# Expressing 1-e^{-z} as a Fourier integral

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 $$F$$?).

• $e^{-|t|} = C\int_0^\infty \frac{\cos(2tx)}{4x^2+1}dx$ Dec 12, 2019 at 15:25
• How did you come up with this? Apologies if this is all common konwledge. (Also, how would you come up with such a formula given some arbitrary function F?) Dec 12, 2019 at 17:21
• @TimothyChu, you want #207 from this table. Dec 12, 2019 at 19:09
• By (inverse) Fourier transform?
– lcv
Dec 12, 2019 at 19:09
• Ah, great. Thanks! I feel silly for asking such a basic question :). Dec 14, 2019 at 3:13