Can Weierstrass's theorem be specialized to symmetric functions and symmetric polynomials?

Question

Weierstrass's theorem says that continuous functions can be uniformly approximated by polynomials.

Can one have a similar theorem saying that symmetric functions can be uniformly approximated by symmetric polynomials?

Answer 1

The answer is YES. If $p$ approximates a symmetric continuous function of $f$ then so does $q$ where $q(x_1,x_2,..,x_n)=\frac 1 {n!} \sum_{\sigma} p(x_{\sigma_1},x_{\sigma_2},...,x_{\sigma_n})$ where the sum is over all permutations of $\{1,2...,n\}$.