Polynomial approximation of symmetric functions

We study the polynomial approximation of symmetric multivariate functions and multi-set functions. Specifically, we consider f ( x 1 , … N stretchy="false">) f(x_1, \dots , x_N) , where alttext="x i element-of double-struck R Superscript d"> i ∈<!-- ∈ <mml:msup> R d encoding="application/x-tex">x_i \in \mathbb {R}^d alttext="f"> encoding="application/x-tex">f is invariant under permutations its alttext="upper N"> encoding="application/x-tex">N arguments. demonstrate how these symmetries can be exploited to improve cost versus error ratio in a function particular dependence that on alttext="d encoding="application/x-tex">d, N degree. These results are then used construct approximations prove rates for defined multi-sets becomes parameter input.