intriguing and so far we have no explanation), but for fixed $d$ we need to optimize $R$ to obtain a good bound.

A polynomial $p$ that is nonnegative on $[R,\infty)$ can be written as $s_1(x) + (x-R) s_2(x)$, where $s_1$ and $s_2$ are sum-of-squares polynomials with $\mathrm{deg}(s_1),\mathrm{deg}(s_2(x))+1 \leq \deg(p)$; see, e.g., [@PS]. This shows that functions of the form that are non-positive on $[R,\infty)$ can be written as $$f(x) = -\big(s_1(x^2) + (x^2-R^2) s_2(x^2)\big) e^{-\pi x^2}.$$

Let $v(x)$ be a vector whose entries form a basis of the univariate polynomials of degree at most $d$. The polynomials $s_1$ and $s_2$ are sum-of-squares if and only if they can be written as $s_i(x) = v(x)^{\sf T} X_i v(x)$ for some positive semidefinite matrices $X_i$ of size $d+1$. That is, we can parameterize functions of the form that are non-positive on $[R,\infty)$ by two positive semidefinite matrices $X_1$ and $X_2$ of size $d+1$.

The space of functions of the form is invariant under the Fourier transform. Since a polynomial of degree $2d$ that is nonnegative on $[0,\infty)$ can be written as $s_3(x) + x s_4(x)$, where $s_i(x) = v(x)^{\sf T} X_i v(x)$ for $i=3,4$ are sum-of-squares polynomials of degree $2d$, we have 