Let $f$ be a positive integrable function, and $q$ a polynomial. we assume that for every polynomial $p$ : $$ \int_{\mathbb{R}} p(t) f(t) dt <\infty $$ Knowing only that for all polynomial $p\neq 0$ $$ \int_{\mathbb{R}} p^2(t) q(t) f(t) dt > 0. $$ How to prouve that $q$ is a non negative polynomial. And what can we say if we assume the equality for a subspace of polynomial? (for example for polynomials up to some finite degree)
Edit : If $f$ is with compact support, using Weierstrass approximation theorem we get that, for all non negative continuous function $h$ : $$ \int_{\mathbb{R}} h(t) q(t) f(t) dt > 0. $$ So we consider $K$ the compact subset of the support where $q$ is negative then by Uryson Lemma for all $\epsilon$ we can find a non negative function $h$ such that $h(x) =1$ on $K$ and vanishing outside $K+\epsilon$ and less than one on the third part. So we have $$ \int_{\mathbb{R}} h(t) q(t) f(t) dt \leq \int_{K} q(t) f(t) dt + 2M\epsilon \int_{\mathbb{R}} f(t) dt $$ Which is negative if we assume that $q$ change the sign for $\epsilon$ small enough.