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：我们研究对称多元函数与多重集函数的多项式逼近问题。具体而言，我们考虑$f(x_1, \dots, x_N)$，其中$x_i \in \mathbb{R}^d$，且函数$f$在其$N$个自变量的置换下保持不变。我们阐明如何利用这些对称性来优化函数$f$多项式逼近中的成本-误差权衡，并特别分析了该权衡对参数$d$、$N$及多项式阶数的依赖关系。基于这些结果，我们进一步构建逼近方法并证明当$N$成为输入参数时，定义在多重集上的函数的逼近阶。