### 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 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Kyushu Journal of Mathematics*,

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