of $K$. We begin by proving such results for the Fourier transforms of a general class of functions. In the next subsection, we then show that the Whitham symbol $m$ belongs to this class. Much of the general theory discussed in this section is adapted from the monograph [@SchillingSongVondracek12], although we slightly extend some of it. Most importantly, we relate it to the theory of positive definite functions and the kernel $K$.

A function $g\colon (0, \infty)\to {\mathbb R}$ is called [*completely monotone*]{} if it is of class $C^\infty$ and $$\label{eq:completely monotone}
(-1)^n g^{(n)}(\lambda)\ge 0$$ for all $n \in {\mathbb Z}_0$ and all $\lambda >0$.

We shall sometimes say that a function is completely monotone on some interval (typically, a half-period), meaning that holds on that interval. Moreover, if $g\colon {\mathbb R}\setminus \{0\} \to {\mathbb R}$ is even, we shall say that $g$ is completely monotone if it is completely monotone on the interval $(0,\infty)$. One of the main reasons for introducing completely monotone functions is that they are precisely the functions which arise as Laplace transforms of measures. This is known as the Bernstein, or Bernstein–Hausdorff–Widder, theorem. We adopt here the convention that *a measure is always countably additive 