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.