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.

  • $\begingroup$ 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. $\endgroup$ – mathreadler Mar 17 '15 at 9:32

I upvoted your question, since, following your line of reasoning, it brought me to ask whether the summands (squares of rational functions) can be guaranteed to be continuous, (since otherwise one is not allowed to use Weierstrass approximation theorem to approximate the rational functions on compact sets). However the answer to your original question is simple: take the square root of your function; it is continuous and positive as well. On any compact set this square root can be approximated by a polynomial P; P^2 is an approximation to your original function (on the same compact set).

As for approximation on all of R^n, for a counterexample it suffices to take any bounded nonconstant continuous positive function.


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