follows that it is possible to modify the corresponding entries in $X_3$ and $X_4$ such that these matrices stay positive definite and such that $I(0,X_2,X_3,X_4) = 0$ holds exactly [@Lo]. This shows that the Fourier transform of the function $f(x) = (R^2 - x^2) v(x^2)^{\sf T} X_2 v(x^2)e^{-\pi x^2}$ is nonnegative.

We use interval arithmetic to compute $f(0) = R^2 s_2(0)$, $\mathcal T((R^2-x^2) s_2(x^2))(0)$, and ${\mathcal{Z}}(f)$, $\tilde {\mathcal{Z}}(f)$, ${\mathcal{Z}}_1(f)$, or ${\mathcal{L}}(f)$. We can then compute rigorous bounds by observing that, for example, the first part of Theorem \[thmNstar\] can be written as follows: Suppose $f$ is a continuous $L^1({\mathbb{R}})$ function with $f(x) \leq 0$ for $|x| \geq R$ and with nonnegative Fourier transform, then $$N^*(T) \leq \left(\frac{f(0)}{\hat f(0)} {\mathcal{Z}}(f) + o(1)\right) N(T).$$

In the arXiv version of this paper we attach the files ‘Z-$40$.txt’, ‘tildeZ-$40$.txt’, ‘L-$40$.txt’, and ‘Z1-$40$.txt’ that contain the value of $R$ on the first line and the matrices $X_2, X_3$ and $X_4$ on the next $3$ lines (all in $100$ decimal floating point values). For convenience it also contains the coefficients of $f$ in the monomial basis on the last line (but these are not used in the verification procedure). We include a script to perform the 