We show that generalized multiquadric radial basis functions (RBFs) on $\mathbb{R}^d$ have a mean dimension that is $1+O(1/d)$ as $d\to\infty$ with an explicit bound for the implied constant, under moment conditions on their inputs. Under weaker moment conditions the mean dimension still approaches $1$. As a consequence, these RBFs become essentially additive as their dimension increases. Gaussian RBFs by contrast can attain any mean dimension between 1 and d. We also find that a test integrand due to Keister that has been influential in quasi-Monte Carlo theory has a mean dimension that oscillates between approximately 1 and approximately 2 as the nominal dimension $d$ increases.

翻译：我们证明，在输入满足矩条件的情况下，定义在$\mathbb{R}^d$上的广义多二次径向基函数（RBFs）的平均维数当$d\to\infty$时为$1+O(1/d)$，并给出了隐含常数的显式界。在更弱的矩条件下，平均维数仍趋近于$1$。因此，这些RBFs随维数增加本质上趋近于可加函数。相比之下，高斯RBFs的平均维数可以取$1$到$d$之间的任意值。我们还发现，由Keister提出的一个在准蒙特卡洛理论中具有重要影响的测试被积函数，其平均维数随名义维数$d$增加而在约$1$和约$2$之间振荡。