关于特殊类∞变量函数的平均和最坏情况下的截断问题
Truncation in Average and Worst Case Settings for Special Classes of $\infty$-Variate Functions
AI总结：
本文研究了特殊类∞变量函数的截断误差，提供了在平均和最坏情况下的精确界，适用于希尔伯特空间和霍尔德条件下的函数。
摘要：
本文考虑了形式为f(x₁,x₂,…)=g(∑_{j=1}^∞x_jξ_j)的函数的截断误差，即通过f_k(x₁,…,x_k)=g(∑_{j=1}^kx_jξ_j)近似f的误差，其中ξ_j收敛到零足够快，且x_j是独立同分布的随机变量。文中指出，此类函数出现在许多重要应用中。为获得可能对大量此类函数的正结果，本文在平均和最坏情况下提供了截断误差的精确界。在前者情况下，函数g来自一个带有给定协方差核的希尔伯特空间G。在后者情况下，函数g来自再生核希尔伯特空间或满足霍尔德条件的函数空间。
英文摘要：
The paper considers truncation errors for functions of the form $f(x_1,x_2,\dots)=g(\sum_{j=1}^\infty x_j\,ξ_j)$, i.e., errors of approximating $f$ by $f_k(x_1,\dots,x_k)=g(\sum_{j=1}^k x_j\,ξ_j)$, where the numbers $ξ_j$ converge to zero sufficiently fast and $x_j$'s are i.i.d. random variables. As explained in the introduction, functions $f$ of the form above appear in a number of important applications. To have positive results for possibly large classes of such functions, the paper provides sharp bounds on truncation errors in both the average and worst case settings. In the former case, the functions $g$ are from a Hilbert space $G$ endowed with a zero mean probability measure with a given covariance kernel. In the latter case, the functions $g$ are from a reproducing kernel Hilbert space, or a space of functions satisfying a Hölder condition.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
