We study the polynomial approximation of symmetric multivariate functions and of multi-set functions. Specifically, we consider $f(x_1, \dots, x_N)$, where $x_i \in \mathbb{R}^d$, and $f$ is invariant under permutations of its $N$ arguments. We demonstrate how these symmetries can be exploited to improve the cost versus error ratio in a polynomial approximation of the function $f$, and in particular study the dependence of that ratio on $d, N$ and the polynomial degree. These results are then used to construct approximations and prove approximation rates for functions defined on multi-sets where $N$ becomes a parameter of the input.

翻译：我们研究对称多变量函数和多元函数的多元近似值。 具体地说, 我们考虑$f( x_ 1,\ dots, x_N), $x_ i\ in\ mathbb{R ⁇ d$ 和$f$ 在 $N 参数的变换下是无差异的。 我们演示如何利用这些对称来提高函数多元函数多元近值的成本比和误差率, 特别是研究该比值对 $d, N$和多元学位的依赖性。 这些结果被用来构建近似值, 并证明多元函数定义的近似率, 在多元函数中, $成为输入的参数 。