where $c_k=2^{2k+1}\frac{(k-1)!}{(2k)!}$. The proof then follows the same strategy as the proof for $\zeta(s)$ and we leave the details to the reader.

Numerically optimizing the bounds
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Going back to the sphere packing problem, since we obviously have $\Delta({\mathbb{R}}^1)=1$, this shows $r(f)\geq 1$ for all $f\in {\mathcal{A}}_{LP}$. The last sign change equals $1$ for two (suspiciously) well-known functions: the hat function $$H(x)=(1-|x|)_+,$$ whose Fourier transform is ${\widehat}H(x)=\frac{\sin^2(\pi x)}{(\pi x)^2}$, and Selberg’s function $$S(x)=\frac{\sin^2(\pi x)}{(\pi x)^2(1-x^2)},$$ whose Fourier transform is supported in $[-1,1]$ and given by ${\widehat}S(x)=1-|x|+\frac{\sin(2\pi x)}{2\pi}$ for $|x|<1$. In particular, we can use these two functions to evaluate the functionals derived in Section \[prelims\] to obtain bounds, but this does not result in the best possible bounds. To obtain better bounds we use the class of functions used in the linear programming bounds by Cohn and Elkies [@CE] for sphere packing. That is, we consider the subspace $\mathcal A_{LP}(d)$ consisting of the functions $f \in \mathcal A_{LP}$ of the form $$\label{eq:subspace} 
f(x) = p(x) e^{-\pi x^2},$$ where $p$ is an even polynomial of degree $2d$.

In [@CE], optimization over a closely related class of functions is done by specifying the functions by their real roots and optimizing the root 