of completely monotone functions in general, whereafter the applications to the Whitham symbol $m$ are investigated. Among other things, we obtain complete monotonicity and an explicit series expression for the Whitham kernel $K$.

Regularity properties {#regularity-properties .unnumbered}
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Recall . We consider in this paper the Fourier transform as a continuous isomorphism $\operatorname{{\mathcal F}}\colon {\mathcal{S}}^\prime({\mathbb R}) \to {\mathcal{S}}^\prime({\mathbb R})$ on the space ${\mathcal{S}}^\prime({\mathbb R})$ of tempered distributions, defined by duality from the Fourier transform on the Schwartz space ${\mathcal{S}}({\mathbb R})$ of smooth and rapidly decaying functions. Our normalisation of $\operatorname{{\mathcal F}}$ is $$(\operatorname{{\mathcal F}}f)(\xi) = \int_{{\mathbb R}} f(x) \exp(-ix\xi) \,{{\mathrm d}}x \quad\text{ for }\quad f \in {\mathcal{S}}({\mathbb R}),$$ which implies that $(\operatorname{{\mathcal F}}^{-1} f)(x)=\frac1{2\pi} (\operatorname{{\mathcal F}}f)(-x)$. Clearly $m\in {\mathcal{S}}^\prime({\mathbb R})$, whence $K$ exists at least as an element of ${\mathcal{S}}^\prime({\mathbb R})$. However, since $m$ is smooth and all of its derivatives are integrable, $K$ is actually smooth for $x\ne 0$, and all its derivatives have rapid decay. In fact, since $m$ is analytic in a strip containing the real axis, $K$ and all of its derivatives are exponentially decaying.

\[prop:decay\] For any fixed $s_0 \in (0, \pi/2)$, $n\ge 0$, one has $$|\operatorname{D}_x^{n} K(x)| \lesssim \exp(-s_0 |x|)$$ for all $|x|\ge 1$.

