High-discrepancy sequences

Shu Tezuka

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


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 languageEnglish
Pages (from-to)431-441
Number of pages11
JournalKyushu Journal of Mathematics
Issue number2
Publication statusPublished - 2007

All Science Journal Classification (ASJC) codes

  • Mathematics(all)


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