Motivated by numerical methods for solving parametric partial differential equations, this paper studies the approximation of multivariate analytic functions by algebraic polynomials. We introduce various anisotropic model classes based on Taylor expansions, and study their approximation by finite dimensional polynomial spaces $\cal{P}_{\Lambda}$ described by lower sets $\Lambda$. Given a budget $n$ for the dimension of $\cal{P}_{\Lambda}$, we prove that certain lower sets $\Lambda_n$, with cardinality $n$, provide a certifiable approximation error that is in a certain sense optimal, and that these lower sets have a simple definition in terms of simplices. Our main goal is to obtain approximation results when the number of variables $d$ is large and even infinite, and so we concentrate almost exclusively on the case $d=\infty$. We also emphasize obtaining results which hold for the full range $n\ge 1$, rather than asymptotic results that only hold for $n$ sufficiently large. In applications, one typically wants $n$ small to comply with computational budgets.

中文翻译：

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多变量各向异性解析函数的多项式逼近

受求解参数偏微分方程的数值方法的启发，本文研究了代数多项式对多元解析函数的逼近。我们介绍了基于泰勒展开的各种各向异性模型类，并通过由较低集 $\Lambda$ 描述的有限维多项式空间 $\cal{P}_{\Lambda}$ 研究它们的近似。给定 $\cal{P}_{\Lambda}$ 维度的预算 $n$，我们证明某些较低的集合 $\Lambda_n$，基数为 $n$，提供了一个可证明的近似误差感觉是最优的，并且这些较低的集合在单纯形方面有一个简单的定义。我们的主要目标是在变量 $d$ 的数量很大甚至无限时获得近似结果，因此我们几乎只专注于 $d=\infty$ 的情况。我们还强调获得适用于全范围 $n\ge 1$ 的结果，而不是仅适用于足够大的 $n$ 的渐近结果。在应用程序中，人们通常希望 $n$ 小以符合计算预算。