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| 概要 |
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 functi...ons 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 $ (infty, 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.続きを見る
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Mendeley出力
