points.

元の言語 英語 431-441 11 Kyushu Journal of Mathematics 61 2 https://doi.org/10.2206/kyushujm.61.431 出版済み - 1 1 2007

### Fingerprint

Discrepancy
High-dimensional
Walsh Functions
Proper subset
Uniform distribution
Value Function
Numerical integration
Convergence Rate
Infinity
Necessary Conditions
Series
Class

### All Science Journal Classification (ASJC) codes

• Mathematics(all)

### これを引用

High-discrepancy sequences. / Tezuka, Shu.

：: Kyushu Journal of Mathematics, 巻 61, 番号 2, 01.01.2007, p. 431-441.

Tezuka, Shu. / High-discrepancy sequences. ：: Kyushu Journal of Mathematics. 2007 ; 巻 61, 番号 2. pp. 431-441.
@article{56b9a1cef85b4791b2dcd5160d782826,
title = "High-discrepancy sequences",
abstract = "First, it is pointed out that the uniform distribution of points in [0, 1]d is not always a necessary condition for every function in a proper subset of the class of all Riemann integrable functions to have the arithmetic mean of function values at the points converging to its integral over [0, 1]d as the number of points goes to infinity. We introduce a formal definition of the d-dimensional high-discrepancy sequences, which are not uniformly distributed in [0, 1]d, and present motivation for the application of these sequences to high-dimensional numerical integration. Then, we prove that there exist non-uniform (∞, d)-sequences which provide the convergence rate O(N-1) for the integration of a certain class of d-dimensional Walsh function series, where N is the number of points.",
author = "Shu Tezuka",
year = "2007",
month = "1",
day = "1",
doi = "10.2206/kyushujm.61.431",
language = "English",
volume = "61",
pages = "431--441",
journal = "Kyushu Journal of Mathematics",
issn = "1340-6116",
publisher = "Kyushu University, Faculty of Science",
number = "2",

}

TY - JOUR

T1 - High-discrepancy sequences

AU - Tezuka, Shu

PY - 2007/1/1

Y1 - 2007/1/1

N2 - First, it is pointed out that the uniform distribution of points in [0, 1]d is not always a necessary condition for every function in a proper subset of the class of all Riemann integrable functions to have the arithmetic mean of function values at the points converging to its integral over [0, 1]d as the number of points goes to infinity. We introduce a formal definition of the d-dimensional high-discrepancy sequences, which are not uniformly distributed in [0, 1]d, and present motivation for the application of these sequences to high-dimensional numerical integration. Then, we prove that there exist non-uniform (∞, d)-sequences which provide the convergence rate O(N-1) for the integration of a certain class of d-dimensional Walsh function series, where N is the number of points.

AB - First, it is pointed out that the uniform distribution of points in [0, 1]d is not always a necessary condition for every function in a proper subset of the class of all Riemann integrable functions to have the arithmetic mean of function values at the points converging to its integral over [0, 1]d as the number of points goes to infinity. We introduce a formal definition of the d-dimensional high-discrepancy sequences, which are not uniformly distributed in [0, 1]d, and present motivation for the application of these sequences to high-dimensional numerical integration. Then, we prove that there exist non-uniform (∞, d)-sequences which provide the convergence rate O(N-1) for the integration of a certain class of d-dimensional Walsh function series, where N is the number of points.

UR - http://www.scopus.com/inward/record.url?scp=43049102421&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=43049102421&partnerID=8YFLogxK

U2 - 10.2206/kyushujm.61.431

DO - 10.2206/kyushujm.61.431

M3 - Article

AN - SCOPUS:43049102421

VL - 61

SP - 431

EP - 441

JO - Kyushu Journal of Mathematics

JF - Kyushu Journal of Mathematics

SN - 1340-6116

IS - 2

ER -