Find out functions of the form $g(x,y) = \int f(x,t) f(y,t) dt$ https://ift.tt/eA8V8J

I am interested in the following question.

Given a symmetric function $g: \mathbb R^n \times \mathbb R^n \rightarrow \mathbb R$ or $\mathbb R_{+}^{n}\times \mathbb R_{+}^{n} \rightarrow 0$. I am interested in finding out whether $g$ can be written as the following form:

$$g(x,y) = \int f(x,t) f(y,t) \lambda(dt),$$ where $\lambda$ is some non-necessary Lebesgue measure.

For example, $g(x,y) = \min\{|x|,|y|\}$ can be written as the above form for $f(x,t) = \mathbb I(0<t<|x|)$. $g(x,y) = \frac{1}{|x|+|y|}$ can also be written as above for $f(x,t) = e^{-|x|t}\mathbb I(t>0)$.

I am wondering if there is any necessary or sufficient condition to describe the set of functions which satisfies the above assumption. One necessary condition is $g(x,y)$ needs to be positive symmetric definite.

Thanks a lot!!

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