Polynomial Approximation of Anisotropic Analytic Functions of Several Variables

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.