### 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.

Original language | English |
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Pages (from-to) | 431-441 |

Number of pages | 11 |

Journal | Kyushu Journal of Mathematics |

Volume | 61 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 1 2007 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Kyushu Journal of Mathematics*,

*61*(2), 431-441. https://doi.org/10.2206/kyushujm.61.431

**High-discrepancy sequences.** / Tezuka, Shu.

Research output: Contribution to journal › Article

*Kyushu Journal of Mathematics*, vol. 61, no. 2, pp. 431-441. https://doi.org/10.2206/kyushujm.61.431

}

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

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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 -