# Approximate a positive multivariate function with a sum of squares of polynomials?

I am constructing approximation to a multivariate function which I know is positive. My question is the following:

Let $f(x)$ be a multivariate positive and continuous function. Can we approximate it arbitrarily well with a sum of squares of multivariate polynomials?

I think it is true because if we let $g(x)$ be a sufficiently good positive polynomial approximation to $f$, due to Artin's result, $g$ can be written as a sum of squares of rational functions and then I approximate each rational function with polynomials, which gives me a sufficiently good approximation in the desired form (sum of squares of polynomials) to the original function $f$.

I am wondering if this makes sense?

I am very unfamiliar with this field, and I apologize if the question is trivial to experts or has been answered. Thanks in advance.

• The monomials are linearly independent so each new monomial term should contribute something. Although how well it contributes probably depends heavily on what function you want to approximage. – mathreadler Mar 17 '15 at 9:32