According to the theory of screw functions and screw lines by John Von Neumann and Issai Schoenberg (see here), any function $F:\mathbb{R} \rightarrow \mathbb{R}$ such that $F(|x_i - x_j|) = \|f(x_i)-f(x_j)\|_2^2$ for some function $f:\mathbb{R} \rightarrow H$ (for some Hilbert space $H$) can be written as:

$F(t) = \int_0^\infty \frac{\sin^2(tx)}{x^2} d\gamma (x)$

where $\int_1^\infty x^{-2} \gamma'(x) dx$ is finite.

Experimentally, I've determined that the function $F(t) = 1-e^{-|t|}$ has the property that $F(|x_i-x_j|)$ can be written as $\|f(x_i)-f(x_j)\|_2^2$. How would I go about finding $\gamma$ for this function (or any such function $F$?).