basis $\{L_n^{-1/2}(2\pi x^2)\}$, as this seems natural and performs well in practice (it multiplied by $e^{-\pi x^2}$ is the complete set of even eigenfunctions of the Fourier transform). We solve the semidefinite programs using sdpa-gmp [@Nakata], which is a primal-dual interior point solver using high precision floating point arithmetic. For the code to generate the semidefinite programs and to perform the post processing we use Julia [@BEKV], Nemo [@FHHJ], and Arb [@J] (where we use Arb for the ball arithmetic used in the verification procedure). For all computations we use $d=40$. In solving the systems we observe that $X_1$ can be set to zero everywhere without affecting the bounds, so that $r(f) = R$ holds exactly for the function $f(x) = (R^2 - x^2) v(x^2)^{\sf T} X_2 v(x^2) e^{-\pi x^2}$ defined by $X_2$.

The above optimization approach uses floating point arithmetic and a numerical interior point solver. This means the identity $I(0, X_2, X_3, X_4) = 0$ will not be satisfied exactly, and, moreover, because the solver can take infeasible steps the matrices $X_2$, $X_3$, and $X_4$ typically have some eigenvalues that are slightly negative. In practice this leads to incorrect upper bounds if the floating point precision is not 