利用纠错码的数值求积
Numerical cubature using error-correcting codes
AI总结：
本文利用线性纠错码构造等权重数值求积公式，在低阶时用扩展BCH码，获得多种空间上显式正权内部公式，渐近改进Tchakaloff界。
摘要：
我们提出了一种利用线性纠错码来改进具有等权重和卷积结构的数值求积公式（特别是等权重乘积公式）的构造方法。该构造在低阶且使用扩展BCH码时最为有效。利用它，我们获得了若干显式、正权、内部求积公式序列，对于每个固定阶数$t$，当维数$n \to \infty$时具有良好的渐近性质。利用区间上的一个特殊求积公式[arXiv:math.PR/0408360]，我们在$n$维立方体上获得了一个具有$O(n^{\floor{t/2}})$个点的等权重$t$-求积公式，该公式在常数因子内达到Stroud下界。当$t$为奇数时，我们还在$n$维球面、$n$维球体和高斯空间$\R^n$上获得了具有$O(n^{t-2})$个点的$t$-求积公式。当$\mu$为球对称且$t=5$时，我们获得$O(n^2)$个点。对于每个$t \ge 4$，我们还获得了$n$维单纯形上的显式、正权、内部公式，具有$O(n^{t-1})$个点；对于$t=3$，我们获得$O(n)$个点。这些构造在渐近意义上改进了非构造性的Tchakaloff界。最近，Victoir独立地发现了一些相关结果，他还指出基本构造更直接地使用了正交阵列。
英文摘要：
We present a construction for improving numerical cubature formulas with equal weights and a convolution structure, in particular equal-weight product formulas, using linear error-correcting codes. The construction is most effective in low degree with extended BCH codes. Using it, we obtain several sequences of explicit, positive, interior cubature formulas with good asymptotics for each fixed degree $t$ as the dimension $n \to \infty$. Using a special quadrature formula for the interval [arXiv:math.PR/0408360], we obtain an equal-weight $t$-cubature formula on the $n$-cube with $O(n^{\floor{t/2}})$ points, which is within a constant of the Stroud lower bound. We also obtain $t$-cubature formulas on the $n$-sphere, $n$-ball, and Gaussian $\R^n$ with $O(n^{t-2})$ points when $t$ is odd. When $μ$ is spherically symmetric and $t=5$, we obtain $O(n^2)$ points. For each $t \ge 4$, we also obtain explicit, positive, interior formulas for the $n$-simplex with $O(n^{t-1})$ points; for $t=3$, we obtain O(n) points. These constructions asymptotically improve the non-constructive Tchakaloff bound. Some related results were recently found independently by Victoir, who also noted that the basic construction more directly uses orthogonal arrays.
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