Polynomial approximation of symmetric functions.

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. [ABSTRACT FROM AUTHOR]