Throughout this paper, $\lesssim$ and $\gtrsim$ shall indicate inequalities that hold up to a uniform positive factor. When the factor involved depends on some additional parameter or function, this will be indicated with subscripts such as $\gtrsim_\mu$.

More precise asymptotics for $K(x)$ as $|x|\to \infty$ is given in Corollary \[cor:improved asymptotics\] below.

Since $K$ is even it suffices to consider $x\ge 1$. Note that the integral $$\int_{\mathbb R}\exp(ix\xi) m(\xi)\, {{\mathrm d}}\xi$$ converges conditionally. Indeed, $$\begin{aligned}
\int_{-R}^R \exp(ix\xi) m(\xi)\, {{\mathrm d}}\xi&=2\int_{0}^R \cos(x\xi) m(\xi)\, {{\mathrm d}}\xi\\
&=\frac{2\sin(Rx)m(R)}{x} -\frac{2}{x} \int_0^R \sin(x\xi) m^\prime(\xi)\, {{\mathrm d}}\xi\\
&\to -\frac{2}{x} \int_0^\infty \sin(x\xi) m^\prime(\xi)\, {{\mathrm d}}\xi\end{aligned}$$ as $R\to \infty$; the latter integral converges absolutely since $m^\prime(\xi)=\mathcal{O}(|\xi|^{-3/2})$ as $|\xi|\to  \infty$. The function $\zeta \mapsto m(\zeta)$ is analytic in ${\mathbb C}\setminus S$, where $S=\cup_{k=1}^\infty i[k\pi-\pi/2, k\pi]\cup i[-k\pi, -k\pi+\pi/2]$. Furthermore, $$\label{eq:tanh bound}
\begin{aligned}
|\tanh(\zeta)|^2
&=\left|\frac{\exp(\xi+is)-\exp(-\xi-is)}{\exp(\xi+is)+\exp(-\xi-is)}\right|^2\\
&=\frac{\exp(2\xi)-2\cos(2s)+\exp(-2\xi)}{\exp(2\xi)+2\cos(2s)+\exp(-2\xi)}\\
&\le \coth^2 \xi \\
&\le \coth^2 \xi_0
\end{aligned}$$ when $|\xi|\ge \xi_0>0$, in which $\zeta=\xi+is$. Noting that $\exp(ix\zeta)$ is bounded for $x > 0$ when $\operatorname{Im}\zeta \ge 0$, we therefore obtain that $$\label{eq:deformation estimate}
\lim_{|\xi|\to \infty} \sup_{s \ge 0} |\exp(ix\zeta)m(\xi+is)|=0.$$ Fix a number $s_0\in (0, \pi/2)$. Using Cauchy’s theorem on a bounded rectangle with vertices $\pm R$, $\pm R+is_0$, and letting $R\to \infty$, it follows that 